\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 250, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/250\hfil Self-similar solutions]
{Self-similar solutions for a superdiffusive heat equation
 with gradient nonlinearity}

\author[M. F. de Almeida, A. Viana \hfil EJDE-2016/250\hfilneg]
{Marcelo Fernandes de Almeida, Arl\'ucio Viana}

\address{Marcelo Fernandes de Almeida \newline
Universidade Federal de Sergipe, DMA - Departamento de Matem\'atica \\
Avenida Rosa Else, S\~ao Crist\'ov\~ao, Sergipe, Brazil}
\email{nucaltiado@gmail.com}

\address{Arl\'ucio Viana \newline
Universidade Federal de Sergipe,
DMAI - Departamento de Matem\'atica \\
Avenida Vereador Ol\'impio Grande,
Itabaiana, Sergipe, Brazil}
\email{arlucioviana@ufs.br}

\thanks{Submitted July 8, 2016. Published Septembere 19, 2016.}
\subjclass[2010]{35A01, 35R11, 35R09, 35B06, 35C06, 35K05, 35L05}
\keywords{Fractional partial differential equations; self-similarity;
\hfill\break\indent radial symmetry; Sobolev-Morrey spaces}

\begin{abstract}
 This article studies the existence, stability, self-similarity and
 symmetries of solutions for a superdiffusive heat equation with superlinear
 and gradient nonlinear terms with initial data in new homogeneous Besov-Morrey
 type spaces. Unlike in previous works on such time-fractional partial
 differential equations of order $\alpha\in(1,2)$, we take non null initial
 velocities into consideration, where new difficulties arise from.
 We overcome them by developing an appropriate decomposition of the
 two-parametric Mittag-Leffler function  to obtain Mikhlin-type estimates
 and obtain our existence theorem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

 Let $\Delta_x=\sum_{i=1}^{N}\frac{\partial^2}{\partial x_i^{2}}$
be the Laplace operator, 
$u:\mathbb{R}^{1+N}\to \mathbb{R}$, and $\partial^{\alpha}_t$ be
the Caputo's fractional derivative of order $1<\alpha<2$ 
(see subsection \ref{Duhamel}). 
In this article, we study the equation
 \begin{equation}\label{heat-wave}
  \partial^{\alpha}_{t}u = \Delta_x u +\kappa_1| \nabla_x u|^{q}
+ \kappa_2| u|^{\rho-1}u,\quad \kappa_1\neq 0,\; \kappa_2\in\mathbb{R} ,
 \end{equation}
subject to the initial data
 \begin{equation}\label{initial-data}
  u(0,x)=\varphi(x), \quad \partial_tu(0,x)=\psi(x),
 \end{equation}
where $q>1$ and $\rho>1$. Note that the rescaled function 
$u_{\gamma}(t,x):=\gamma^{\frac{2}{\rho-1}}u(\gamma^{\frac{2}{\alpha}}t,\gamma x)$ 
solves  \eqref{heat-wave} with initial data
 \begin{equation}\label{key_au}
  \varphi_{\gamma}(x)=\gamma^{\frac{2}{\rho-1}}\varphi(\gamma x),\quad
\psi_{\gamma}(x)=\gamma^{\frac{2}{\rho-1}+\frac{2}{\alpha}}\psi(\gamma x) ,
 \end{equation}
 provided that $q=\frac{2\rho}{\rho+1}$ and $u(t,x)$ solves
\eqref{heat-wave}-\eqref{initial-data}. Hence, we obtain a \emph{scaling map}
of solutions,
 \begin{equation}\label{scal-map}
  u(t,x)\mapsto u_{\gamma}(t,x),\quad \text{ for all }\gamma>0,
 \end{equation}
and solutions invariant by \eqref{scal-map} will be called 
\emph{self-similar solutions}, that is,
 \begin{equation}\label{self-similar}
  u(t,x)=u_{\gamma}(t,x).
 \end{equation}
In the study of self-similar solutions, the natural candidates to be 
initial data are the homogeneous functions,
 \[ %\label{hom-data}
  \varphi(\gamma x) =\gamma^{-\frac{2}{\rho-1}}\varphi(x),\quad 
 \psi(\gamma x) =\gamma^{-\frac{2}{\rho-1}-\frac{2}{\alpha}}\psi(x).
 \]
 In this work we are interested in existence of self-similar solutions 
to  \eqref{heat-wave}-\eqref{initial-data}. For this purpose, we study 
\eqref{heat-wave}-\eqref{initial-data} through its integral formulation
 \begin{equation}\label{int}
  u(t,x) = G_{\alpha,1}(t)\varphi(x)+G_{\alpha,2}(t)\psi(x)
+\mathcal{N}_{\alpha}(u)(t,x),
 \end{equation}
 where
 \begin{gather}\label{Mit-Lef}
  \widehat{G_{\alpha,j}(t)f}(\xi) 
= t^{j-1}E_{\alpha,j}(-4\pi^2t^\alpha|\xi|^2)\widehat{f}(\xi),\quad
 j=1,2,f\in\mathcal{S}'(\mathbb{R}^N) , \\
  \mathcal{N}_{\alpha}(u)=\int_{0}^{t}G_{\alpha,1}(t-s)\int_{0}^{s}r_{\alpha}(s-\tau) 
\left(\kappa_2| u|^{\rho-1} u+\kappa_1| \nabla_x u|^{q}\right)
d\tau ds,  \label{N-alpha} \\
r_{\alpha}(t)=t^{\alpha-2}/\Gamma(\alpha-1) . 
\end{gather}
 Hereafter a solution $u$ will be understood as a distribution $u(t,\cdot)$ 
satisfying \eqref{int}, for each $t>0$.

 The presence of the gradient requires suitable estimates in certain
 Sobolev-Morrey spaces $\mathcal{M}^{1}_{r,\mu}$ and  this motivated us to study 
the problem in the space $X_\beta$ of all bounded continuous functions 
$u:(0,\infty)\to  \mathcal{M}_{r,\mu}$ endowed with the norm
 \begin{equation}\label{norm}
  \|u \|_{X_{\beta}} = \sup_{t>0} t^{\frac{\alpha}{2}+\beta}
\|u(t)\|_{\mathcal{M}_{r,\mu}^1 } +\sup_{t>0} t^{\beta}\|u(t)\|_{\mathcal{M}_{r,\mu}},
 \end{equation}
where $\beta,r$ and $\mu$ will be chosen later (see \eqref{param1}).
Assuming $\psi\not \equiv 0$ brings new difficulties because we need to 
obtain suitable estimates for two-parametric Mittag-Leffler function 
$E_{\alpha,2}(4\pi^2t^{\alpha}|\xi|^2)$. More precisely, we
develop an appropriate decomposition for Mittag-Leffler function 
 to obtain a suitable estimate (see \eqref{point1} and \eqref{point2})
 which enables us to introduce the space
 \begin{equation}
  \mathcal{I}=\{(\varphi,\psi)\in\mathcal{S}'\times\mathcal{S}';
 (\varphi,\psi)\in D(\alpha,\beta)\times \tilde{D}(\alpha,\beta)\},
 \end{equation}
 where
\[
  D(\alpha,\beta):=\{\varphi\in \mathcal{S}': G_{\alpha,1}(t)\varphi\in X_{\beta}\},
\quad
\widetilde{D}(\alpha,\beta):=\{\psi\in\mathcal{S}: G_{\alpha,2}(t)\psi\in X_{\beta}\},
\]
 for all $t>0$. Hence, applying Lemma \ref{galpha} we obtain 
(see Remark \ref{rem1}-(B)) that 
$\mathcal{M}_{p,\mu}\times\mathcal{M}_{p,\mu}^{-2/\alpha}
\subseteq D(\alpha,\beta)\times\widetilde{D}(\alpha,\beta)$. 
It is remarkable that the investigation of self-similarity and symmetries 
for \eqref{heat-wave}-\eqref{initial-data} allows us to deal with  
following prototype functions
\[
  \varphi(x)=\epsilon_1| x| ^{-\frac{2}{\rho-1}}, \quad
\psi(x)=\epsilon_2| x|^{-\frac{2}{\rho-1}-\frac{2}{\alpha}} ,
\]
 which belong to  $D(\alpha,\beta)\times\widetilde{D}(\alpha,\beta)$
 but can be  arbitrarily large in the space
 $L^2(\mathbb{R}^N)\times \dot{H}^{\frac{2}{\alpha}}(\mathbb{R}^N)$. 
See Remark \ref{rem2}-(A).

 Our symmetry result, roughly speaking, says that if the initial data
 $\varphi$ and $\psi$ are invariant on the orthogonal group acting on 
$\mathbb{R}^N$ so the solution is. In particular, we show the existence of 
radial self-similar solutions (see Remark \ref{rem3}-(A)).

 We point out that our results hold  for $\alpha=1$ and $\psi=0$ and, 
in this case,  the upper bound  
$(\gamma_2-\gamma_1)+\frac{N-\mu}{p_1}-\frac{N-\mu}{p_2}$ 
in Lemma \ref{galpha} can be removed. On the other hand, for  
$1<\alpha<2$ the Mikhlin theorem yields more restrictive constraints 
to Lemma \ref{galpha} than the usual estimates for the heat semigroup 
in such a way that Theorem \ref{gw} cannot come near to $\alpha=2$.

Now, let us to review some works. Fujita \cite{YF} remarked that the 
linear counterpart of equation \eqref{heat-wave} has similarities with wave 
and heat equations and presents certain qualitative properties which 
qualifies it as a reasonable interpolation between these equations. 
When $\kappa_2=0$, $\alpha=1$ and $\psi=0$, \eqref{heat-wave}-\eqref{initial-data} 
turns into the viscous Hamilton-Jacobi equation. Using scaling technique, 
Ben-Artzi {\it et al} \cite{bsweissler} found the number $r _c=\frac{N(q-1)}{2-q}$ 
and showed that it is a critical exponent for existence of solutions in $L^r$. 
In particular, the problem is well-posed when $r\geq r_c$ and $1<q<2$. 
In Remark \ref{rem1}-(C) we provide an existence result  for this problem 
in Morrey spaces $\mathcal{M}_{p,\,N-\frac{N}{r_c}p}$ which are strictly 
larger than the Lebesgue spaces, namely,
 \begin{equation} \label{inv-spaces1}
  L^r(\mathbb{R}^N)\subsetneq\mathcal{M}_{p,\mu} (\mathbb{R}^N)
\subsetneq D(1,\beta),
 \end{equation}
provided that $\frac{N}{r}=\frac{N-\mu}{p}$ and $p<r$.  
Our existence result is then compatible with \cite[Theorem 2.1]{bsweissler}, 
in view of $1< p\leq r_c\leq r$. In particular, the initial data taken 
in Theorem \ref{gw} is larger than those considered in \cite{bsweissler}.

Recently several authors have addressed the study of global existence, 
self-similarity, asymptotic self-similarity  and radial symmetry of 
solutions for the semilinear heat equation with gradient nonlinear terms. 
See e.g.  \cite{Chipot,gild,STW,Weissler-heat,Souplet1}. 
In \cite{STW} it is assumed that  $\varphi$ belongs to homogeneous Besov 
space $ \dot{B}_{r_1,\infty}^{-\beta_1}$ and
\[
  \| \varphi\|_{\dot{B}_{r_1,\infty}^{-\beta_1}}
=\sup_{t>0}t^{\beta_1/2}\| e^{t\Delta}\varphi\|_{L^{r_1}(\mathbb{R}^N)}
\leq \epsilon,\quad \beta_1=\frac{2}{\rho-1}-\frac{N}{r_1} .
\]
 By employing the Gagliardo-Nirenberg inequality, the authors studied 
the existence and asymptotic behavior of global mild solutions. 
On the other hand, our functional approach enables us to control the gradient
estimates without making use of the Gagliardo-Nirenberg inequality and allows 
us to deal with a larger class of functions space for initial data.

 Let us now review some works concerning to \eqref{heat-wave}-\eqref{initial-data}
with $\psi=0$ and $\kappa_1=0$. In \cite{HMiao} the authors established 
$L^p-L^q$ estimates for $\{G_{\alpha,1}(t)\}_{t\geq 0}$ and showed a 
blowup alternative and local well-posedness in $L^q(\mathbb{R}^N)$-framework 
for any $\varphi\in L^q(\mathbb{R}^N)$, where $q\geq \frac{N\alpha (\rho+1)}{2}$. 
Using Mikhlin-Hormander's type theorem on Morrey spaces, de Almeida and 
Ferreira \cite{Marcelo} studied self-similarity, symmetry, antisymmetry and 
positivity of global solutions with small data $\varphi\in\mathcal{M}_{p,\mu}$,
 $\mu=N-\frac{2p}{\rho-1}$.  In \cite{MJ}, the authors established existence, 
self-similarity, symmetries and asymptotic behavior of solutions in 
Besov-Morrey spaces $\mathcal{N}^{\sigma}_{p,\mu,\infty}$ and provided a 
maximal class of existence in the sense that there is no known results in 
$X\supsetneq \mathcal{N}^{\sigma}_{p,\mu,\infty}$. Indded, we notice that
 \begin{equation}\label{inv-spaces2}
  \mathcal{M}_{p,\lambda}\subsetneq
  \mathcal{N}_{p,\mu ,\infty }^{\sigma }\quad \text{and}\quad
\dot{B}_{r,\infty  }^{k}\subset\mathcal{N}_{p,\mu ,\infty }^{\sigma },
 \end{equation}%
where $\frac{N-\lambda }{p}=-\sigma +\frac{N-\mu }{p}=-k+\frac{N}{r}$, 
$\sigma =\frac{N-\mu }{p}-\frac{2}{\rho -1}$, $k=\frac{N }{r}-\frac{2}{\rho -1}$ 
and $1\leq p<r$. All spaces in \eqref{inv-spaces1} and \eqref{inv-spaces2} 
are invariant by scaling.

 We still observe that problem \eqref{heat-wave}--\eqref{initial-data} can be 
studied with a Fourier multiplier $\sigma(D)$ in place of $\Delta_x$, 
where $|\sigma(\xi)|\leq C|\xi|^k$ due to estimates
\eqref{point1} and \eqref{point2} into Propositions \ref{fund-lemma} 
and \ref{fund-lemma2}. Example of such an operator is the Riesz potential 
$(-\Delta _{x})^{{k }/{2}}f=\mathcal{F}^{-1}|\xi |^{k}\mathcal{F}f$, 
where $\mathcal{F}$ denote the Fourier transform in $\mathcal{S}'$.


This manuscript is organized as follows. Some basic properties of the 
Sobolev-Morrey spaces and Mittag-Leffler functions are reviewed in 
Section \ref{pre}. We state and make some remarks on our results in 
Section \ref{func-settings} and their proofs are performed in 
Section \ref{proofs}. Sections \ref{technical} and \ref{sme} are reserved 
to a careful study of the several estimates which are crucial to yield our results.

 \section{Preliminaries}\label{pre}

 In this section we review some well-known properties
of the Morrey spaces and  Sobolev-Morrey spaces, more details can be found in
\cite{Sawano, Kato-Morrey,Yamazaki2,Mazzucato,Miyakawa1}. Also, 
we obtain an integral equation which is formally equivalent to  
\eqref{heat-wave}-\eqref{initial-data} in the lines of \cite{Kilbas2}.

 \subsection{Sobolev-Morrey spaces}\label{sms}

 Let $Q_{r}(x_{0})$ be the open ball in $\mathbb{R}^{N}$ centered at $x_{0}$
and with radius $r>0$. Given two parameters $1\leq p<\infty $ and 
$0\leq \mu  <N$, the Morrey space 
$\mathcal{M}_{p,\mu }=\mathcal{M}_{p,\mu }(\mathbb{R} ^{N})$ is defined to be 
the set of all functions $f\in L^{p}(Q_{r}(x_{0}))$ such  that
 \begin{equation*}
  \| f\| _{\mathcal{M}_{p,\mu}}:=\sup_{x_{0}\in \mathbb{R}^{n},\,r>0}r^{-\frac{%
    \mu }{p}}\| f\| _{L^{p}(Q_{r}(x_{0}))}<\infty,   %\label{norm-Morrey}
 \end{equation*}
 which is a Banach space endowed with this norm. 
For $s\in \mathbb{R}$ and $1\leq p<\infty$, the homogeneous Sobolev-Morrey space 
$ \mathcal{M}_{p,\mu }^{s}=(-\Delta_x)^{-s/2}\mathcal{M}_{p,\mu }$ is the Banach 
space of all tempered distributions $f\in\mathcal{S}'(\mathbb{R}^N)/\mathcal{P}$ 
modulo polynomials $\mathcal{P}$ with $N$ variables.
 If $s<\frac{N-\mu}{p}$ and $p>1$, from  \cite[Theorem 1.1]{Sawano} or 
\cite{Mazzucato}, it holds that
 \begin{equation*}
  \| f\|_{\mathcal{M}_{p,\mu}}\sim \big\|\Big(\sum_{\nu\in\mathbb{Z}}
| \mathcal{F}^{-1}\psi_{\nu}(\xi)\mathcal{F}f|^2\Big)^{1/2}
\big\|_{\mathcal{M}_{p,\mu}}\label{Littewood-Paley-Morrey},
 \end{equation*}
where $\sim$ denotes norm equivalence and $\{\psi_{\nu}\}_{\nu\in\mathbb{Z}}$ 
is a homogeneous Littlewood-Paley resolution of unity, that is,
\[
  \psi_{\nu}(\xi)=\phi_{\nu}(\xi)-\phi_{\nu-1}(\xi),\quad
 \phi_{\nu}(\xi)=\phi_0(\xi/2^{\nu}),
\]
 for $\phi_0\in C^{\infty}_{0}(\mathbb{R}^N)$ such that $\phi_0=1$ on the ball 
$Q_1(0)$ and $\operatorname{supp}\phi_0\,\subset Q_{2}(0)$. 
In particular, using \eqref{Littewood-Paley-Morrey}  and that 
$| \xi| ^s\sim2^{s\nu}$ on the 
$\operatorname{supp}\psi_{\nu}(\xi)\subset \{ \xi\in\mathbb{R}^N:
 2^{\nu-1}< | \xi|< 2^{\nu+1}\}$,  we obtain
 \begin{equation}
\begin{aligned}
&\big\|\Big(\sum_{\nu\in\mathbb{Z}}| 2^{s\nu} \mathcal{F}^{-1}\psi_{\nu}
(\xi)\mathcal{F}f|^2\Big)^{1/2}\big\|_{\mathcal{M}_{p,\mu}}\\
&\sim \big\|\Big(\sum_{\nu\in\mathbb{Z}}|  
\mathcal{F}^{-1}\psi_{\nu}(\xi)| \xi|^s\mathcal{F}f|^2\Big)^{1/2}
 \big\|_{\mathcal{M}_{p,\mu}} \\
&= \big\|\Big(\sum_{\nu\in\mathbb{Z}}|  2^{\nu N}
\check{\psi}(2^{\nu}\cdot)\ast (|\xi|^s\widehat{f})^{\vee}|^2
\Big)^{1/2}\big\|_{\mathcal{M}_{p,\mu}} \\
&\sim \| (|\cdot|^s\widehat{f})^{\vee}
\|_{\mathcal{M}_{p,\mu}}.
\end{aligned} \label{norm-key-SM}
 \end{equation}
 Given $f\in\mathcal{M}_{p,\mu}^s$, the quantity \eqref{norm-key-SM} 
define two equivalent norms on Sobolev-Morrey space, namely,
 \begin{equation}
\begin{gathered}
 \| f\| _{\mathcal{M}_{p,\mu }^{s}}=\| (|\cdot|^s\widehat{f})^{\vee}
\| _{\mathcal{M}_{p,\mu}}, \\ 
\| f\| _{\mathcal{M}_{p,\mu }^{s}}
=\big\|\Big(\sum_{\nu\in\mathbb{Z}}| 2^{s\nu} \mathcal{F}^{-1}\psi_{\nu}(\xi)
\mathcal{F}f|^2\Big)^{1/2}\big\|_{\mathcal{M}_{p,\mu}}
\end{gathered} \label{norm-SM}.
 \end{equation}
It follows from  Littlewood-Paley decomposition of the Lebesgue space 
$L^p(\mathbb{R}^N)$ and homogeneous Sobolev space $H^{s}_p(\mathbb{R}^N)$ that 
$\mathcal{M}_{p,0}=L^{p}(\mathbb{R}^N)$ and 
$\mathcal{M}_{p,0}^{s}=\dot{H}_{p}^{s}(\mathbb{R}^N)$, respectively. 
 Also, Morrey and Sobolev-Morrey spaces present the following scaling
\[
  \| f(\gamma \cdot )\| _{\mathcal{M}_{p,\mu}}=\gamma^{-\frac{N-\mu }{p}}\|
  f\| _{\mathcal{M}_{p,\mu}} \quad \text{and}\quad
 \| f(\gamma\cdot )\| _{\mathcal{M}_{p,\mu }^{s}}
=\gamma  ^{s-\frac{N-\mu }{p}}\| f\| _{\mathcal{M}_{p,\mu }^{s}},
 \label{scal-SM}
 \]
where the exponents $s$ and $s-\frac{N-\mu }{p}$ are called 
\emph{scaling index} and \emph{regularity index}, respectively.

 \begin{lemma}   \label{lem:2.1} 
Suppose that $s\in\mathbb{R}$, $1\leq   p_1,p_2,p_3<\infty$ and 
$0\leq\mu_i<N$, $i=1,2,3$.
\begin{itemize}
   \item[(i)] (Inclusion) If $\frac{N-\mu_1}{p_1}=\frac{N-\mu_2}{p_2}$ and $%
   p_1\leq p_2$,
   \begin{equation}
    \mathcal{M}_{p_2,\mu_2}\subset\mathcal{M}_{p_1,\mu_1}.  \label{emb1}
   \end{equation}

\item[(ii)] (Sobolev-type embedding) Let $p_1\leq
   p_2$,
   \begin{equation}
    \mathcal{M}_{p_1,\mu}^{s}\subset\mathcal{M}_{p_2,\mu}^{s
 - (\frac{N-\mu}{p_1}-\frac{N-\mu}{p_2})}.  \label{sobolev-emb}
   \end{equation}

   \item[(iii)] (H\"oder inequality) Let 
$\frac{1}{p_3}=\frac{1}{p_2}+   \frac{1}{p_1}$ and 
$\frac{\mu_3}{p_3}=\frac{\mu_2}{p_2}+\frac{\mu_1}{p_1}$.
   If $f_j\in \mathcal{M}_{p_j,\mu_j}$ with $j=1,2$, then 
$f_1f_2\in\mathcal{M} _{p_3,\mu_3}$ and
   \begin{equation}
    \| f_1f_2\|_{p_3,\mu_3}\leq \| f_1\|_{p_1,\mu_1}\|  f_2\|_{p_2,\mu_2}.
  \label{eq:holder}
   \end{equation}
  \end{itemize}
 \end{lemma}

 Finally, notice that the following homogeneous functions, of degree $-d$ 
and $s-d$, belong to Morrey and Sobolev-Morrey spaces, respectively:
 \begin{equation}\label{data-stand}
  \rho_0(x)=Y_{k}(x)|x|^{-d-k}\in \mathcal{M}_{p,\mu} \quad\text{and} \quad
\rho_s(x)=Y_{k}(x)|x|^{s-d-k}\in \mathcal{M}_{p,\mu}^s,
 \end{equation}
where $Y_{k}(x)\in L^{p}(\mathbb{S}^{N-1})$ is a harmonic homogeneous polynomial 
of degree $k,$ $\mu=N-dp$, $0<d-s<N$ and $1<p<N/d$. Indeed, using 
\cite[Theorem 4.1]{Stein1} we obtain 
$\widehat{\rho}_s(\xi)=\gamma_{k,s}Y_{k}(\xi)| \xi|^{d-s-k-N}$ provided $0<d-s<N$, 
where $\gamma_{k,s}$ is a positive constant.  It follows  from \eqref{norm-SM} that
 \begin{align*}
  \| \rho_s\|_{\mathcal{M}_{p,\mu}^s}
&=\big| \Big(\sum_{\nu=-\infty}^{+\infty}|2^{s\nu} \mathcal{F}^{-1}
\psi_{\nu}(\xi)\gamma_{k,s}Y_{k}(\xi)|\xi|^{d-s-k-N}|^2\Big)^{1/2}
 \big\|_{\mathcal{M}_{p,\mu}} \\
&\sim\big\| \Big(\sum_{\nu=-\infty}^{+\infty}| \mathcal{F}^{-1}\psi_{\nu}(\xi)
| \xi|^{s}\gamma_{k,s}Y_{k}(\xi)|\xi|^{d-s-k-N}|^2\Big)^{1/2}
\big\|_{\mathcal{M}_{p,\mu}} \\
&=\| \rho_0\|_{\mathcal{M}_{p,\mu}},
 \end{align*}
which is finite. In fact,  polar coordinates in $\mathbb{R}^N$ and homogeneity 
of $Y_{k}(x)\in L^{p}(\mathbb{S}^{N-1})$ yield
 \begin{align*}
  \| \rho_0\|_{L^{p}(Q_r)}^p=\int_{\mathbb{S}^{N-1}}| Y_k(x')|^p
\int_0^r t^{N-dp-1}dt\,d\sigma(x')=\| Y_k\|_{L^p(\mathbb{S}^{N-1})}^p\;r^{\mu} ,
\end{align*}
where $\mu=N-dp$, $1<p<N/d$.

\subsection{Duhamel formula}\label{Duhamel}

We consider the partial fractional differential equation
 \begin{equation}\label{eq_gen1}
  \begin{gathered}
   \partial_t^{\alpha}u(t,x)=\Delta_x u(t,x)-f(t,x),\quad
 x\in\mathbb{R}^N,\, t>0,\\
   u(t,x)\bigr|_{t=0}= \varphi(x), \quad 
\frac{\partial }{\partial t}u(t,x)\bigr|_{t=0}= \psi(x),
  \end{gathered}
 \end{equation}
for $\alpha\in (1,2)$ and $\partial_t^{\alpha}$ stands for partial 
fractional derivative given by
\[
  \partial^{\alpha}_{t}f(t,x)=\frac{1}{\Gamma(m-\alpha)}
\int_0^t\frac{\partial_s^mf(s,x)}{(t-s)^{\alpha+1-m}}ds, \quad
 m-1<\alpha \leq m,\; m\in\mathbb{N}.
\]
 Formally, applying the Fourier transform in \eqref{eq_gen1}, we obtain 
the fractional ordinary differential equation
 \begin{gather*}
\partial_t^{\alpha}\widehat{u}(t,\xi)+4\pi^2|\xi|^2\widehat{u}(t,\xi)
 =\widehat{f}(t,\xi),\\
\widehat{u}(t,\xi)|_{t=0}= \widehat{\varphi}(\xi), \quad 
 \partial_t\widehat{u}(t,\xi)\bigr|_{t=0}= \widehat{\psi}(\xi)
  \end{gather*}
which, by \cite[Example 4.10]{Kilbas2}, is equivalent to
 \begin{equation}
\begin{aligned}
  \widehat{u}(t,\xi)
&= E_{\alpha,1}(-4\pi^2t^{\alpha}|\xi|^2)\widehat{\varphi}(\xi) 
 + tE_{\alpha,2}(-4\pi^2t^{\alpha}|\xi|^2)\widehat{\psi}(\xi) \\
&\quad +\int_0^t E_{\alpha,1}(-4\pi^2(t-s)^{\alpha}|\xi|^2)
\int_0^{s}r_{\alpha}(s-\tau)\widehat{f}(\tau,\xi)d\tau ds,
\end{aligned}
\end{equation}
where $E_{\alpha,\beta}(z)$ denotes the two-parametric Mittag-Leffler function
 \begin{equation}
  E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty}\frac{z^k}{\Gamma(\alpha k+\beta)}
\quad\text{and}\quad
 E_{\alpha}(z):=E_{\alpha,1}(z),\; \text{for all } \alpha, \beta>0.
  \label{mit1}
 \end{equation}
 Hence, in original variables, we have
 \begin{equation*}
  u(t,x) = G_{\alpha,1}(t)\varphi(x)+G_{\alpha,2}(t)\psi(x)
+\mathcal{N}_{\alpha}(u)(t,x),
 \end{equation*}
where $ \widehat{G_{\alpha,j}(t)f}(\xi)$ is defined by \eqref{Mit-Lef} 
and $\mathcal{N}_{\alpha}$ is defined by \eqref{N-alpha}.


Note that  $G_{2,2}(t)$ is the  wave group 
$\big(\frac{\sin(4\pi^2t| \xi|)}{4\pi^2t| \xi|}\big)^{\vee}$,
$G_{2,1}(t)=\big(\cos (4\pi^2t|\xi|)\big)^{\vee}$ and 
$G_{1,1}(t)=(e^{-4\pi^2t|\xi|^2})^{\vee}$ is the heat semigroup.

 \section{Functional setting and theorems}\label{func-settings}

Before starting our theorems, let $\beta>0$ and $0\leq\mu<N$ be such that
 \begin{equation}
  \beta=\frac{\alpha}{2}\Big(\frac{N-\mu}{p}-\frac{N-\mu}{r}\Big) 
\quad\text{and}\quad
 \mu = N -\frac{2p}{\rho-1}\label{param1} ,
 \end{equation}
which make $\|\cdot\|_{X_\beta}$ invariant by \emph{scaling map} 
\eqref{scal-map}.


 \subsection{Existence of solutions}

 Given a Banach space $Y$, we will denote $B_Y(\varepsilon)$ a closed ball 
of radius $\varepsilon$ centered at the origin of the space $Y$.
Our existence and stability result is stated as follows.

 \begin{theorem}\label{gw} 
Let $N\geq2$, $1<\alpha<2$, $q=\frac{2\rho}{\rho+1}$,
 and $0\leq \mu=N-\frac{2p}{\rho-1}$, for $p>1$. Suppose that 
$\frac{N-\mu}{p}-\frac{N-\mu}{r}<2$, $r>\rho>1+\alpha$,
  \begin{equation}
   \frac{p}{r}<\frac{1}{\alpha}-\frac{1}{2},\quad 
\frac{\alpha}{2-\alpha}<q<\frac{2}{\alpha},\quad 
\big(1-\frac{p}{r}\big) < \frac{\rho-1}{\alpha}\big(\frac{1}{q}-\frac{\alpha}{2}\big)
\label{param-hip2} .
  \end{equation}

(i) (Global solution)
 There exist $\varepsilon>0$ such that if
 $\| \varphi\|_{D(\alpha,\beta)} +\| \psi\|_{\widetilde{D}(\alpha,\beta)}
\leq \varepsilon$, then problem  \eqref{heat-wave}-\eqref{initial-data} 
has a unique global-in-time mild solution $u\in B_{X_{\beta}}(2\varepsilon)$ 
satisfying
   \begin{equation}
    \| u(t,\cdot)\|_{\mathcal{M}_{r,\mu}}\leq C t^{-\beta}\quad \text{and}\quad
 \| \nabla_x u(t,\cdot)\|_{\mathcal{M}_{r,\mu}}\leq C t^{-\beta-\alpha/2} .
   \end{equation}

 (ii) (Stability in $X_{\beta}$) 
The solution $u$ in Theorem \ref{gw}(i) is stable with respect to the 
initial data $\varphi$ and $\psi$, that is, the data-map solution 
$(\varphi,\psi)\mapsto u$ is locally Lipschitz continuous from 
$D(\alpha,\beta)\times\widetilde{D}(\alpha,\beta)$ into $X_{\beta}$:
   \begin{equation}
    \| u-\tilde{u}\|_{X_{\beta}}
\leq C\big(\| \varphi-\tilde{\varphi} \|_{D(\alpha,\beta)}
+\| \psi-\tilde{\psi} \|_{\widetilde{D}(\alpha,\beta)}\big) ,
   \end{equation}
where $u$ and $\tilde{u}$ are solutions of \eqref{heat-wave} with 
initial values $( \varphi,\psi)$ and $(\tilde{\varphi},\tilde{\psi})$, 
respectively.
 \end{theorem}

 \begin{remark}\label{rem1} \rm
Let us compare our theorem with some previous results.

(A) If $\psi=0$, we may take $\varphi\in \mathcal{M}_{p,\mu}$ in 
Theorem \ref{gw}-(i) with smallness on $\|\varphi\|_{\mathcal{M}_{p,\mu}}$.

(B) Theorem \ref{gw}-(i) holds for $\alpha=1$ and $\psi=0$. 
Hence, the space $D(1,\beta)$ strictly includes the space 
$\mathcal{N}(\varphi)$ taken in \cite{STW}. Indeed, let $r<r_1$ and 
$\mu_2=0$ in Lemma \ref{lem:2.1}-(i) to get
   \begin{align*}
\| \varphi\|_{D(1,\beta)}
&=\sup_{t>0} t^{\beta}\| e^{t\Delta}\varphi\|_{\mathcal{M}_{r,\mu}} 
+ \sup_{t>0} t^{\frac{1}{2}+\beta}\| e^{t\Delta}\varphi\|_{\mathcal{M}_{r,\mu}^1},
\\
&\leq \sup_{t>0} t^{\beta}\| e^{t\Delta}\varphi\|_{L^{r_1}} 
+ \sup_{t>0} t^{\frac{1}{2}+\beta}\| e^{t\Delta}\varphi\|_{\dot{H}_{r_1}^1}
=\| \varphi\|_{\mathcal{N}(\varphi)}.
   \end{align*}
On the one hand (see \cite[(2.56)]{Mazzucato2}), homogeneous Besov-Morrey 
spaces can be defined by
\[
    \mathcal{N}_{r,\mu,\infty}^{-2s}=\big\{ f\in\mathcal{S}':
 \| f\|_{\mathcal{N}_{r,\mu,\infty}^{-2s}}
=\sup_{t>0}t^{-s}\| e^{t\Delta}f\|_{\mathcal{M}_{r,\mu}}<\infty\big\},\quad
 s>0.
\]
   Hence, the space $D(1,\beta)$ is a kind of Besov-Morrey spaces. 
On the other hand, when $\alpha\neq 1$ the norms 
$\| \varphi\|_{D(\alpha,\beta)}=\| G_{\alpha,1}(t)\varphi\|_{X_{\beta}}$ and 
$\| \psi\|_{\widetilde{D}(\alpha,\beta) }=\| G_{\alpha,2}(t)\psi\|_{X_{\beta}}$ 
satisfy
\[
    \| \varphi\|_{D(\alpha,\beta)}\leq C  \| \varphi\|_{\mathcal{M}_{p,\mu}}\quad
 \text{and}\quad
    \| \psi\|_{\widetilde{D}(\alpha,\beta)}
\leq  C \| \psi\|_{\mathcal{M}_{p,\mu}^{-{2}/{\alpha}}}
\]
in view of Lemma \ref{galpha}. So $\mathcal{M}_{p,\mu}\subset D(\alpha,\beta)$ 
and $\mathcal{M}_{p,\mu}^{-{2}/{\alpha}}\subset \widetilde{D}(\alpha,\beta)$.

(C) (Viscous Hamilton-Jacobi) Let $\kappa_2=0$, $\psi=0$ in 
\eqref{heat-wave}-\eqref{initial-data}, $\mu=N-\frac{q-1}{2-q}p$ and
 $\|\varphi\|_{D(\alpha,\beta)}$ small enough. Using the proof of Theorem \ref{gw}, 
the problem \eqref{heat-wave}-\eqref{initial-data} has a unique solution 
$u\in C((0,\infty);\mathcal{M}_{r,\mu})\cap C((0,\infty);\mathcal{M}_{r,\mu}^1)$ 
such that
   \begin{gather*}
\sup_{t>0}t^{\frac{(N-\mu)}{2}\alpha(\frac{1}{p}-\frac{1}{r})}
\|u(t)\|_{\mathcal{M}_{r,\mu}} \leq C, 
\\
\sup_{t>0}t^{\frac{\alpha}{2}+\frac{(N-\mu)}{2}\alpha(\frac{1}{p}-\frac{1}{r})}
\|\nabla u(t)\|_{\mathcal{M}_{r,\mu}} \leq C,
\end{gather*}
under the assumptions in Theorem \ref{gw} with the change $\rho=\frac{2}{2-q}$.
In other words, we obtain a version of 
Theorem 2.1 and Proposition 2.3 of \cite{bsweissler} when  $1<\alpha<2$. 
If $\alpha=1$, the assumption $\frac{N-\mu}{p}-\frac{N-\mu}{r}<2$ 
is not necessary  because of the smoothing effect of the heat semigroup 
in $\mathcal{M}_{p,\mu}$ (see e.g. \cite{Kato-Morrey}).
 \end{remark}

 \subsection{Self-similar solutions}

 As we commented before, a necessary condition for initial data to produce 
self-similar solutions is homogeneity and simplest candidates are the radial 
functions  
\begin{equation}\label{prototype}
 \varphi(x)= \varepsilon_1 | x| ^{-\frac{2}{\rho-1}} \quad\text{and}\quad
 \psi(x)= \varepsilon_2 | x| ^{-\frac{2}{\rho-1}-\frac{2}{\alpha}} .
 \end{equation}
 Hence, we need $D(\alpha,\beta)$ and $\widetilde{D}(\alpha,\beta)$ to satisfy
 \begin{equation}
  \| \psi_{\gamma}\|_{\widetilde{D}(\alpha,\beta)} 
= \| \psi\|_{\widetilde{D}(\alpha,\beta)}\quad \text{and}\quad
 \| \varphi_{\gamma}\|_{D(\alpha,\beta)} = \| \varphi\|_{D(\alpha,\beta)} ,
 \end{equation}
 and it comes from the scaling invariance of $X_\beta$.
 Moreover,  \eqref{data-stand} and Remark \ref{rem1}-(B) permit us to take 
the {\it singular functions} \eqref{prototype} as initial data, since 
$\varphi\in \mathcal{M}_{p,\mu}\subset D(\alpha,\beta)$ and 
$\psi\in \mathcal{M}^{-2/\alpha}_{p,\mu}\subset \widetilde{D}(\alpha,\beta)$ 
provided $\mu = N -\frac{2p}{\rho -1}$, 
$\rho>\max \{1+\frac{2}{N}, 1+\frac{2\alpha}{\alpha N-2}\}$ and $1<p<r$.


 \begin{theorem}[Self-similarity]\label{selfsimilarity} 
Under the assumptions of Theorem \ref{gw}, let $\varphi$ and $\psi$ be
 homogeneous functions of degree $-\frac{2}{\rho-1}$ and 
$-\frac{2}{\rho-1}-\frac{2}{\alpha}$, respectively. 
Then the solution $u$ of Theorem \ref{gw}-(i) is self-similar.
 \end{theorem}


 \begin{remark}\label{rem2} \rm
Let us remark some consequences of this theorem.

(A) (Infinity energy data) In Theorem \ref{selfsimilarity} we can build 
singular initial data $(\psi,\varphi)$ which can be arbitrarily large in 
 $L^2(\mathbb{R}^N)\times \dot{H}^{2/\alpha}(\mathbb{R}^N)$, provided that 
 $\frac{2}{\alpha}+\frac{2}{\rho-1}<\frac{N}{2}$ and $1<p<\frac{N(\rho-1)}{2}$.
   Indeed, let  $\varphi\in\mathcal{S}'(\mathbb{R}^N)$ and 
$\psi\in\mathcal{S}'(\mathbb{R}^N)/\mathcal{P}$ be given by \eqref{prototype}. 
Using $\widehat{\varphi}(\xi)=\gamma_{0,0}
\varepsilon_1 | \xi| ^{\frac{2}{\rho-1}-N}$, we see that $\varphi$ and $\psi$ 
are arbitrarily large in $\dot{H}^{2/\alpha}$ and $L^2$ in view of
\begin{align*}
    \| \psi\|_{L^2(\mathbb{R}^N)}^2
&=\varepsilon_2^2 \int_{\mathbb{R}^N}| x| ^{-\frac{4}{\rho-1}
 -\frac{4}{\alpha}}dx \\
&=\varepsilon_2^2\lim_{\omega_2\to \infty}
 \int_{0}^{\omega_2}\int_{\mathbb{S}^{N-1}}r ^{-\frac{4}{\rho-1}
 -\frac{4}{\alpha}}r^{N-1}d\sigma dr \\
&=C\lim_{\omega_2\to \infty}\omega_2 ^{-\frac{4}{\rho-1}
 -\frac{4}{\alpha}+N}=+\infty
\end{align*}
   and
\begin{align*}
\| \varphi\|_{\dot{H}^{\frac{2}{\alpha}}(\mathbb{R}^N)}^2
&=\int_{\mathbb{R}^N}|\xi|^{4/\alpha}|\widehat{\varphi}(\xi)|^2d\xi
 =\gamma_{0,0}^2\varepsilon_1^2\int_{\mathbb{R}^N}|\xi|^{4/\alpha
 +\frac{4}{\rho-1}-2N}d\xi \\
&=C\lim_{\omega_1\to  0}\int_{\omega_1}^{\infty}
 \int_{\mathbb{S}^{N-1}}r ^{4/\alpha+\frac{4}{\rho-1}-N-1}d\sigma dr\\
&=C\lim_{\omega_1\to  0}\omega_1 ^{\frac{4}{\alpha}+\frac{4}{\rho-1}-N}
=+\infty.
\end{align*}
Then, even the initial data $\varphi$ and $\psi$ are in the Morrey spaces 
$\mathcal{M}_{p,\mu}$ and $\mathcal{M}_{p,\mu}^{-{2}/{\alpha}}$, 
respectively, they may be arbitrarily large in $\dot{H}^{2/\alpha}(\mathbb{R}^N)$ 
and $L^2(\mathbb{R}^N)$.

(B) Inspired by \cite{Ribaud}, we use a  Littlewood-Paley decomposition of 
the Sobolev-Morrey spaces (see subsection \ref{sms}) to build general 
singular functions for Theorem \ref{selfsimilarity}. In fact,
   let $Y_{k_1}(x)$, $Y_{k_2}(x)$  be homogeneous harmonic polynomials  
of degree $k_1$ and  $k_2$, respectively. Consider $S(\varphi,\psi)$ the set
 of functions $(\varphi,\psi)\in\mathcal{S}'(\mathbb{R}^N)\times \mathcal{S}'(\mathbb{R}^N)/\mathcal{P}$ such that
\[ %\label{harmon-initial}
    \varphi(x)=\epsilon_1\frac{Y_{k_1}(x)}{| x|^{\frac{2}{\rho-1}+k_1} }
\quad\text{and}\quad
  \psi(x)=\epsilon_2\frac{Y_{k_2}(x)}{| x|^{\frac{2}{\rho-1}+\frac{2}{\alpha}+k_2}}.
\]
By \eqref{data-stand}, the set $S(\varphi,\psi)$ gives us a class of data  
for existence of self-similar solutions for \eqref{heat-wave}-\eqref{initial-data}.
\end{remark}

 \subsection{Symmetries}

 This subsection concerns with symmetries of solutions obtained in 
Theorems \ref{gw} and \ref{selfsimilarity}. It is straightforward to check 
that $E_{\alpha,1}(4\pi^2t^{\alpha}|\xi|^2)$ and 
$tE_{\alpha,2}(4\pi^2t^{\alpha}|\xi|^2)$ are invariant by the set 
$\mathcal{O}(N)$ of all rotations in $\mathbb{R}^{N}$. It follows that 
$G_{\alpha,1}(t)$ and $G_{\alpha,2}(t)$ are  $\mathcal{O}(N)-$ invariant.  
Hence, it is natural to ask whether or not the solutions of the above theorems 
present symmetry properties under certain qualitative conditions on the 
initial data.

Let $\mathcal{A}$ be a subset of $\mathcal{O}(N)$.
 A function $h$ is said symmetric under action $\mathcal{A}$ when 
$h(x)=h(T(x))$ for all $T\in\mathcal{A}$. If $h(x)=-h(T(x))$, the function 
$h$ is said antisymmetric under the action of $\mathcal{A}$.

\begin{theorem}\label{symmetry} 
Let the hypotheses of Theorem \ref{gw} be satisfied. 
The solution $u(\cdot,t)$ is symmetric  for all $t>0$, whenever 
$\varphi$ and $\psi$ are symmetric  under action $\mathcal{A}$.
 \end{theorem}


\begin{remark}\label{rem3} \rm
A radially symmetric solution is a self-similar solution, if the profile 
$\omega$ depends only on $r=| x|$, that is, there is a function $\mathcal{U}$ 
such that   $u(t,x)= t^{-\frac{\alpha}{\rho-1}} 
\mathcal{U}({| x|}/{t^{\frac{\alpha}{2}}})$,  $t>0$.
 
(A) Let $\mathcal{A}=\mathcal{O}(N)$ in Theorem \ref{symmetry}. 
If $\varphi$ and $\psi$ are radial and homogeneous functions of degree 
$-\frac{2}{\rho-1}$ and $-\frac{2}{\rho-1}-\frac{2}{\alpha}$, 
respectively (see Remark \ref{rem2}), then Theorems \ref{gw}, 
\ref{selfsimilarity} and \ref{symmetry} imply that 
\eqref{heat-wave}-\eqref{initial-data} have a unique self-similar solution 
$u\in X_\beta$ which is radially symmetric in $\mathbb{R}^N$.

 (B) Unlike the case $\kappa_1=0$, antisymmetry does not hold in general, 
for $\kappa_1\neq0$.
 \end{remark}

 \section{Technical estimates}\label{technical}

 In this section we prove some Mikhlin-type estimates for Mittag-Leffler 
functions. In spite of the fact that these estimates are necessary in 
the proof of Lemma \ref{galpha}, they are of independent interest. 
We start the section with a suitable  decomposition of 
$E_{\alpha,\beta}(z)$.

 \subsection{Decompositions of $E_{\alpha,\beta}(z)$}

\begin{proposition}\label{dec-gen-E}
Let $z\in\mathbb{C}$ be such that $\mathcal{R}e (z) >0$ and define
  \begin{gather*} %\label{axi0}
   \omega_{\alpha,\beta}(z)
=\frac{z^{\frac{1-\beta}{\alpha}}}{\alpha}\Big[\exp\big(a_\alpha(z)
+\frac{1-\beta}{\alpha}\pi i\big) 
+ \exp\big(b_\alpha(z)-\frac{1-\beta}{\alpha}\pi i\big)\Big],
\\ %\label{axi}
 l_{\alpha,\beta}(z)=\int_0^\infty \mathcal{H}_{\alpha,\beta}(s)
e^{- z^{1/\alpha}s^{1/\alpha}}z^{\frac{1}{\alpha}(1-\beta)}ds,
  \end{gather*}
where
\begin{gather}
   \mathcal{H}_{\alpha,\beta}(s)=\frac{1}{\alpha\pi}
\frac{\sin[(\alpha-\beta)\pi]-s\sin(\beta\pi)}{s^2
+2s\cos(\alpha \pi)+1} s^{\frac{1-\beta}{\alpha}} , \label{axi2}
 \\
a_\alpha(z)=z^{1/\alpha}e^{\frac{\pi i}{\alpha}}, \quad
 b_\alpha(z)= z^{1/\alpha}e^{-\frac{\pi i}{\alpha}} . \nonumber
  \end{gather}
Suppose that $1<\alpha<2$ and $1\leq\beta\leq2$, then
  \begin{equation}\label{decomp}
   E_{\alpha,\beta}(-z)=\omega_{\alpha,\beta}(z)+ l_{\alpha,\beta}(z) .
  \end{equation}
 \end{proposition}

 \begin{proof}
Recall that Mittag-Leffler function can be written as
  \begin{equation}
   E_{\alpha,\beta}(-z) = \frac{1}{2\pi i}\int_{Ha}
\frac{t^{\alpha-\beta}e^t}{t^\alpha+z} dt ,\label{M-L}
  \end{equation}
where  $Ha$ is the Hankel path, i.e. a path starts and ends at $-\infty$ 
and encircles the disk $| t|\leq | z|^{1/\alpha}$ positively.
The integrand $\Phi(t)=\frac{t^{\alpha-\beta}e^t}{t^\alpha+z}$ of \eqref{M-L}
 has two poles $a_\alpha(z)$ and $b_\alpha(z)$, because $1<\alpha<2$. 
Proceeding as in  \cite[Lemma 1.1]{YF}, the residues theorem yields
  \begin{align*}
   2\pi i E_{\alpha,\beta}(-z) 
&= \int_{\infty}^{R}\Phi(te^{-\pi i})d(te^{-\pi i}) 
 + 2\pi i \left(\operatorname{Res}(\Phi,a_\alpha(z)) 
 + \operatorname{Res}(\Phi,b_\alpha(z))\right)\\
&\quad  -\int_{\epsilon}^{R}\Phi(te^{-\pi i})d(te^{-\pi i})
 - \int_{R}^{\epsilon}\Phi(te^{\pi i})d(te^{\pi i})\\
&\quad - \int_{-\pi}^{\pi}\Phi(\epsilon e^{\theta i})d(\epsilon e^{\theta i})
 + \int_{R}^{\infty}\Phi(te^{\pi i})d(te^{\pi i}) \\
&=: I_1(R) +  2\pi i \left(\operatorname{Res}(\Phi,a_\alpha(z)) 
+ \operatorname{Res}(\Phi,b_\alpha(z))\right) -I_2(\epsilon,R) \\
&\quad  - I_3(R,\epsilon) - I_4(\epsilon) + I_5(R).
  \end{align*}
We first get
\[
   \lim_{R\to \infty} I_1(R) 
= \lim_{\epsilon\to 0^+} I_4(\epsilon) 
= \lim_{R\to \infty} I_5(R) = 0.
\]
An easy computation yields
\[
   l_{\alpha,\beta}(z)=-\frac{1}{2\pi i}\lim_{R\to \infty,
\epsilon\to 0^+} I_2(\epsilon,R) +I_3(R,\epsilon).
\]
  Indeed,
  \begin{align}
   &\frac{1}{2\pi i}\lim_{\epsilon\to  0^+,R\to \infty}
\big[I_2(\epsilon,R) + I_3(R,\epsilon)\big]\nonumber\\
&= \frac{1}{2\pi i}\Big(\int_{0}^{\infty} 
 \frac{e^{-t} t^{\alpha-\beta}e^{(\alpha-\beta)\pi i}}{t^\alpha e^{\alpha\pi i} +z}
 dt  - \int_{0}^{\infty} \frac{e^{-t} t^{\alpha-\beta}
 e^{-(\alpha-\beta)\pi i}}{t^\alpha e^{-\alpha\pi i} + z} dt\Big) \nonumber\\
& = -\frac{1}{2\pi i}2i \int_{0}^{\infty} e^{-t} t^{\alpha-\beta}
\frac{z\sin[(\alpha-\beta)\pi] -t^\alpha\sin(\beta\pi) }{t^{2\alpha}
 +2t^\alpha z\cos(\alpha\pi) + z^2} dt \label{E-dec1}
\\
   &=-\int_0^\infty \mathcal{H}_{\alpha,\beta}(s)
\exp{(- z^{1/\alpha}s^{1/\alpha})}
z^{\frac{1}{\alpha}(1-\beta)}ds\label{E-dec2}, \\
   & = -l_{\alpha,\beta}(z) , \nonumber
  \end{align}
where the change $t\mapsto z^{1/\alpha}s^{1/\alpha}$ was used
from \eqref{E-dec1} to \eqref{E-dec2}. Also, we obtain
  \begin{gather*}
   \operatorname{Res}(\Phi,a_\alpha(z)) 
= \frac{z^{\frac{1-\beta}{\alpha}}}{\alpha} \exp(a_\alpha(z)+\pi i(1-\beta)/\alpha),
\\
   \operatorname{Res}(\Phi,b_\alpha(z)) = \frac{z^{\frac{1-\beta}{\alpha}}}{\alpha}
 \exp(b_\alpha(z) -\pi i(1-\beta)/\alpha).
  \end{gather*}
These give us the desired decomposition.
\end{proof}

 In particular, for $\beta=1$ and $\beta=2$ in Proposition 
\ref{dec-gen-E}, we have the decompositions
 \begin{equation}\label{key1}
  E_{\alpha,1}(-z)=\omega_{\alpha,1}(z)+l_{\alpha,1}(z)
 \end{equation}
 in \cite[Lemma 1.1]{YF}, and
 \begin{equation}\label{key2}
  E_{\alpha,2}(-z)=\omega_{\alpha,2}(z)+l_{\alpha,2}(z),
 \end{equation}
 in \cite[Lemma 1.2-(IV)]{YF1}. 
Notice that $\omega_{\alpha,1}(z)$ oscillates with frequency $\sin(\pi/\alpha)$ 
and  amplitude decaying exponentially with rate $| \cos(\pi/\alpha)|$, in view of
\[
  \omega_{\alpha,1}(z)=\frac{2}{\alpha}\exp (z^{1/\alpha}
\cos(\pi/\alpha))\cos(z^{1/\alpha}\sin (\pi/\alpha)).
\]
 On the other hand, the function $l_{\alpha,1}(z)$ exhibits the relaxation 
phenomena of $E_{\alpha,1}(-z)$, namely,
\[
  l_{\alpha,1}(z)=\int_{0}^{\infty}\mathcal{H}_{\alpha,1}(s)
\exp(-s^{1/\alpha}z^{1/\alpha})ds
=\int_{0}^{\infty}\exp(-s^{1/\alpha}z^{1/\alpha})
d\mu_{\alpha}(s),
\]
 where
\[
  \mathcal{H}_{\alpha,1}(s)=\frac{\sin (\alpha \pi)}{\alpha\pi}
\frac{1}{s^{2}+2s\cos(\alpha\pi)+1}
\]
 and  $d\mu_{\alpha}(s)=\mathcal{H}_{\alpha,1}(s)ds$ is a finite measure 
in $\mathbb{R}_{+}$ such that $\mu_{\alpha}(\mathbb{R}_{+})=2-\frac{2}{\alpha}$.
 Furthermore, when $\beta=\alpha$, the decomposition \eqref{decomp} 
is useful to show that the map $G_{\alpha,\beta}(\cdot)$, $\beta=1,2$, 
is differentiable for $t>0$. Indeed, see \eqref{cgal1} and \eqref{cgal2} below.

 \subsection{Mikhlin estimates for $E_{\alpha,\beta}(-\sigma(\xi))$}\label{Mikhlin}

We provide estimates for $E_{\alpha,1}(\sigma(\xi))$,  
$E_{\alpha,2}(\sigma(\xi))$ and  $E_{\alpha,\alpha}(\sigma(\xi))$, 
 where $\sigma \in C^{\infty}(\mathbb{R}^{N}\backslash\{0\} ; (-\infty,0))$ 
is the symbol of the Fourier multiplier
\[
  \sigma(D)f=\mathcal{F}^{-1}\sigma(\xi)\mathcal{F}f(\xi),\quad
 f\in\mathcal{S}(\mathbb{R}^N).
\]
 Consider the change $z\mapsto \sigma(\xi)$ into \eqref{key1} and write 
it as follows:
 \begin{equation}\label{key-decomp1}
  E_{\alpha,1}(\sigma(\xi))=
  \omega_{\alpha,1}(-\sigma(\xi))+l_{\alpha,1}(-\sigma(\xi)).
 \end{equation}

\begin{proposition}\label{fund-lemma}
  Let $\sigma(\xi)\in C^{\infty}(\mathbb{R}^{N}\backslash\{0\})$ be a 
function homogeneous of degree $k>0$ and such that
  \begin{equation}\label{key-sym}
   \big| \frac{\partial^{\gamma}}{\partial \xi^{\gamma}}
[\sigma(\xi)] \big| 
\leq    A| \xi| ^{k-|\gamma| },\quad \xi\neq0
  \end{equation}
for all multi-index $\gamma\in(\mathbb{N}\cup\{0\})^{N}$ with 
$|\gamma|   \leq\lbrack N/2]+1$. Let $1<\alpha<2$ and $0\leq\delta<k$, 
there exists $C>0 $ such that
  \begin{equation}
\big| \frac{\partial^{\gamma}}{\partial \xi^{\gamma}}
[ | \xi| ^{\delta}E_{\alpha,1}(\sigma(\xi))] \big| 
\leq    CA | \xi| ^{-|\gamma| },\quad \xi\neq0.
   \label{point1}
  \end{equation}
 \end{proposition}

 \begin{proof} 
Taking \eqref{key-sym} into account we obtain
  \begin{equation}
\big| \frac{\partial^{\gamma} }{\partial \xi^{\gamma}} 
[-\sigma(\xi)]^{l}\big|\leq CA\, |\xi|^{-|\gamma|}| \xi|^{k l}, \quad
\text{for all } l\in  \mathbb{R}.\label{aux-deriv-symbol}
  \end{equation}
Hence, the $\gamma^{th}$-order derivative of the parcel 
$|\xi|^{\delta}\omega_{\alpha,1}(\sigma(\xi))$ can be estimated by
\begin{equation}
  \begin{aligned}
   &\big|\frac{\partial^{\gamma} }{\partial \xi^{\gamma}}
 [|\xi|^{\delta}\omega_{\alpha,1}(-\sigma(\xi))]\big| \\
   &=\big|\frac{\partial^{\gamma} }{\partial \xi^{\gamma}}
 \big[|\xi|^{\delta}\exp (e^{\frac{i\pi}{\alpha}}(-\sigma(\xi))^{1/\alpha})
 +|\xi|^{\delta}\exp (e^{-\frac{i\pi}{\alpha}}(-\sigma(\xi))^{1/\alpha})\big]
 \big| \\
&\leq C|\xi|^{-|\gamma|}\big[c_0|\xi|^{\delta}+c_1|\xi|^{\delta+\frac{k}{\alpha}}
+\dots +c_{|\gamma|}|\xi|^{\delta+\frac{|\gamma| k}{\alpha}}
\big] e^{\cos(\frac{\pi}{\alpha})(-\sigma(\xi))^{1/\alpha}} \\
   &\leq CA|\xi|^{-|\gamma|}. 
  \end{aligned} \label{w-est}
\end{equation}
 To estimate $l_{\alpha,1}(\sigma(\xi))$, recall that
\[
   l_{\alpha,1}(\sigma(\xi))=\int_0^{\infty}
\mathcal{H}_{\alpha,1}(s)\exp(-s^{1/\alpha}
(-\sigma(\xi))^{1/\alpha})ds.
\]
  Using the homogeneity $\sigma(\lambda \xi)=\lambda^{k}\sigma(\xi)$, we have
\begin{equation}
  \begin{aligned}
   &\big|\frac{\partial^{\gamma}}{\partial\xi^{\gamma}}
[|\xi|^{\delta}e^{-s^{1/\alpha}(-\sigma(\xi))^{1/\alpha}}]\big|
 \\
&\leq C |\xi|^{-|\gamma|}\big[c_0|\xi|^{\delta}+c_1|\xi|
 ^{\delta+\frac{k}{\alpha}}s^{1/\alpha}+\dots
+c_{|\gamma|}|\xi|^{\delta+\frac{|\gamma| k}{\alpha}}s^{\frac{|\gamma|}{\alpha}}
\big] e^{-s^{1/\alpha}(-\sigma(\xi))^{1/\alpha}} \\
&=C s^{-\frac{\delta}{k}}|\xi|^{-|\gamma|}
\big[c_0| s^{\frac{1}{k}}\xi|^{\delta}+c_1| s^{\frac{1}{k}}\xi|^{\delta
+\frac{k}{\alpha}}+\dots \\
&\quad +c_{|\gamma|}| s^{\frac{1}{k}}\xi|
 ^{\delta+\frac{|\gamma| k}{\alpha}}\big] 
e^{-[-\sigma(s^{1/k}\xi)]^{1/\alpha}}  \label{aux-est4}\\
&\leq CAs^{-\frac{\delta}{k}}|\xi|^{-|\gamma|} .
  \end{aligned}
\end{equation}
Then
  \begin{align*}
&\big|\frac{\partial^{\gamma}}{\partial\xi^{\gamma}}
 [|\xi|^{\delta}l_{\alpha,1}(-\sigma(\xi))]\big| \\
&=\frac{\sin (\alpha \pi)}{\alpha\pi}\int_{0}^{\infty}\frac{1}{s^{2}
 +2s\cos(\alpha\pi)+1}\big|\frac{\partial^{\gamma}}{\partial\xi^{\gamma}}
[|\xi|^{\delta}e^{ -s^{1/\alpha}(-\sigma(\xi))^{1/\alpha}}]\big| ds
 \nonumber\\
&\leq CA\Big(\frac{\sin (\alpha \pi)}{\alpha\pi}
\int_{0}^{\infty}\frac{s^{-\frac{\delta}{k}}}{s^{2}+2s\cos(\alpha\pi)+1}ds\Big) 
|\xi|^{-|\gamma|}\nonumber\\
   &\leq CA\, |\xi|^{-|\gamma|} ,\nonumber
  \end{align*}
because $0\leq \delta<k$. These estimates prove the proposition.
  \end{proof}

 
 In general, we obtain the following proposition for the two-parametric 
Mittag-Leffler function.

 \begin{proposition}\label{fund-lemma2}
  Let $\sigma(\xi)\in C^{\infty}(\mathbb{R}^{N}\backslash\{0\})$ be a
 homogeneous function of degree $k>0$ satisfying \eqref{key-sym}, 
for all multi-index $\gamma\in(\mathbb{N}\cup\{0\})^{N}$ with $|\gamma|
  \leq\lbrack N/2]+1$.  Then, there exists a positive constant $C$ 
(independent of $\delta$ and $k$)  such that
  \begin{equation}
   \big| \frac{\partial^{\gamma}}{\partial \xi^{\gamma}}
[ |\xi| ^{\delta}E_{\alpha,\beta}(\sigma(\xi))] \big| 
\leq    CA|\xi| ^{-|\gamma| },\quad  \xi\neq0
   \label{point2}
  \end{equation}
  provided that $1<\alpha<2$ and  $\,k(\frac{\beta}{\alpha}-\frac{1}{\alpha})
\leq\delta<k$.
 \end{proposition}

 \begin{proof} 
The proof is similar the proof of Proposition \ref{fund-lemma}. Indeed, 
proceeding as in \eqref{w-est}, it follows that
  \begin{equation}
\begin{aligned}
&\big| \frac{\partial^{\gamma_1}}{\partial \xi^{\gamma_1}} 
h_{\alpha,\beta}(\xi)\big|  \\
&:=\Big|\frac{\partial^{\gamma_1} }{\partial \xi^{\gamma_1}}
\Big[|\xi|^{\delta}\exp \big(a_\alpha(\sigma(\xi))+\frac{1-\beta}{\alpha}\pi i\big) \\
&\quad +|\xi|^{\delta}\exp \big(b_\alpha(\sigma(\xi))
-\frac{1-\beta}{\alpha}\pi i\big)\Big]\Big|  \\
&\leq CA|\xi|^{-|\gamma_1|}\big[c_0|\xi|^{\delta}
+c_1|\xi|^{\delta+\frac{k}{\alpha}}+\dots 
+c_{|\gamma_1|}|\xi|^{\delta+\frac{|\gamma_1| k}{\alpha}}\big] 
e^{\cos(\pi/\alpha) (-\sigma( \xi))^{1/\alpha}},
\end{aligned}\label{halpha}
  \end{equation}
for all multi-index $\gamma_1$. Hence, Leibniz's rule, \eqref{aux-deriv-symbol} 
and \eqref{halpha} give us
\begin{align*}
& \big|\frac{\partial^{\gamma_1}}{\partial \xi^{\gamma}}
[|\xi|^{\delta}\omega_{\alpha,\beta}(-\sigma(\xi))]\big|  \\
&\leq  \sum_{\gamma_1\leq \gamma}\binom{\gamma}{\gamma_1}
\big| \frac{\partial^{\gamma_1}}{\partial \xi^{\gamma_1}}
[\sigma(\xi)]^{\frac{1-\beta}{\alpha}}\big|\,
\big| \frac{\partial^{\gamma-\gamma_1}}{\partial \xi^{\gamma-\gamma_1}} 
[h_{\alpha,\beta}(\xi)]\big| \\
&\leq CA|\xi|^{-|\gamma_1| -|\gamma-\gamma_1|}
\big[c_0|\xi|^{\delta+k\left(\frac{1}{\alpha}-\frac{\beta}{\alpha}\right)}
+\dots +c|\xi|^{\delta+k\left(\frac{1}{\alpha}-\frac{\beta}{\alpha}\right)
+\frac{|\gamma-\gamma_1| k}{\alpha}}\big] \\
&\quad\times e^{\cos(\pi/\alpha)(- \sigma(\xi))^{1/\alpha}}\\
&\leq C A |\xi|^{-|\gamma|},\nonumber
\end{align*}
  in view of $\delta+k(\frac{1}{\alpha}-\frac{\beta}{\alpha})\geq0$. 
Also, using \eqref{aux-deriv-symbol} and \eqref{aux-est4}, the Leibniz's 
rule yields
 \begin{equation}
   \big| \frac{\partial^{\gamma}}{\partial \xi^{\gamma}} 
\big[\big((-\sigma(\xi))^{\frac{1-\beta}{\alpha}}\big)
\big(|\xi|^{\delta}\exp (-s^{1/\alpha}(-\sigma(\xi))^{1/\alpha})\big)\big]
\big|\nonumber
   \leq C A|\xi|^{-|\gamma|}s^{-\frac{\delta}{k}
+\frac{\beta}{\alpha}-\frac{1}{\alpha}}.\nonumber
  \end{equation}
  Hence, we estimate
  \begin{align*}
   \big|\frac{\partial^{\gamma}}{\partial \xi^{\gamma}} 
[|\xi|^{\delta}l_{\alpha,\beta}(-\sigma(\xi))] \big| 
&\leq   \int_{0}^{\infty} | \mathcal{H}_{\alpha,\beta}(s) | 
\big| \frac{\partial^{\gamma}}{\partial \xi^{\gamma}} 
\big[(-\sigma(\xi))^{\frac{1-\beta}{\alpha}}|\xi|^{\delta}
\exp (-s^{1/\alpha}(-\sigma(\xi))^{1/\alpha})
\big]\big| ds \\
&= CA(I +II)|\xi|^{-|\gamma|}  \\
&\leq C |\xi|^{-|\gamma|},
\end{align*}
where the  integrals $I$ and $II$ are defined by (see \eqref{axi2})
  \begin{gather}
   I=\frac{\sin [(\alpha -\beta) \pi]}{\alpha\pi}\int_{0}^{\infty}
\frac{s^{-\frac{\delta}{k}}}{s^{2}+2s\cos(\alpha\pi)+1} ds, \\
   II=-\frac{\sin (\beta \pi)}{\alpha\pi}
\int_{0}^{\infty}\frac{s^{1-\frac{\delta}{k}}}{s^{2}+2s\cos(\alpha\pi)+1} ds.
  \end{gather}
Those integrals are finite in view of $\delta<k$. This completes the proof 
of the proposition.
 \end{proof}

\section{Sobolev-Morrey estimates}\label{sme}

 In this section we obtain fundamental estimates which will be important 
to prove  Theorem \ref{gw}.

\subsection{Linear estimates}\label{linear-est}

Here, we present some estimates of the  Mittag-Leffler operators 
$\{G_{\alpha,\beta}(t)\}_{t\geq 0}$ in  Sobolev-Morrey spaces. 
Indeed, based on Propositions \ref{fund-lemma} and \ref{fund-lemma2} 
with the homogeneous symbol $\sigma(\xi)=-4\pi^2|\xi|^2$ of degree $2$, 
the following lemma can be proved by proceeding as in \cite[Lemma 3.1-(i)]{MJ}.

 \begin{lemma}\label{galpha}
  Let $\gamma_1\leq \gamma_2\in\mathbb{R}$, $ 1<p_1\leq p_2<\infty$,
$0\leq \mu< N$,  $1< \alpha<2$ and 
$\lambda=(\gamma_2-\gamma_1)+\frac{N-\mu}{p_1}-\frac{N-\mu}{p_2}$. 
There is a constant $C$ such that
  \begin{gather}
   \| G_{\alpha,1}(t) f\|_{\mathcal{M}_{p_2,\mu}^{\gamma_2}} 
\leq Ct^{-\frac{\alpha}{2}\lambda}\|f\|_{\mathcal{M}_{p_1,\mu}^{\gamma_1}},\quad
 \text{if }\lambda<2,\label{item-i}\\
   \| G_{\alpha,2}(t) f\|_{\mathcal{M}_{p_2,\mu}^{\gamma_2}} 
\leq Ct^{-\frac{\alpha}{2}\lambda} \,
\|f\|_{\mathcal{M}_{p_1,\mu}^{\gamma_1-\frac{2}{\alpha}}},\quad \text{if } 
\lambda +\frac{2}{\alpha}<2,\label{item-ii}  \\
   \| G_{\alpha,\alpha}(t) f\|_{\mathcal{M}_{p_2,\mu}^{\gamma_2}} 
\leq Ct^{\alpha-1-\frac{\alpha}{2}\lambda}\|f\|_{\mathcal{M}_{p_1,\mu}^{\gamma_1}},\quad
 \text{if }\big(2-\frac{2}{\alpha}\big)<\lambda<2,\label{item-iii}
  \end{gather}
for all $f\in\mathcal{S}'(\mathbb{R}^N)$.
 \end{lemma}

We finish this subsection by noticing that 
$\{\partial_tG_{\alpha,1}(t)\}_{t\geq 0}$ and
 $\{\partial_tG_{\alpha,2}(t)\}_{t\geq 0}$ are bounded in Morrey spaces. 
Indeed, a straightforward computation gives us
\[
  \frac{d}{dt} E_{\alpha,1}(-4\pi^2|\xi|^2t^{\alpha})
 = -4\pi^2|\xi|^2\left[t^{\alpha-1}E_{\alpha,\alpha}(-4\pi^2|\xi|^2t^{\alpha})\right],
\]
for $t>0$  and $\xi\neq 0$.
It follows from Lemma \ref{galpha}-(iii) that
 \begin{equation} \label{cgal1}
  \|  \partial_tG_{\alpha,1}(t)f\|_{\mathcal{M}_{p_2,\mu}}
\leq C\|  G_{\alpha,\alpha}(t)f\|_{\mathcal{M}_{p_2,\mu}^{2}}
\leq C t^{-\frac{\alpha}{2}
\left(\frac{N-\mu}{p_1}-\frac{N-\mu}{p_2}\right)-1}
\|  f\|_{\mathcal{M}_{p_1,\mu}}.
 \end{equation}
 Using
 \begin{equation}
  tE_{\alpha,2}(-4\pi^2|\xi|^2t^\alpha)
=\int_{0}^{t}E_{\alpha,1}(-4\pi^2|\xi|^2s^{\alpha}) ds ,
 \end{equation}
Lemma \ref{galpha}-(i) yields
\begin{equation} \label{cgal2}
  \|  \partial_tG_{\alpha,2}(t)f\|_{\mathcal{M}_{p_2,\mu}}
=\|  G_{\alpha,1}(t)f\|_{\mathcal{M}_{p_2,\mu}}
\leq C t^{-\frac{\alpha}{2}\left(\frac{N-\mu}{p_1}-\frac{N-\mu}{p_2}\right)}
\|  f\|_{\mathcal{M}_{p_1,\mu}}.
\end{equation}

 \subsection{Nonlinear estimates}\label{non_estmate}

 This subsection is devoted to estimate the nonlinear term 
$\mathcal{N}_{\alpha}(u)$ on the functional space $X_{\beta}$. 
Firstly, let us denote $\mathbf{B}(\nu,\eta)$ by \emph{special beta function} 
$\mathbf{B}(\nu,\eta)=\int_0^1(1-t)^{\nu-1}t^{\eta-1}dt$ which is finite, 
for all $\eta,\nu>0$. Let $k_1,k_2,k_3<1$, for  $t>0$ and $s>0$ the 
changes of variable $\tau \mapsto \tau s$ and $s\mapsto st$ give us
 \begin{equation}
\begin{aligned}
  I(t)&=\int_0^t(t-s)^{-k_1}\int_0^s(s-\tau)^{-k_2}\tau^{-k_3}d\tau ds  \\
  &=\mathbf{B}(1-k_2,1-k_3)\int_0^t(t-s)^{-k_1}s^{-k_2-k_3+1}ds \\
  &=\mathbf{B}(1-k_2,1-k_3)\mathbf{B}(1-k_1, 2-k_2-k_3)t^{2-k_1-k_2-k_3}.
\end{aligned}\label{beta}
 \end{equation}
We freely use \eqref{beta} in the next proof.

 \begin{lemma} \label{est-Non}
  Under assumptions of Theorem \ref{gw}, there is a positive constant 
$K=K(\kappa_1,\kappa_2)$ such that
  \begin{equation}
 \|\mathcal{N}_{\alpha}(u)-\mathcal{N}_{\alpha}(v)\|_{X_{\beta}}
\leq K\| u-v\|_{X_{\beta}}\big[\| u\|_{X_{\beta}}^{\rho-1}
+\| v\|_{X_{\beta}}^{\rho-1}+\| u\|_{X_{\beta}}^{q-1}+\| v\|_{X_{\beta}}^{q-1}\big] .
  \end{equation}
 \end{lemma}

 \begin{proof} 
Recall $\mathcal{N}_{\alpha}(u)$ and rewrite it as follows:
  \begin{equation}
\begin{aligned}
   \mathcal{N}_{\alpha}(u)(t)
&=\int_{0}^{t}G_{\alpha,1}(t-s)\int_{0}^{s}r_{\alpha}(s-\tau) 
 \left(\kappa_2| u|^{\rho-1} u+\kappa_1| \nabla_x u|^{q}\right) d\tau ds \\
   & =: \mathcal{N}_{\alpha}^1(u)(t)+\mathcal{N}_{\alpha}^2(u)(t).
\end{aligned}
  \end{equation}
The proof is divided in three steps.
\smallskip

\noindent\textbf{First step:} 
Estimates for $\mathcal{N}_{\alpha}^1(u)$. In \eqref{item-i}, 
let $(\gamma_1,\gamma_2,p_1,p_2)=(0,1,r/\rho,r)$ and  $1<\rho<r$ to obtain
  \begin{align*}
   &\| \mathcal{N}_{\alpha}^1(u)(t)-\mathcal{N}_{\alpha}^1(v)(t) 
\|_{\mathcal{M}^1_{r,\mu}}  \\
   &\leq C \int_0^t (t-s)^{-\lambda_1}\int_0^sr_{\alpha}(s-\tau)
\|f(u)-f(v)\|_{\mathcal{M}_{r/\rho,\mu}}d\tau ds  ,
  \end{align*}
where $f(u)(\tau)=\kappa_1| u(\tau)|^{\rho-1}u(\tau)$ and 
$\lambda_1=\frac{\alpha}{2}+\frac{\alpha}{2}
\left(\frac{N-\mu}{r/\rho}-\frac{N-\mu}{r}\right)$. Using that
  \begin{equation}
   |\, | a|^{\rho-1} a-| b|^{\rho-1}b|\leq C| a-b|
\left(| a|^{\rho-1}+ | b|^{\rho-1}\right), \text{ for all }  \rho>1 \label{iq-fund}
  \end{equation}
and $\frac{\rho}{r}=\frac{1}{r}+\frac{\rho-1}{r}$,  
the H\"older inequality \eqref{eq:holder} yields
  \begin{equation}
   \| \mathcal{N}_{\alpha}^1(u)(t)-\mathcal{N}_{\alpha}^1(v)(t) 
\|_{\mathcal{M}^1_{r,\mu}}
\leq C|\kappa_2| \int_0^t (t-s)^{-\lambda_1}\theta(s)ds\label{B_1} ,
  \end{equation}
where $\theta(s)$ is given by
  \begin{equation}
\begin{aligned}
   \theta(s)
&=\int_0^s(s-\tau)^{\alpha-2}\| u(\tau)-v(\tau)\|_{\mathcal{M}_{r,\mu}}
\big(\| u(\tau)\|_{\mathcal{M}_{r,\mu}}^{\rho-1}
+\| v(\tau)\|_{\mathcal{M}_{r,\mu}}^{\rho-1}\big)d\tau \\
&\leq C\int_0^s(s-\tau)^{\alpha-2}\tau^{-\rho\beta}
 \tau^{\beta}\| u(\tau)-v(\tau)\|_{\mathcal{M}_{r,\mu}} \times  \\
&\quad \times\tau^{\beta(\rho-1)}
 \big(\| u(\tau)\|_{\mathcal{M}_{r,\mu}}^{\rho-1}+\| v(\tau)
 \|_{\mathcal{M}_{r,\mu}}^{\rho-1}\big)d\tau \\
&\leq C\int_0^s(s-\tau)^{\alpha-2}\tau ^{-\rho\beta}d\tau
 \| u-v\|_{X_{\beta}}\big(\| u\|_{X_{\beta}}^{\rho-1}+\| v\|_{X_{\beta} }^{\rho-1}
\big).
\end{aligned}\label{theta1}
  \end{equation}
Notice that $\alpha (\rho-1)\frac{N-\mu}{2r}=\alpha -(\rho-1)\beta$ yields
\[
   -\lambda_1+\alpha-\rho\beta
=-\frac{\alpha}{2}+(\rho -1)\beta -\alpha +\alpha- \rho\beta=-\frac{\alpha}{2}-\beta.
\]
  It follows that \eqref{B_1} can be bounded by
  \begin{equation}
\begin{aligned}
\| \mathcal{N}_{\alpha}^1(u)(t)-\mathcal{N}_{\alpha}^1(v)(t)
 \|_{\mathcal{M}^1_{r,\mu}} 
&\leq C|\kappa_2| I_1(t) \,\| u-v\|_{X_{\beta}}
\big(\| u\|_{X_{\beta}}^{\rho-1}+\| v\|_{X_{\beta} }^{\rho-1}\big) \\
&\leq C|\kappa_2| t^{-\beta-\frac{\alpha}{2}} \| u-v\|_{X_{\beta}}
\big(\| u\|_{X_{\beta}}^{\rho-1}+\| v\|_{X_{\beta} }^{\rho-1}\big),
\end{aligned}
\end{equation}
because the integral $I_1(t)$ (see \eqref{beta}) satisfies
\[
   I_1(t)=\int_0^t (t-s)^{-\lambda_1}
\Big(\int_0^s(s-\tau)^{\alpha-2}\tau ^{-\rho\beta}d\tau\Big) ds
=C t^{-\lambda_1+\alpha -\rho\beta}=Ct^{-\beta -\frac{\alpha}{2}} .
\]
Proceeding in a similar fashion, we obtain
  \begin{equation}
\begin{aligned}
\|\mathcal{N}_{\alpha}^1(u)(t)-\mathcal{N}_{\alpha}^1(v)(t)
 \|_{\mathcal{M}_{r,\mu}} 
&\leq C|\kappa_2|  J_{1}(t) \,\| u-v\|_{X_{\beta}}
 \big(\| u\|_{X_{\beta}}^{\rho-1}+\| v\|_{X_{\beta} }^{\rho-1}\big) \\
&\leq C|\kappa_2|  t^{-\beta} \,\| u-v\|_{X_{\beta}}
\big(\| u\|_{X_{\beta}}^{\rho-1}+\| v\|_{X_{\beta}}^{\rho-1}\big),
 \end{aligned} 
\end{equation}
where $J_1(t)$ is given by
\[
   J_1(t)=\int_0^t (t-s)^{-\vartheta_{1}}
\int_0^s(s-\tau)^{\alpha-2}\tau ^{-\rho\beta}d\tau ds,
\]
  and  $\vartheta_1=\frac{\alpha}{2}
\left(\frac{N-\mu}{r/\rho}-\frac{N-\mu}{r}\right)$.
 In view of
\[
   -\vartheta_1+\alpha-\rho\beta
=\frac{\alpha}{2}
\big(\frac{N-\mu}{r/\rho}-\frac{N-\mu}{r}\big)+\alpha-\rho\beta 
=(\rho -1)\beta -\alpha +\alpha- \rho\beta=-\beta  ,
\]
one has $J_1(t)=Ct^{-\beta}$.

The convergence of $I_{1}(t)$ and $J_{1}(t)$ follows from \eqref{param-hip2} 
for  $\lambda_1= \frac{\alpha}{2}+\vartheta_1<1$ and because 
$\frac{p}{r}<\left(\frac{1}{\alpha}-\frac{1}{2}\right)$ is equivalent to 
$\vartheta_1=\alpha\frac{p}{r}<1-\frac{\alpha}{2}$. Further,
 $\left(1-\frac{p}{r}\right)
 < \frac{\rho-1}{\alpha}\left(\frac{1}{q}-\frac{\alpha}{2}\right)
=\frac{\rho-1}{\alpha\rho}\left(\frac{\rho+1}{2}-\frac{\alpha\rho}{2}\right)
<\frac{\rho-1}{\alpha\rho}$ 
leads to $\rho\beta<1$.
\smallskip

\noindent\textbf{Second step:}
 Estimates for $\mathcal{N}_{\alpha}^2(u)$. Using inequality \eqref{item-i} 
with $(\gamma_1,\gamma_2,p_1,p_2)=(0,1,r/q,r)$, in view of $1<q<\rho<r$, we obtain
\begin{align*}
   &\|\mathcal{N}_{\alpha}^2(u)(t)-\mathcal{N}_{\alpha}^2(v)(t)
 \|_{\mathcal{M}_{r,\mu}^1} \\
   &\leq C \int_0^t (t-s)^{-\lambda_2}\int_0^sr_{\alpha}(s-\tau)
\|g(u)-g(v)\|_{\mathcal{M}_{r/q,\mu}}d\tau ds,
  \end{align*}
where $g(f)(\tau)=\kappa_2| \nabla_xf(\tau)|^q$ and 
$\lambda_2=\frac{\alpha}{2}+\frac{\alpha}{2}
\left(q\frac{N-\mu}{r}-\frac{N-\mu}{r}\right)$. 
It follows that
\[
   \|\,|\nabla_x u(t)|^q-|\nabla_x v(t)|^q
\|_{\mathcal{M}_{r,\mu}}
\leq C  \| u(t)- v(t)\|_{\mathcal{M}_{r,\mu}^1}
\big(\|  u(t)\|_{\mathcal{M}_{r,\mu}^1}^{q-1}
+\| v(t)\|_{\mathcal{M}_{r,\mu}^1}^{q-1}\big)\,.
\]
Hence,
\[
   \|\mathcal{N}_{\alpha}^2(u)(t)-\mathcal{N}_{\alpha}^2(v)(t)
\|_{\mathcal{M}_{r,\mu}^1}
\leq C|\kappa_1| \int_0^t (t-s)^{-\lambda_2}\tilde{\theta}(s)ds %\label{B_2},
\]
where $\tilde{\theta}(s)$ is bounded as
\begin{equation}
\begin{aligned}
   \tilde{\theta}(s)
&= C\int_0^s(s-\tau)^{\alpha-2}\| u(\tau)
 -v(\tau)\|_{\mathcal{M}_{r,\mu}^1}
 \left(\| u(\tau)\|_{\mathcal{M}_{r,\mu}^1}^{q-1}
 +\| v(\tau)\|_{\mathcal{M}_{r,\mu}^1 }^{q-1}\right)d\tau \\
&\leq C\int_0^s(s-\tau)^{\alpha-2}\tau ^{-q(\beta +\frac{\alpha}{2})}d\tau
 \| u-v\|_{X_{\beta}}\left(\| u\|_{X_{\beta}}^{q-1}+\| v\|_{X_{\beta} }^{q-1}\right).
\end{aligned} \label{theta2}
  \end{equation}
Hence, we estimate
  \begin{align*}
\left\|\mathcal{N}_{\alpha}^2(u)(t)-\mathcal{N}_{\alpha}^2(v)(t)
\right\|_{\mathcal{M}_{r,\mu}^1}
&\leq C| \kappa_1| I_2(t) \,\| u-v\|_{X_{\beta}}
\left(\| u\|_{X_{\beta}}^{q-1}+\| v\|_{X_{\beta}}^{q-1}\right) \\
&\leq C|\kappa_1| t^{-\beta-\frac{\alpha}{2}}\,\| u-v\|_{X_{\beta}}
\left(\| u\|_{X_{\beta}}^{q-1}+\| v\|_{X_{\beta}}^{q-1}\right),
\end{align*}
where $I_2(t)$ is given by
\[
   \int_0^t (t-s)^{-\lambda_2}\int_0^s(s-\tau)^{\alpha-2}
\tau ^{-q(\beta +\frac{\alpha}{2})}d\tau ds
=Ct^{-\lambda_2+\alpha-q(\beta+\alpha/2)}=Ct^{-\beta -\frac{\alpha}{2}}.
\]
Indeed, in view of $q=\frac{2\rho}{\rho+1}$ and
  \begin{equation}
   (q-1)\beta=\alpha \frac{q-1}{\rho-1}-\alpha (q-1)\frac{N-\mu}{2r}
=\frac{\alpha}{2}(2-q)-\alpha (q-1)\frac{N-\mu}{2r},\label{eq-auxN}
  \end{equation}
we obtain
  \begin{align*}
   -\lambda_2+\alpha-q(\beta+\alpha/2)
&=-\alpha (q-1)\frac{N-\mu}{2r}-\frac{\alpha}{2} 
+ \alpha-q(\beta+\alpha/2) \\
   &=(q-1)\beta-\frac{\alpha}{2}(2-q)-\frac{\alpha}{2} 
+ \alpha-q(\beta+\alpha/2) \\
   &=-\frac{\alpha}{2}-\beta. 
\end{align*}
  It remains to obtain estimates for  
$\sup_{0<t<T}t^{\beta}\|\mathcal{N}_{\alpha}^2(u)(t)
-\mathcal{N}_{\alpha}^2(v)(t)\|_{\mathcal{M}_{r,\mu}}$. 
Proceeding as before, one has
  \begin{align*}
&\|\mathcal{N}_{\alpha}^2(u)(t)-\mathcal{N}_{\alpha}^2(v)(t)
\|_{\mathcal{M}_{r,\mu}} \\
&\leq C|\kappa_1| J_{2}(t)\sup_{0<t<T}t^{\beta +\frac{\alpha}{2}}
\| \nabla_x u(t)-\nabla_xv(t)\|_{\mathcal{M}_{r,\mu}} \\
&\quad \times \sup_{0<t<T}t^{(\beta +\frac{\alpha}{2})(q-1)}
 \left(\| \nabla_x u(t)\|_{\mathcal{M}_{r,\mu}}^{q-1}
 +\| \nabla_x v(t)\|_{\mathcal{M}_{r,\mu}}^{q-1}\right) \\
&\leq C|\kappa_1| J_{2}(t)\,\| u-v\|_{X_{\beta}}
\left(\| u\|_{X_{\beta}}^{q-1}+\| v\|_{X_{\beta}}^{q-1}\right) \\
&\leq C| \kappa_1| t^{-\beta}\,\| u-v\|_{X_{\beta}}
 \left(\| u\|_{X_{\beta}}^{q-1}+\| v\|_{X_{\beta}}^{q-1}\right),
\end{align*}
where $J_2(t)$ is given by
\[
  \int_0^t (t-s)^{-\vartheta_2}\int_0^s(s-\tau)^{\alpha-2}
\tau ^{-q(\beta +\frac{\alpha}{2})}d\tau ds
=Ct^{-\vartheta_2+\alpha -q(\beta +\alpha/2)}=Ct^{-\beta}.
\]
In fact, the inequality \eqref{item-i} with 
$(\gamma_1,\gamma_2, p_1,p_2)=(0, 0,r/q, r)$ implies that  
$\vartheta_2=\frac{\alpha}{2}\big(q\frac{N-\mu}{r}-\frac{N-\mu}{r}\big)
=\alpha (q-1)\frac{N-\mu}{2r}$, which by \eqref{eq-auxN} give us
  \begin{equation}
   -\vartheta_2+\alpha -q(\beta+\frac{\alpha}{2}) 
= (q-1)\beta -\frac{\alpha}{2}(2-q)+\alpha -q(\beta+\frac{\alpha}{2})=-\beta.
  \end{equation}
The convergence of the beta functions $I_2(t), J_2(t)$ follows by our 
hypotheses in \eqref{param-hip2}, because  
$\left(1-\frac{p}{r}\right) 
< \frac{\rho-1}{\alpha}\left(\frac{1}{q}-\frac{\alpha}{2}\right)$ 
is equivalent to $q(\beta+\alpha/2)<1$ and 
$\frac{p}{r}<\left(\frac{1}{\alpha}-\frac{1}{2}\right)$ and 
$q=\frac{2\rho}{\rho+1}$ yields to 
$\vartheta_2=\frac{\alpha}{\rho+1}\frac{p}{r}<\left( 1-\frac{\alpha}{2}\right)$
 which is equivalent $ \lambda_2=\frac{\alpha}{2}+\vartheta_2<1$.
\smallskip

\noindent\textbf{Third step:} The two steps above lead to
  \begin{align*}
   &\|  \mathcal{N}_{\alpha}(u)-\mathcal{N}_{\alpha}(v)\|_{X_{\beta}} \\
  &\leq  \|  \mathcal{N}_{\alpha}^1(u)-\mathcal{N}_{\alpha}^1(v)\|_{X_{\beta}}
 +\|  \mathcal{N}_{\alpha}^2(u)-\mathcal{N}_{\alpha}^2(v)\|_{X_{\beta}}\\
  &\leq K(\kappa_1,\kappa_2)\| u-v\|_{X_{\beta}}
\left[\left(\| u\|_{X_{\beta}}^{\rho-1}+\| v\|_{X_{\beta}}^{\rho-1}
\right)+\left(\| u\|_{X_{\beta}}^{q-1}+\| v\|_{X_{\beta}}^{q-1}\right)\right].
  \end{align*}
This completes the proof.
 \end{proof}


 \section{Proofs of theorems}\label{proofs}


Now, we put all estimates of Section \ref{sme} together to prove our theorems.

 \subsection{Proof of Theorem \ref{gw}}
\textbf{Part (i):} Notice that
 \begin{equation}
  \|G_{\alpha,1}(t)\varphi\|_{X_{\beta}} 
+ \|G_{\alpha,2}(t)\psi\|_{X_{\beta}}
=\| \varphi\|_{D(\alpha,\beta)}+\| \psi\|_{\widetilde{D}(\alpha,\beta)}
\leq \varepsilon,\label{est-proof4}
 \end{equation}
where $\varepsilon>0$ will be chosen such that
 \begin{equation}
  \left(2^{\rho}\varepsilon^{\rho-1}+2^{q}\varepsilon^{q-1}\right)
<\frac{1}{2K}.\label{rest}
 \end{equation}
Consider the complete metric $d_B(\cdot,\cdot)$ defined by 
$d_{B}(u,v)=\| u-v\|_{X_{\beta}}$ on the ball $B_{X_{\beta}}(2\varepsilon)$ 
and let $\Lambda:B_{X_{\beta}}(2\varepsilon)\to  B_{X_{\beta}}(2\varepsilon)$ 
be the operator
\[
  \Lambda (u)(t)= G_{\alpha,1}(t)\varphi +G_{\alpha,2}(t)\psi 
+\mathcal{N}_{\alpha}(u)(t).
\]
 We would like to show that  
$\Lambda(B_{X_{\beta}}(2\varepsilon))\subset B_{X_{\beta}}(2\varepsilon)$ 
and $\Lambda$ is a contraction on metric space $(B_{X_{\beta}}(2\varepsilon), d_B)$. 
In fact,  the continuity of 
$G_{\alpha,j}(\cdot):(0,\infty)\to \mathcal{M}_{r,\mu}$, $j=1,2$, was
 proved at the final of subsection \ref{linear-est}, and the regularization 
property of the convolution imply that 
$(\Lambda u):(0,\infty)\to \mathcal{M}_{r,\mu}$ is continuous, whenever 
$u\in X_\beta$. Further, from Lemma \ref{est-Non} we obtain
 \begin{equation}
\begin{aligned}
  \| \Lambda(u)-\Lambda(v)\|_{X_{\beta}}
&=\|\mathcal{N}_{\alpha}(u) -\mathcal{N}_{\alpha}(v)\|_{X_{\beta}} \\
&\leq K\| u-v\|_{X_{\beta}}\left[\| u\|_{X_{\beta}}^{\rho-1}
 +\| v\|_{X_{\beta}}^{\rho-1}+\| u\|_{X_{\beta}}^{q-1}
 +\| v\|_{X_{\beta}}^{q-1}\right] \\
&\leq K\left(2^{\rho}\varepsilon^{\rho-1}+2^{q}\varepsilon^{q-1}\right)
 \| u-v\|_{X_{\beta}}
\end{aligned} \label{est-proof3}
 \end{equation}
 for all $u,v\in B_{X_{\beta}}(2\varepsilon)$. Now, 
\eqref{est-proof4} and \eqref{est-proof3} give us
 \begin{equation}
\begin{aligned}
  \| \Lambda(u)\|_{X_{\beta}}
&\leq\| G_{\alpha,1}(t)\varphi+G_{\alpha,2}(t)\psi
  \|_{X_{\beta}} + \left\|\mathcal{N}_{\alpha}(u)
 -\mathcal{N}_{\alpha}(0)\right\|_{X_{\beta}} \\
  &\leq \varepsilon + K\left(2^{\rho}\varepsilon^{\rho-1}
+2^{q}\varepsilon^{q-1}\right)2\varepsilon < 2\varepsilon\,,
\end{aligned}
 \end{equation}
 in view of \eqref{rest} and provided that $u\in B_{X_{\beta}}(2\varepsilon)$. 
Hence, $\Lambda(B_{X_{\beta}}(2\varepsilon))\subset B_{X_{\beta}}(2\varepsilon)$ 
and $\Lambda$ is a contraction in $B_{X_{\beta}}(2\varepsilon)$. 
From the Banach fixed theorem there is a \emph{mild solution} 
$u\in X_{\beta}$ for \eqref{heat-wave}-\eqref{initial-data} which is unique 
in the ball $B_{X_{\beta}}(2\varepsilon)$.
\smallskip

\noindent\textbf{Part (ii):}
 Let $u$ and $\tilde{u}$ be two \emph{mild solutions} in 
$B_{X_{\beta}}(2\varepsilon)$,  obtained in the Part (i), subject to initial data 
$(\varphi,\psi)$ and $(\tilde{\varphi}, \tilde{\psi})$, respectively. Then
 \begin{align*}
  \| u-\tilde{u}\|_{X_{\beta}}
&\leq \| G_{\alpha,1}(\varphi-\tilde{\varphi})\|_{X_{\beta}} 
 +\| G_{\alpha,2}(\psi-\tilde{\psi})\|_{X_{\beta}} 
 + \|\mathcal{N}_{\alpha}(u)-\mathcal{N}_{\alpha}(\tilde{u})\|_{X_{\beta}} \\
  &\leq  \| \varphi-\tilde{\varphi}\|_{D(\alpha,\beta)}
 +\| \psi-\tilde{\psi}\|_{\widetilde{D}(\alpha,\beta)}
 +K\left( 2^\rho\varepsilon^{\rho-1}+2^q\varepsilon^{q-1}\right) 
 \| u-\tilde{u}\|_{X_{\beta}}\,,
\end{align*}
 which yields the Lipschitz continuity
 \begin{equation}
  \| u-\tilde{u}\|_{X_{\beta}}\leq \frac{1}{\left(1-\frac{1}{4K}\right)}
\left(\| \varphi-\tilde{\varphi}\|_{D(\alpha,\beta)}
+\| \psi-\tilde{\psi}\|_{\widetilde{D}(\alpha,\beta)}\right).
 \end{equation}
 This completes the proof.

 \subsection{Proof of Theorem \ref{selfsimilarity}}

 Let $\delta_{\lambda}f(x)=f(\lambda x)$ and notice that 
$\widehat{\delta_{\lambda}f}(\xi)=\lambda^{-N}\widehat{f}(\xi/\lambda)$. 
Recall that $G_{\alpha, 2}(t)\psi$ and $G_{\alpha,1}(t)\varphi$ are given by
 $$
G_{\alpha,2}(t)\psi:=t\,k_{\alpha,2}(t,\cdot)\ast \psi \quad\text{and}\quad
G_{\alpha,1}(t)\varphi:=k_{\alpha,1}(t,\cdot)\ast \varphi,
$$
 where $\widehat{k}_{\alpha,2}(t,\xi)=tE_{\alpha,2}(-4\pi^2t^{\alpha}|\xi|^2)$
 and $\widehat{k}_{\alpha,1}(t,\xi)=E_{\alpha,1}(-4\pi^2t^{\alpha}|\xi|^2)$.
 Notice that
 \begin{align*}
  [\delta_{\gamma}G_{\alpha,2}(\gamma^{\frac{2}{\alpha}}t)\psi]^{\wedge}(\xi)
&=\gamma^{-N} \gamma^{\frac{2}{\alpha}}t\widehat{k}_{\alpha,2}
(\gamma^{\frac{2}{\alpha}}t,\xi/\gamma)\widehat{\psi}(\xi/\gamma) \\
  &=\gamma^{\frac{2}{\alpha}}[tE_{\alpha,2}(-4\pi^2t^{\alpha}|\xi|^2)]
 \gamma^{-N}\widehat{\psi}(\xi/\gamma) \\
  &=\gamma^{-\frac{2}{\rho-1}}[G_{\alpha,2}(t)\psi]^{\wedge}(\xi),
 \end{align*}
 in view of $\delta_{\gamma}\psi(x)
=\gamma^{-\frac{2}{\rho-1}-\frac{2}{\alpha}}\psi(x)$. Hence,
\[
  [G_{\alpha,2}(t)\psi]_{\gamma}=G_{\alpha,2}(t)\psi_{\gamma}\quad
\text{and}\quad
 [G_{\alpha,1}(t)\varphi]_{\gamma}=G_{\alpha,1}(t)\varphi_{\gamma}.
\]
 We can easily check that $\mathcal{N}_{\alpha}(u_{\gamma})
=[\mathcal{N}_{\alpha}(u)]_{\gamma}$, for all $\gamma>0$. Here, we denoted 
 $[\mathcal{N}_{\alpha}(u)]_{\gamma}(t,x)
=\gamma^{\frac{2}{\rho-1}}\mathcal{N}_{\alpha}(u)
(\gamma^{\frac{2}{\alpha}}t,\gamma x) $. Therefore,
\[
  u_\gamma(t) = [G_{\alpha,1}(t)\varphi]_\gamma+ [G_{\alpha,2}(t)\psi]_\gamma 
+ [\mathcal{N}_\alpha(u)]_\gamma = G_{\alpha,1}(t)\varphi_\gamma 
+ G_{\alpha,2}(t)\psi_\gamma + \mathcal{N}_\alpha(u_\gamma)
\]
 is a mild solution $u_\gamma\in X_{\beta}$ of \eqref{heat-wave}-\eqref{initial-data}. 
From $\| u_\gamma\|_{X_\beta} = \| u \|_{X_\beta}$ and the uniqueness proved 
in Theorem \ref{gw}-(i), we have
\[
  u(t,x)=u_\gamma(t,x), \quad\text{a.e. $x\in\mathbb{R}^N$ and for all }\gamma,t>0.
\]
This completes the proof.



 \subsection{Proof of Theorem \ref{symmetry}}\label{prof-symmetry}

Let $G_{\alpha,2}(t)\psi:=k_{\alpha,2}(t,\cdot)\ast \psi$ and 
$G_{\alpha,1}(t)\varphi:=k_{\alpha,1}(t,\cdot)\ast \varphi$,
 be as above. For $T\in\mathcal{A}$, we have
 \begin{align*}
  k_{\alpha,1}(t, T(x))
&=\int_{\mathbb{R}^N} e^{2\pi i \,\langle T(x),
\xi\rangle} E_{\alpha,1}(-4\pi^2t^{\alpha}|\xi|^{2}))d\xi \\
  &=\int_{\mathbb{R}^N} e^{2\pi i \,\langle T(x),
T(\xi)\rangle} E_{\alpha,1}(-4\pi^2t^{\alpha}|T(\xi)|^{2}))| \det T| d\xi \\
  &=\int_{\mathbb{R}^N} e^{2\pi i \langle x,\xi\rangle} 
E_{\alpha,1}(-4\pi^2t^{\alpha}|\xi|^{2}))d\xi
=k_{\alpha,1}(t, x),
 \end{align*}
 where we used the change of variable $\xi\mapsto T(\xi)$ and the fact that 
$| \det T| = 1$. In a  similar fashion one has
 \[
k_{\alpha,2}(t, T(x))=k_{\alpha,2}(t, x).
\]
 It follows that
 \begin{align*}
  u_1(t, T(x))
&:=\int_{\mathbb{R}^N} k_{\alpha,1}(t, T(x)-y)\varphi(y)dy 
+\int_{\mathbb{R}^N} k_{\alpha,2}(t, T(x)-y)\psi(y)dy\\
  &=\int_{\mathbb{R}^N} k_{\alpha,1}(t, T(x-z))\varphi(Tz)dz 
+\int_{\mathbb{R}^N} k_{\alpha,2}(t, T(x-z))\psi(Tz)dz\\
  &=-\int_{\mathbb{R}^N} k_{\alpha,1}(t, x-z)\varphi(z)dz 
-\int_{\mathbb{R}^N} k_{\alpha,2}(t, x-z)\psi(z)dz\\
  &=u_1(t,x)
 \end{align*}
 when $\varphi$ and $\psi$ are symmetric under action $\mathcal{A}$. Let
 \[
\theta(s,x)=\int_{0}^{s}r_{\alpha}(s-t) \kappa_2| u(t,x)|^{\rho-1} u(t,x)
-\kappa_1 | \nabla_xu(t,x)|^{q}
\]
 and notice that
 \begin{align*}
  \theta(s,Tx)
&=\int_{0}^{s}r_{\alpha}(s-t) \left(\kappa_2| u(t, Tx)|^{\rho-1} u(t,Tx)-\kappa_1| \nabla_x u(t,Tx)|^{q}\right) d\tau\nonumber\\
  &=\int_{0}^{s}r_{\alpha}(s-t) \left(\kappa_2| u(t,x)|^{\rho-1}u(t,x) -\kappa_1 | T \,\nabla_x u(t,T(x))|^{q}\right)d\tau\\
  &=\theta(s,x),
 \end{align*}
 that is, $\theta(t,\cdot)$ is symmetric whenever $u(t,\cdot)$ is. Hence,
\[
  \mathcal{N}_{\alpha}(u)(t,x)
=\int_{0}^{t}\int_{\mathbb{R}^N}k_{\alpha,1}(t-s,x-y)\theta(s,y)dyds
\]
 is symmetric if $u(t,\cdot)$ is, for each $t>0$. From now on, employing 
an induction argument  in the Picard's sequence
 \begin{gather*}
  u_{1}(t,x) =G_{\alpha,2}(t)\psi +G_{\alpha,1}(t)\varphi  \\
  u_{k}(t,x) =u_{1}(t,x)+\mathcal{N}_{\alpha }(u_{k-1})(t,x),\;k=2,3,\dots
 \end{gather*}
 one can prove that $(u_k)$ is symmetric. It follows that $u(t,x)$ is symmetric, for
 all $t>0$.
The proof is complete.


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\end{document}