\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 17, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/17\hfil Klein-Gordon equation in curved spacetime] {Decay estimates for the Klein-Gordon equation in curved spacetime} \author[M. Yaz{\i}c{\i} \hfil EJDE-2018/17\hfilneg] {Muhammet Yaz{\i}c{\i}} \address{Muhammet Yaz{\i}c{\i} \newline Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, Trabzon, 61080, Turkey} \email{m.yazici@ktu.edu.tr} \thanks{Submitted June 16, 2017. Published January 13, 2018.} \subjclass[2010]{35L15, 35C15, 35Q75} \keywords{De sitter spacetime; Klein-Gordon eqution; fundamental solutions, \hfill\break\indent $L^\infty$ estimates} \begin{abstract} We consider the initial-value problem for the Klein-Gordon equation in de Sitter spacetime. We derive $L^{\infty}$ decay estimates for the solution to the linear Klein-Gordon equation in de Sitter spacetime with and without source term. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In this article, we consider the following initial value problem for the Klein-Gordon equation in de Sitter spacetime, \begin{equation} \begin{gathered} \partial^{2}_{t}\Phi+n\Phi_{t}-e^{-2t}\Delta\Phi+m^{2}\Phi=f(x,t), \quad (x,t)\in \mathbb{R}^n\times\mathbb{R},\\ \Phi(x,0)=\varphi_0(x) , \quad \partial_{t}\Phi(x,0)=\varphi_{1}(x), \quad x\in\mathbb{R}^n, \end{gathered}\label{eq03.2} \end{equation} where $f\in C^{\infty}(\mathbb{R}^{n+1})$, $\varphi_0, \varphi_{1}$ are in Sobolev space $W^{[n/2]+1,1}(\mathbb{R}^n)$, and $m>0$. In Minkowski spacetime, the initial value problem for the semilinear Klein-Gordon equation $$ u_{tt}-\Delta u+ m^{2}u=|u|^{\alpha}u, $$ has been extensively investigated. The existence of global weak solutions has been obtained by J\"orgens \cite{Jor61}, Pecher \cite{Pec76}, Brenner \cite{Bren79}, Ginibre and Velo \cite{Gin85, Gin89}. In order for the total energy is well-defined in the energy space, one needs the assumption $\alpha < 4/(n-1)$. On the other hand, the initial value problem for so-called Higgs boson equation $$ u_{tt}-\Delta u-m^{2}u=-|u|^{\alpha}u, \quad (x,t)\in \mathbb{R}^n\times\mathbb{R} $$ in Minkowski spacetime, and $$ \partial^{2}_{t}\Phi+nH\Phi_{t}-e^{-2Ht}\Delta\Phi-m^{2}\Phi =-|\Phi|^{\alpha}\Phi, \quad (x,t)\in \mathbb{R}^n\times\mathbb{R} $$ in de Sitter spacetime are studied by Yagdjian \cite{Yag012}, and some qualitative property of the solution revealed if the global solution exists. In addition, it was shown by Baskin \cite{Bas13} that the initial value problem for $$ \partial_{t}^2 u+n\partial_{t}u +\frac{\partial_{t}\sqrt{h_{t}}}{\sqrt{h_{t}}}\partial_{t}u +e^{-2t}\Delta_{h_{t}}u+\lambda u +|u|^\alpha u=0, \quad(y,t)\in Y\times \mathbb{R} $$ admits a small amplitude global solution in the energy space $H^1\oplus L^{2}$, provided $\lambda>n^{2}/4$ and $\alpha=4/(n-1)$. Here $h_{t}$ is a smooth family of Riemannian metrices on compact $n$-dimensional manifold $Y$, which is characterized as an asymptotically de Sitter spacetime. In Nakamura \cite{Nak14}, the assumption on the regularity of the initial data is weakened in the case of $m\geq n/2$. Turning back to the initial value problem (\ref{eq03.2}), the following theorem obtained by Yagdjian \cite{Yag12} states the estimate in the Sobolev space $H^{s}(\mathbb{R}^n)$. \begin{theorem}[\cite{Yag12}] \label{thm1} Let $\Phi=\Phi(x,t)$ be the solution of the initial value problem $$ \Phi_{tt}+n\Phi_t+e^{-2t}\Delta \Phi+m^2\Phi=f, \quad \Phi(x,0)=\varphi_0(x), \quad\Phi_t(x,0)=\varphi_{1}(x) $$ for $(x,t)\in\mathbb{R}^n\times(0,\infty)$, where $\varphi_0, \varphi_{1}\in C^{\infty}_0(\mathbb{R}^n)$ and $f\in C^{\infty}(\mathbb{R}^{n+1})$. Let $l$ be a nonnegative integer, $m<\sqrt{n^2-1}/2$ and $n\geq2$. Then there exists a constant $C>0$ such that \begin{equation} \begin{split} &\|(-\Delta)^{-s}\Phi(\cdot,t)\|_{W^{l,q}(\mathbb{R}^n)}\\ &\leq C e^{(M-\frac{n}{2})t}(1-e^{-t})^{\left(2s-n\left(\frac{1}{p} -\frac{1}{q}\right)\right)} \left\{\|\varphi_0\|_{W^{l,p}(\mathbb{R}^n)}+(1-e^{-t})\|\varphi_{1}\|_{W^{l,p} (\mathbb{R}^n)}\right\}\\ &\quad +C e^{-(\frac{n}{2}-M)t}\int_0^{t}e^{(\frac{n}{2}-M)b} e^{-b\left(2s-n\left(\frac{1}{p}-\frac{1}{q}\right)\right)} \|f(\cdot,b)\|_{W^{l,p}(\mathbb{R}^n)}db \end{split} \label{pre3.5} \end{equation} for all $t>0$, provided that $10,\; x\in \mathbb{R}^n, \end{equation} in the Besov space $B_p^{s,q}$, where $A(x,\partial_x)=\sum_{|\alpha|\leq2}a_{\alpha}(x)\partial_x^\alpha$ is a second-order negative elliptic differential operator with real coefficients $a_{\alpha}\in \mathcal{B}^{\infty}$ and $m$ in the set $(0,\sqrt{n^2-1}/2)\cup[n/2,\infty)$. Here, $\mathcal{B}^{\infty}$ denotes the space of all $C^{\infty}$ functions with uniformly bounded derivatives of all orders. The case $m\in(\sqrt{n^2-1}/2,n/2)$ is also considered by Yagdjian \cite{Yag17_1} in the Besov space. In this article, we are interested in the case of $00$ such that \begin{equation} \begin{split} \|\Phi(\cdot,t)\|_{L^{\infty}(\mathbb{R}^n)} &\leq C e^{(M-\frac{n}{2})t}\left\{\|\varphi_0\|_{W^{[n/2]+1,1} (\mathbb{R}^n)}+\|\varphi_{1}\|_{W^{[n/2]+1,1}(\mathbb{R}^n)}\right\}\\ &\quad +C e^{-(\frac{n}{2}-M)t}\int_0^{t}e^{(\frac{n}{2}-M)b} \|f(\cdot,b)\|_{W^{[n/2]+1,1}(\mathbb{R}^n)}db, \end{split} \label{pre3.5b} \end{equation} for all $t>0$. Here we have set $M=\sqrt{\frac{n^2}{4}-m^2}$. \end{theorem} Here, $W^{k,p}(\mathbb{R}^n)=\{u\in L^{p}(\mathbb{R}^n): D^{\alpha}u\in L^{p}(\mathbb{R}^n), \; |\alpha|\leq k\}$, denotes a Sobolev space with the norm \begin{gather*} \|u\|_{W^{k,p}(\mathbb{R}^n)} =\Big(\sum_{|\alpha|\leq k}\int_{\mathbb{R}^n}|D^{\alpha}u|^{p}\Big)^{1/p}, \quad(1\leq p<\infty),\\ \|u\|_{W^{k,\infty}(\mathbb{R}^n)} =\sum_{|\alpha|\leq k}\operatorname{ess\,sup}_{\mathbb{R}^n}|D^{\alpha}u|. \end{gather*} \section{Preliminaries} Throughout this article, the positive constants which may change, are denoted by the same letters $C$. We prepare some inequalities for proving Theorem \ref{thm2}. First of all, we introduce the hypergeometric function $F(a,b;c;\zeta)$ and study its property. It is defined by the power series $$ F(a,b;c;\zeta)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}} \frac{\zeta^n}{n!}, \quad |\zeta|<1, $$ where $a,b,c\in\mathbb{C}$ with $c\neq 0, -1,-2,\dots $, and we denote \begin{gather*} (a)_0=1,\\ (a)_{n}=\Gamma(a+n)/\Gamma(a)=a(a+1)\dots (a+n-1), \quad n=1,2,3,\dots . \end{gather*} Here $\Gamma$ is the gamma function (see, e.g. \cite{Bat53}). We remark that there exists a constant $C>0$ such that \begin{equation} | F(a,b;c; \zeta)|\leq C \label{pre3.1} \end{equation} for all $\zeta\in[0,1]$ if $\mbox{Re}(c-b-a)>0$ for $a,b,c\in \mathbb{C}$ with $c\neq 0,-1,-2,..$. (see e.g. \cite{Yag009} and references therein). \section{Fundamental solutions of the linear Klein-Gordon equation} We separate the initial value problem \eqref{eq03.2} into two parts. First, we consider the Klein-Gordon equation without source term: \begin{equation} \begin{gathered} \partial^{2}_{t}\Phi+n\Phi_{t}-e^{-2t}\Delta\Phi+m^{2}\Phi=0, \quad (x,t)\in \mathbb{R}^n\times\mathbb{R},\\ \Phi(x,0)=\varphi_0(x) , \quad \partial_{t}\Phi(x,0)=\varphi_{1}(x), \quad x\in\mathbb{R}^n, \\ \end{gathered}\label{e1} \end{equation} where $\Phi(x,0)=\varphi_0,\Phi(x,0)=\varphi_{1}\in C_0(\mathbb{R}^n)$. Next, we consider the Klein-Gordon equation with source term, \begin{equation} \begin{gathered} \partial^{2}_{t}\Phi+n\Phi_{t}-e^{-2t}\Delta\Phi+m^{2}\Phi=f(x,t), \quad (x,t)\in \mathbb{R}^n\times\mathbb{R},\\ \Phi(x,0)=0, \quad \partial_{t}\Phi(x,0)=0, \\ \end{gathered} \label{e2} \end{equation} where $f\in C^{\infty}(\mathbb{R}^{n+1})$. For $(x_0,t_0)\in \mathbb{R}^{n+1}$, the forward and backward light cones are defined as \begin{gather*} D_{+}(x_0,t_0):=\left\{(x,t)\in\mathbb{R}^{n+1}: t\geq t_0,\; |x-x_0|\leq e^{-t_0}-e^{-t} \right\},\\ D_{-}(x_0,t_0):=\left\{(x,t)\in\mathbb{R}^{n+1}: t\leq t_0,\; |x-x_0|\leq e^{-t}-e^{-t_0} \right\}. \end{gather*} The function introduced by Yagdjian \cite{Yag009}, \cite{Yag12} is \begin{align*} E(x,t;x_0,t_0;M) &:= (4e^{-t_0-t})^{-M}\left((e^{-t}+e^{-t_0})^2-|x-x_0|^2 \right)^{-\frac{1}{2}+M}\\ &\quad\times F\left(\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(e^{-t_0} -e^{-t})^2-|x-x_0|^2}{(e^{-t_0}+e^{-t})^2-|x-x_0|^2}\right), \end{align*} for $(x,t)\in D_{+}(x_0,t_0)\cup D_{-}(x_0,t_0)$, where $M=\sqrt{\frac{n^2}{4}-m^2}$ and $(x-x_0)^2=(x-x_0).(x-x_0)$ for $x,x_0\in \mathbb{R}^n$. The kernels $K_0(z,t;M)$ and $K_{1}(z,t;M)$ are given by Yagdjian \cite{Yag009}, \cite{Yag12} as follows \begin{align*} &K_0(z,t;M)\\ :&=-\Big[\frac{\partial}{\partial b}E(z,t;0,b;M)\Big]_{b=0}\\ &=(4e^{-t})^{-M}\left((1+e^{-t})^2-z^2\right)^{M-\frac{1}{2}} \left((1-e^{-t})^2-z^2\right)^{-1} \Big[\big(e^{-t}-1 \\ &\quad +M(e^{-2t}-1-z^{2})\big) F\Big(\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(1-e^{-t})^2-z^2} {(1+e^{-t})^2-z^2}\Big)\\ &\quad + (1-e^{-2t}+z^{2})\big(\frac{1}{2}+M\big) F\Big(-\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(1-e^{-t})^2-z^2}{(1+e^{-t})^2-z^2}\Big)\Big] \end{align*} and \begin{align*} &K_{1}(z,t;M)\\ :&=E(z,t;0,0;M)\\ &=(4e^{-t})^{-M}\left((1+e^{-t})^2-z^2\right)^{-\frac{1}{2}+M} F\Big(\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(1-e^{-t})^2-z^2}{(1+e^{-t})^2-z^2}\Big), \end{align*} where $0\leq z \leq 1-e^{-t}$. The solution $\Phi=\Phi(x,t)$ of the initial value problem \begin{equation} \Phi_{tt}+n\Phi_{t}-e^{-2t}\Delta\Phi+m^{2}\Phi=0, \quad \Phi(x,0)=\varphi_0(x), \hspace{4mm} \Phi_{t}(x,0)=\varphi_{1}(x), \label{eq03.7} \end{equation} with $\varphi_0$, $\varphi_{1}\in C_0^{\infty}(\mathbb{R}^n)$ is given by Yagdjian-Galstian \cite{Yag009, Yag09} as follows \begin{equation} \begin{split} &\Phi(x,t) \\ &=e^{-\frac{n-1}{2}t}v_{\varphi_0}(x,\phi(t))\\ &\quad +e^{-nt/2}\int_0^1v_{\varphi_0}(x,\phi(t)s) \left(2K_0(\phi(t)s,t;M)+nK_{1}(\phi(t)s,t;M)\right)\phi(t)ds\\ &\quad+e^{-nt/2}\int_0^1v_{\varphi_{1}}(x,\phi(t)s)(2K_{1} (\phi(t)s,t;M))\phi(t)ds, \end{split} \label{eq03.8} \end{equation} where $\phi(t):=1-e^{-t}$ with $t>0$. Here, for $\varphi \in C_0^{\infty}(\mathbb{R}^n)$, $v_{\varphi}(x,t)$ denotes the solution of \begin{equation} v_{tt}-\Delta v=0, \quad v(x,0)=\varphi(x), \quad v_{t}(x,0)=0, \quad (x,t)\in\mathbb{R}^n\times(0,\infty). \label{eq03.9} \end{equation} Moreover, the solution $\Phi=\Phi(x,t)$ of the initial value problem \begin{equation} \Phi_{tt}+n\Phi_{t}-e^{-2t}\Delta\Phi+m^{2}\Phi=f, \quad \Phi(x,0)=0, \quad \Phi_{t}(x,0)=0, \label{eq03.4} \end{equation} with $f\in C^{\infty}(\mathbb{R}^{n+1})$ is given by Yagdjian-Galstian \cite{Yag009, Yag09} as follows \begin{equation} \Phi(x,t)= 2e^{-nt/2}\int_0^{t}db\int_0^{e^{-b}-e^{-t}}dr e^{\frac{n}{2}b}v(x,r;b)E(r,t;0,b;M), \label{eq03.5} \end{equation} where $v(x,t;b)$ is the solution to the following initial value problem for the wave equation \begin{equation} v_{tt}-\Delta v=0, \hspace{4mm } v(x,0;b)=f(x,b), \quad v_{t}(x,0;b)=0, \quad (x,t)\in\mathbb{R}^n\times(0,\infty), \label{eq03.6} \end{equation} where $b>0$. \section{Proof of Theorem \ref{thm2}} We derive $L^\infty$ estimates for the linear Klein-Gordon equation in de Sitter spacetime. We apply the following two lemmas to prove the theorem. \begin{lemma} \label{lem6.1} Let $M>1/2$ and $\phi(t)=1-e^{-t}$. Then \begin{equation} \int_0^1 (1+ \phi(t)s)^{-\frac{n-1}{2}}|K_1(\phi(t)s,t;M)|\phi(t)ds\leq C_{M}e^{Mt} \label{eq03.11} \end{equation} for all $t>0$. \end{lemma} \begin{proof} Changing the variable by $1+\phi(t)s=r$ and using the definition of the kernel $K_1$, we obtain \begin{align*} &\int_0^1 (1+ \phi(t)s)^{-\frac{n-1}{2}}|K_1(\phi(t)s,t;M)|\phi(t)ds\\ &= 4^{-M}e^{Mt} \int_1^{2-e^{-t}} r^{-\frac{n-1}{2}}((1+e^{-t})^2 -(r-1)^2)^{-\frac{1}{2}+M}\\ &\quad\times| F\left(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(1-e^{-t})^2 -(r-1)^2}{(1+e^{-t})^2-(r-1)^2} \right) |dr\\ &\leq C e^{-Mt} \int_0^{e^{t}-1} ((e^{t}+1)^2-y^2)^{-\frac{1}{2}+M} \Big|F\Big(\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(e^{t}-1)^2-y^2}{(e^{t}+1)^2-y^2} \Big)\Big| dy, \end{align*} where we have changed the variable by $e^t(r-1)=y$ in the last inequality. Since $M>1/2$, by \eqref{pre3.1} the hypergeometric function in the last inequality is bounded, and hence \begin{align*} \int_0^1 (1+ \phi(t)s)^{-\frac{n-1}{2}}|K_1(\phi(t)s,t;M)|\phi(t)ds &\leq C e^{-Mt} \int_0^{e^{t}-1} ((e^{t}+1)^2-y^2)^{-\frac{1}{2}+M}dy\\ &\leq C_{M}e^{-Mt} (e^{t}+1)^{2M-1}(e^{t}-1), \end{align*} which leads to \eqref{eq03.11}. \end{proof} \begin{lemma} \label{lem7.1} Let $M>1/2$ and $\phi(t)=1-e^{-t}$. Then \begin{equation} \int_0^1 (1+ \phi(t)s)^{-\frac{n-1}{2}}|K_0(\phi(t)s,t;M)|\phi(t)ds\leq C_{M}e^{Mt} \label{eq03.12} \end{equation} for all $t>0$. \end{lemma} \begin{proof} Similarly to the proof of Lemma \ref{lem6.1}, we obtain \begin{align*} &\int_0^1 (1+ \phi(t)s)^{-\frac{n-1}{2}}|K_0(\phi(t)s,t;M)|\phi(t)ds\\ &\leq C_{M}e^{-Mt} \int_0^{e^{t}-1}((e^{t}+1)^2-y^2)^{M-\frac{1}{2}} \left((e^{t}-1)^2-y^2\right)^{-1}\\ &\quad\times \Big|\Big[(e^{t}-e^{2t}+M(1-e^{2t}-y^{2})) F\Big(\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(e^{t}-1)^2-y^2}{(e^{t}+1)^2-y^2}\Big)\\ & \quad+ (e^{2t}-1+y^{2})(\frac{1}{2}+M) F\Big(-\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(e^{t}-1)^2-y^2}{(e^{t}+1)^2-y^2}\Big)\Big]\Big|dy. \end{align*} From \cite{Yag12}, we have \begin{equation} \begin{split} & \int_0^{z-1}((z+1)^2-y^2)^{M-\frac{1}{2}}\left((z-1)^2-y^2\right)^{-1}\\ &\quad\times \Big|\Big[(z-z^{2}+M(1-z^{2}-y^{2}))F(\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(z-1)^2-y^2}{(z+1)^2-y^2})\\ &\quad+ (z^{2}-1+y^{2})(\frac{1}{2}+M)F(-\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(z-1)^2-y^2}{(z+1)^2-y^2})\Big]\Big|dy \\ &\leq C_{M}(z+1)^{2M} \end{split} \label{eq03.13} \end{equation} for all $z \in [1,\infty)$. Hence \eqref{eq03.13} leads to \eqref{eq03.12}. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] First we consider the solution of the initial value problem \eqref{e1}. In the case of $\varphi_{1}=0$, from \eqref{eq03.8}, we have \begin{align*} \Phi(x,t)&=e^{-\frac{n-1}{2}t}v_{\varphi_0} (x,\phi(t)) +e^{-nt/2}\int_0^1v_{\varphi_0}(x,\phi(t)s) (2K_0(\phi(t)s,t;M) \\ &\quad +nK_{1}(\phi(t)s,t;M)) \phi(t)ds. \end{align*} Then, we obtain \begin{equation} \begin{split} &\| \Phi(\cdot,t)\|_{L^{\infty}(\mathbb{R}^n)} \\ &\leq e^{-\frac{n-1}{2}t}\|v_{\varphi_0} (\cdot,\phi(t))\|_{L^{\infty} (\mathbb{R}^n)}+e^{-nt/2}\int_0^1\|v_{\varphi_0} (\cdot,\phi(t)s)\|_{L^{\infty}(\mathbb{R}^n)}\\ &\quad\times |(2K_0(\phi(t)s,t;M)+nK_{1}(\phi(t)s,t;M))| \phi(t)ds. \end{split}\label{eq03.18} \end{equation} As is well known, the solution $v(x,t)$ of the initial value problem \eqref{eq03.9} satisfies \begin{equation} \| v (\cdot,t)\|_{L^{\infty}(\mathbb{R}^n)} \leq C(1+t)^{-\frac{n-1}{2}}\|\varphi\|_{W^{[n/2]+1,1}(\mathbb{R}^n)} \label{eq03.18a} \end{equation} for $t\geq0$, if $n\geq2$ (see e.g. \cite{Wahl71}). For all $t\geq0$, we have \begin{align*} e^{-\frac{n-1}{2}t}\| v_{\varphi_0} (\cdot,\phi(t)\|_{L^{\infty}(\mathbb{R}^n)} &\leq Ce^{-\frac{n-1}{2}t}(1+\phi(t))^{-\frac{n-1}{2}}\|\varphi_0\|_{W^{[n/2]+1,1} (\mathbb{R}^n)}\\ &\leq C e^{-\frac{n-1}{2}t}\|\varphi_0\|_{W^{[n/2]+1,1}(\mathbb{R}^n)}. \end{align*} Hence, we obtain \begin{equation} e^{-\frac{n-1}{2}t}\| v_{\varphi_0} (\cdot,\phi(t)\|_{L^{\infty}(\mathbb{R}^n)} \leq C e^{-\frac{n-1}{2}t}\|\varphi_0\|_{W^{[n/2]+1,1}(\mathbb{R}^n)}. \label{eq03.19} \end{equation} On the other hand, we obtain \begin{equation} \begin{split} &e^{-nt/2}\int_0^1\|v_{\varphi_0}(\cdot,\phi(t)s) \|_{L^{\infty}(\mathbb{R}^n)} |(2K_0(\phi(t)s,t;M)+nK_{1}(\phi(t)s,t;M))| \phi(t)ds\\ &\leq C \|\varphi_0\|_{W^{[n/2]+1,1}(\mathbb{R}^n)}e^{-nt/2} \int_0^1(1+\phi(t)s)^{-\frac{n-1}{2}}|(2K_0(\phi(t)s,t;M)\\ &\quad +nK_{1}(\phi(t)s,t;M))| \phi(t)ds. \end{split} \label{eq03.20} \end{equation} From Lemma \ref{lem6.1} and Lemma \ref{lem7.1}, we have \begin{gather} e^{-nt/2}\int_0^1(1+\phi(t)s)^{-\frac{n-1}{2}}|2K_0(\phi(t)s,t;M)| \phi(t)ds \leq Ce^{(M-\frac{n}{2})t}, \label{eq03.21} \\ e^{-nt/2}\int_0^1(1+\phi(t)s)^{-\frac{n-1}{2}}|nK_{1}(\phi(t)s,t;M)| \phi(t)ds \leq Ce^{(M-\frac{n}{2})t}. \label{eq03.22} \end{gather} Hence, from \eqref{eq03.19}, \eqref{eq03.21} and \eqref{eq03.22} we obtain \begin{equation} \|\Phi(\cdot,t)\|_{L^{\infty}(\mathbb{R}^n)} \leq C e^{(M-\frac{n}{2})t}\|\varphi_0\|_{W^{[n/2]+1,1}(\mathbb{R}^n)} \label{es1} \end{equation} when $\varphi_{1}=0$. For the case $\varphi_0=0$, we have \begin{equation} \|\Phi(\cdot,t)\|_{L^{\infty}(\mathbb{R}^n)} \leq C e^{(M-\frac{n}{2})t}\|\varphi_1\|_{W^{[n/2]+1,1}(\mathbb{R}^n)} \label{es2} \end{equation} in a similar way. Next, we consider the solution of the initial value problem \eqref{e2}. From \eqref{eq03.5} and the definition of $E(x,t;x_0,t_0;M)$ we have \begin{align*} &\Phi(x,t) \\ &= 2e^{-nt/2}\int_0^{t}db \int_0^{e^{-b}-e^{-t}} dr e^{\frac{n}{2}b}v(x,r;b)4^{-M}e^{M(b+t)}((e^{-t}+e^{-b})^2-r^2)^{-\frac{1}{2}+M} \\ &\quad\times F(\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(e^{-b}-e^{-t})^2-r^2}{(e^{-b}+e^{-t})^2-r^2}) dr, \end{align*} where $v$ is the solution of \eqref{eq03.6}. From \eqref{eq03.18a}, we obtain \begin{equation*} \| v (\cdot,r;b)\|_{L^{\infty}(\mathbb{R}^n)} \leq C(1+r)^{-\frac{n-1}{2}}\|f(\cdot,b)\|_{W^{[n/2]+1,1}(\mathbb{R}^n)} \end{equation*} for all $r>0$. Hence, \begin{align*} \|\Phi(\cdot,t)\|_{L^{\infty}(\mathbb{R}^n)} &\leq C_{M}e^{-nt/2}e^{Mt}\int_0^{t} e^{\frac{n}{2}b}e^{Mb} \|f(\cdot,b)\|_{W^{[n/2]+1,1}(\mathbb{R}^n)}db\\ &\quad\times \int_0^{e^{-b}-e^{-t}}(1+r)^{-\frac{n-1}{2}} ((e^{-t}+e^{-b})^2-r^2)^{-\frac{1}{2}+M}\\ &\quad\times |F(\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(e^{-b}-e^{-t})^2-r^2}{(e^{-b}+e^{-t})^2-r^2}) |dr\\ &\leq C_{M}e^{-nt/2}e^{Mt}\int_0^{t} e^{\frac{n}{2}b} e^{Mb}\|f(\cdot,b)\|_{W^{[n/2]+1,1}(\mathbb{R}^n)}db\\ &\quad\times \int_0^{e^{-b}-e^{-t}} ((e^{-t}+e^{-b})^2-r^2)^{-\frac{1}{2}+M}\\ &\quad\times |F(\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(e^{-b}-e^{-t})^2-r^2}{(e^{-b}+e^{-t})^2-r^2}) |dr. \end{align*} If we change the variable by $r=e^{-t}y$, then we obtain \begin{equation} \begin{split} \|\Phi(\cdot,t)\|_{L^{\infty}(\mathbb{R}^n)} &\leq C_{M}e^{-nt/2}e^{-Mt}\int_0^{t} e^{\frac{n}{2}b}e^{Mb} \|f(\cdot,b)\|_{W^{[n/2]+1,1}(\mathbb{R}^n)}db\\ &~\quad\times \int_0^{e^{t-b}-1} \left((e^{t-b}+1)^2-y^2\right)^{-\frac{1}{2}+M}\\ &~\quad\times |F(\frac{1}{2}-M,\frac{1}{2}-M;1; \frac{(e^{t-b}-1)^2-y^2}{(e^{t-b}+1)^2-y^2}) |dy. \end{split} \label{eq03.25} \end{equation} Since $M>1/2$, by \eqref{pre3.1}, we have the following estimate for the second integral of \eqref{eq03.25}, \begin{align*} &\int_0^{e^{t-b}-1} \left((e^{t-b}+1)^2-y^2\right)^{-\frac{1}{2}+M} \Big|F(\frac{1}{2}-M,\frac{1}{2}-M;1;\frac{(e^{t-b}-1)^2-y^2}{(e^{t-b}+1)^2-y^2}) \Big|dy\\ &\leq C_{M} \int_0^{e^{t-b}-1} \left((e^{t-b}+1)^2-y^2\right)^{-\frac{1}{2}+M}dy\\ &\leq C_{M} \left(e^{t-b}+1\right)^{2M-1}(e^{t-b}-1)\\ &\leq C_{M} \left(e^{t-b}+1\right)^{2M}\\ &\leq C_{M} e^{2M(t-b)}, \end{align*} for $b