\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 235, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/235\hfil 
Upper semicontinuity of attractors]
{Upper semicontinuity of attractors and continuity of
equilibrium sets for parabolic problems with degenerate
$p$-Laplacian}

\author[S. M.  Bruschi,  C. B. Gentile, M. R. T. Primo \hfil EJDE-2017/235\hfilneg]
{Simone  M. Bruschi, Cl\'audia B. Gentile, Marcos R. T. Primo}

\address{Simone  M. Bruschi (corresponding author)\newline
Departamento de Matem\'atica,
Universidade de Bras\'{\i}lia, Brazil}
\email{sbruschi@unb.br}

\address{Cl\'audia B. Gentile \newline
Departamento de Matem\'atica, Universidade Federal de S{\~a}o
Carlos, Brazil}
\email{gentile@dm.ufscar.br}

\address{Marcos R. T. Primo \newline
Departamento de Matem\'atica,
Universidade Estadual de Maring\'a, Brazil}
\email{mrtprimo@uem.br}


\dedicatory{Communicated by Claudianor O. Alves}

\thanks{Submitted November 9, 2016. Published September 29, 2017.}
\subjclass[2010]{35K92, 35K40}
\keywords{p-Laplacian; continuity properties;
equilibrium sets; global attractors}

\begin{abstract}
 In this work we obtain some continuity properties on the parameter
 $q$ at $p=q$ for the Takeuchi-Yamada problem which is a degenerate
 $p$-laplacian version of the  Chafee-Infante problem.
 We prove the continuity of the flows and the equilibrium sets, and
 the upper semicontinuity of the global attractors.

\end{abstract}

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


 \section{Introduction}  \label{s1}

The inspiration for this study arose from the description by Chafee and Infante of
the bifurcation scheme and stability properties of
 the equilibrium solutions for the semilinear problem
 \begin{equation} \label{eqCI}
\begin{gathered}
 u_{t}=\lambda u_{xx}+ u-u^3 ,\quad (x,t)\in (0,1)\times (0,+\infty)\\
 u(0,t)=u(1,t)=0,\quad  0\leq t<+\infty  \\
 u(x,0)=u_{0}(x),\quad x\in (0,1),
\end{gathered}
\end{equation}
where $\lambda$ is a positive parameter and the initial data are
sufficiently smooth \cite{ci1}. Using a time-map method that adjusts
the initial speed of a Cauchy problem to ensure that the desired
boundary conditions are satisfied, Chafee and Infante proved that for fixed
values of $\lambda>0$ there are a finite number of stationary
solutions to the problem \eqref{eqCI}, which bifurcate in pairs from the null
solution at each point of a decreasing sequence $ \{\lambda_n \}$,
each new pair being symmetrical with respect to the abscissa axis
and containing one more zero than the prior pair in such way
that when $ \lambda_n \to 0$, the number of stationary solutions
of \eqref{eqCI} tends to infinity.

Since this problem belongs to a class of problems in which
trajectories asymptotically tend to the equilibrium points
 when $t \to \infty$, and also when $ t \to - \infty$ in the case of
complete trajectories, detailed knowledge of the stationary
solutions is useful in understanding the
attractor structure, which for gradient systems, is the set of all
equilibrium solutions
with their connecting trajectories, \cite{Hale}.

Subsequently Takeuchi and Yamada published a detailed
description of the bifurcation diagram for equilibria of the
quasilinear problem
\begin{equation} \label{eqTY}
\begin{gathered}
 u_{t}=\lambda
(|u_{x}|^{p-2}u_{x})_{x}+|u|^{q-2}u(1-|u|^{r}),\quad (x,t)\in(0,1)\times
(0,+\infty)\\
 u(0,t)=u(1,t)=0,\quad  0 \leq t<+\infty \\
 u(x,0)=u_{0}(x),\quad  x\in (0,1),
 \end{gathered}
 \end{equation}
 where  $p > 2$, $q\geq 2$, $r>0$ and  $\lambda>0$, taking into
consideration the relations between $p$ and $q$, \cite{TY}. Denoting by
$E_\lambda = E_{\lambda}(p,q)$ the set of equilibria of problem \eqref{eqTY},
which describes a gradient system, we find that:
\begin{itemize}
\item If $ p> q$, $E_{\lambda}=\{0\} \cup_{n=0}^\infty \pm E_{\lambda}^n$,
 where
$E_{\lambda}^n=E_{\lambda}^n(p,q)$ is the set of stationary
solutions $ \phi_n$ with $n$ zeros in $(0,1)$ and the sign $\pm$
indicates the sign of $(\phi_n)_x(0)$, any equilibrium
$\phi \in +E_\lambda = E_\lambda$ has positive initial condition
$\phi_x(0)$ and $-E_\lambda$ is the set of the opposites.
 In this case, there is a relevant sequence $\{ \lambda_{n} \}$ such that,
if $n\geq 1$ and $\lambda\geq \lambda_{n}$, then
$E_{\lambda}^n$ is a single set. $E_{\lambda}^0$ is always a single set.

\item if $ p = q$, $E_{\lambda}$ changes depending on where the parameter
$\lambda$ is located with respect to two sequences, $\{\lambda_n^*\}$ and
$\{\lambda_{n}\}$. The first sequence sets the maximum number of
zeros allowed to an equilibrium. The second sequence, as in the
prior case, states for each $n>0$, if $E_{\lambda}^n$ is a single
or a continuum set. If $\lambda \geq
\lambda_0^*$, then $E_\lambda=\{0\}$. If $\lambda_{M+1}^*\leq \lambda <
\lambda_{M}^*$, then $E_{\lambda}=\{0\} \cup_{n=0}^M \pm
E_{\lambda}^n$. $E_{\lambda}^0$ is always a single set.

\item if $ p < q$, $E_{\lambda}$ also changes according the position of $\lambda$ with respect to two sequences $\{\lambda_n^*(p,q)\}$
and $\{\lambda_{n}\}$. Again, if $\lambda \geq
\lambda_0^*$, $E_\lambda=\{0\}$. If $\lambda_{M+1}^* < \lambda
\leq\lambda_{M}^*$, then $E_{\lambda}=\{0\} \cup_{n=0}^M
\left(\pm \{F_\lambda^n\} \cup \pm E_{\lambda}^n\right)$, where $
F_{\lambda}^n = \{\psi_n\}$ and, if $\lambda = \lambda_n^*(p,q), $
$E_{\lambda}^n= \emptyset$, $E_{\lambda}^n = \{\phi_n\}$ if
$\lambda_n  \leq \lambda < \lambda^*_{n}(p,q)$. Here $\psi_n$ and
$\phi_n$ are equilibria with $n$ zeros in $(0,1)$,
$|\psi_n(x)|<|\phi_n(x)|$ for all $x \in (0,1)$ except for zero
points of $\psi_n$ and $\phi_n$.
\end{itemize}

In any case,  if $n\geq 1$ and $\lambda <
\lambda_{n}$, then $E_{\lambda}^n$ is diffeomorphic to $[0,1]^n$.

When $  p = q $ there are notable
similarities between problems \eqref{eqCI} and \eqref{eqTY},
particularly in regarding the stability properties of the equilibria.
The trivial solution in each case
is asymptotically stable for large values
of the diffusion parameter $\lambda$ and becomes unstable when
the first pair of nontrivial equilibria bifurcates from null solution. These,
in turn, remain asymptotically stable, while
other stationary solutions are unstable.

The principal difference between problems \eqref{eqCI} and \eqref{eqTY}
lies in the following. In the former, semilinear problem, although the
number of elements in the equilibrium set tends to
infinity when the diffusion goes to zero, it remains discrete,
because the equilibria bifurcate from the trivial solution in pairs.
In the later, quasilinear problem, however, the equilibrium
set can contain continuum components if the diffusion coefficient is
insufficiently large since the stationary solutions can reach their
extremes at $ 1 $ and $ -1 $, which are zeros on the right side
of the equation. Thus,
stationary solutions can form flat cores when attaining these values
and, although the sum of the lengths of all flat cores must be
constant, it can be freely distributed among them. Accordingly there
is a continuum of equilibrium solutions with the
same number of zeros. This situation does not occur in the
semilinear problem, as the ``$x$-time'' required for
equilibria to achieve their extremes in $ 1 $ or $  -1$ is infinite.

Nevertheless, in regard to problem \eqref{eqTY}, for each fixed
value of $\lambda$, there are finite connected components of $
E_\lambda(p,p)$, each composed of solutions containing
the same number of zeros and bifurcating from the
trivial solution. The attractor is the finite union of the
unstable set of the connected components of $ E_\lambda $. In this case,
the attractor is the union of $E_\lambda$
with the complete trajectories joining their connected parties,
\cite{BGP1, TY}. It was proved in \cite{BGP1} that
the problem \eqref{eqCI} can be attained as a limit of
$\eqref{eqTY}$ when $ p \downarrow 2$ and, for each fixed value of
$\lambda$, $ E_\lambda$ behaves continuously with relation to $ p $,
becoming discrete when $p$ is located in some positive distance of
$2$.

Only the case $p=q$ is considered in \cite{BGP1}, as in other
cases, the complex configuration of the equilibrium sets diverges
significantly. The purpose here is
to prove the continuity of the equilibrium set of problem
\eqref{eqTY} with respect to $q$ when $q \to p$. To this end, the
following questions must be considered.
 For $\lambda$ fixed, when $q
\uparrow p$, even when $q$ is close to $p$, given $n>0$ there
are at least two solutions in $E_\lambda(p,q)$ having $n$ zeros in
$(0,1)$. When $p=q$, however, there exists a maximum value $M$ such
that any solution in $E_\lambda(p,p)$ have a number of zeros less
than or equal to $M$. The value of $M$ is determined as a function of
the position of $\lambda$ with respect to the points of the sequence
$\{\lambda_n\}$.  When $q\downarrow p$, the
number of zeros in $(0,1)$ of equilibria is bounded if $p = q$ or $p<q$
 but the sequences that
determine the maximum value of zeros for a stationary solution are
distinct, being $\{\lambda_{n}^*\}$ in the first case and
$\{\lambda_{n}^*(p,q)\}$ in the latter. Further, given
$n$, $E_\lambda(p,q)$ can contain two entirely distinct equilibria
with $n$ zeros, $\pm \psi_\lambda^n $, that do not appear in the
configuration of $E_\lambda(p,p)$. As will be shown in Section 4 these
unanticipated equilibria, i.e., the stationary
solutions which are not supposed to exist in case $p=q$, converges to
the trivial solution when $q\to p$ despite the value of $\lambda$.

The lower continuity of attractors is not an easy problem and there is
no much we know about. In the specific case when $p=q$ and the diffusion
parameter $\lambda$ is such that  $\lambda_1^*\leq\lambda \leq \lambda_0^*$,
then we can say that the attractors $\mathcal{A}_p$ of the problem \eqref{eqTY}
are lower semicontinuous at  $p=2$. This follows from the fact that, in this
case, there exists only two complete trajectories for the Chafee-Infante
problem (case $p=2$), (see \cite{Henry}, p126), and then we can combine
the continuity of the semigroups on $p$ with the continuity of the equilibrium
set to verify that each point on those complete trajectories can be reached
as a limit of points on complete trajectories inside the attractors
$\mathcal{A}_p, p>2$.


Regarding the diffusion parameter $\lambda$, once
$\lambda_n$ depends on $(p,q)$, the question arises if the connected
components of $E_\lambda(p,q)$ and $E_\lambda(p,p)$ have similar
cardinality properties when $q$ is close to $p$, whether when $p\neq q$
the equilibrium components of $E_\lambda(p,q)$ that have natural correspondence with
some component $E_{\lambda}^n(p,p)$ when $p=q$ are discrete or continuum
according to the respective cardinality of $E_{\lambda}^n(p,p)$.
The answer to this query is no, as detailed in Section 4. There is but one
situation described in Case 3, Section 4,
in which this fact must be addressed, but the continuity of $E_\lambda(p,q)$ is not
affected. Similarly, despite the fact that sequence
$\lambda_{n}^*(p,q)$ depends on $q$, the same maximum
value $M$ for the amount of zeros allowed to an equilibrium in
$E_{\lambda}(p,p)$ and $E_{\lambda}(p,q)$, $p<q$, is found consistently.


Based on the preceding, the continuity of the sets $E_\lambda(p,q)$ is studied
via the continuity properties of equilibria. In Section 4, the ordinary
differential equation, which describes the
stationary solutions of \eqref{eqTY}, is reviewed and its dependence on
initial conditions and parameters is analyzed. Section 2 presents the required
uniform estimates and locates the dynamics of problem \eqref{eqTY} in
$W_0^{p}(0,1)$. Subsequently it is proven that the family
 of attractors $\mathcal{A}_{pq}$ is upper semicontinuous with respect to
 $(p,q)$ in $C[0,1]$ topology. Additionally, if $p$ remains fixed,
$\mathcal{A}_{pq}$ is upper
semicontinuous with respect to $q$ in $W_0^{1,p}(0,1)$.

\section{Uniform estimates and asymptotic properties}

 The asymptotic behavior of solutions of problem \eqref{eqTY}
is a well known issue, and it is not difficult to prove that \eqref{eqTY}
defines a semigroup
which has a global attractor when set in $L^2(0,1)$ or even in
$W_0^{1,p}(0,1)$, since it enjoys good properties of
compactness and dissipativity. In this section we list all the
necessary estimates to guarantee the existence of these attractors.
Most of the results below is shown in \cite{BGP1}, so we will just
explicit the uniformity of the upper bounds with respect to
parameters $p$ and $q$ when $(p,q)$ is in a bounded subset $R$ of
$(2,\infty)\times [2,\infty)$. We will denote by $u_{pq}$ a solution
of \eqref{eqTY}.

We first need to obtain estimates for the $L^2(0,1)$ norm of
solutions. This is done exactly as in \cite[Lemma 2.1]{BGP1},
whose statement is repeated here
  properly fitted to the context of this work.

 \begin{lemma}\label{estunifh}
Let $u_{pq}$ be a solution of \eqref{eqTY} with
$u_{pq}(0)=u_{0}\in L^2(0,1)$. Given $T_0>0$ there exists
$\widetilde{K}_{1}>0$ such that $\|u_{pq}(t)\|_2 < \widetilde{K}_1$
for  $t\geq T_0$ and $(p,q) \in R$. Furthermore, given $B\subset
L^2(0,1)$, $B$ bounded, there exists ${K}_{1}>0$  such that
$\|u_{pq}(t)\|_2 < K_1$ for  $t\geq 0$, $(p,q) \in R$ and $u_{0} \in
B$. The positive constants $\widetilde{K}_1, K_1$ are independent of
$(p,q) \in R$, $r>0$ and $\lambda >0$.
 \end{lemma}

\begin{remark} \label{rmk2.2} \rm
We note that the constant $\widetilde{K}_1$ gives us a $L^2(0,1)$
estimate after some time has elapsed from the origin,
and it is uniform on $(p, q) \in R$, completely independent of the
initial data and uniform on bounded sets with respect to the
parameter $r$. The constant $K_1$,
 which estimates solutions since the origin, carries, as expected,
a dependence on the initial data which is uniform however on bounded
subsets of $L^2(0,1)$.
\end{remark}

To establish the estimates on $W_0^{1,p}(0,1)$ we introduce
the following notation:
$\varphi^1_{pq},\varphi^2_q:L^2(0,1)\to {\mathbb R}$ given by
$$
\varphi^1_{pq} (u) \doteq
\begin{cases}
      \frac{\lambda}{p}\int_0^1 |u_x(x)|^p dx +\frac{1}{q+r}\int_0^1
|u(x)|^{q+r} dx,&  u \in W^{1,p}_0(0,1)\\
      +\infty, &\text{otherwise},
      \end{cases}
 $$
and
$$
\varphi^2_{q} (u) \doteq\
\begin{cases}
       \frac{1}{q}\int_0^1 |u(x)|^{q} dx, &  u \in L^q(0,1)\\
      +\infty, &\text{otherwise}
\end{cases}
$$

It is advantageous to rewrite the equation in \eqref{eqTY} in an
abstract way involving the difference of two subdifferential
operators. Thus the existence of global solutions is easily obtained
as a consequence of \cite{otani} and new estimates can be obtained,
this time in a stronger norm.
\begin{equation}\label{formaabstrata}
  \frac{du}{dt}(t) +
\partial\varphi^1_{pq}(u(t))-\partial\varphi^2_q(u(t)) = 0
\end{equation}
where $\partial\varphi^1_{pq}$ and  $\partial\varphi^2_{q}$ are
subdifferential of $\varphi^1_{pq}$ and $\varphi^2_{q}$
respectively.

\begin{remark} \label{diferencadesubdif} \rm
 Given $c_0$ and $q_M$, $0 < c_0 < 1$, $q_M>2$, there exists $c > 0$
depending only on $r$ and $c_0$ such that
$$
\varphi^2_{q}(u) \leq c_0\varphi^1_{pq}(u) + c
$$
for each $u \in W^{1, p}_0(0,1)$,
$\lambda > 0$ and $2\leq q\leq q_M$. In fact, if $\eta> 0$,
\begin{align*}
\varphi^2_q(u)
&=   \frac{1}{q}\| u
\|^q_{L^q(0,1)} \\
&\leq  \frac{r}{(q + r)(\eta q)^{\frac{q + r}{r}}}
 + q \eta^{\frac{q + r}{q}} \Big(
\frac{\lambda}{p} \| u \|^p_{W^{1, p}_0(0,1)} +
\frac{1}{q + r}\| u \|^{q +
r}_{{L^{q+r}(0,1)}}\Big).
\end{align*}

Let $g(q)= (\frac{c_0}{q})^{\frac{q}{q + r}} =
e^{\frac{q}{q+r}\ln(c_0/q)}$, then  $g'(q) <0$ and  it is
enough to choose $\eta$ such that $0 < \eta <
(\frac{c_0}{q_M})^{\frac{q_M}{ q_M+ r}}$ for
$2\leq q\leq q_M$. Then $q\eta^{\frac{q+r}{q}} \leq c_0$ and
\[
c \doteq \frac{r}{(2 + r)2^{\frac{2+r}{2}}\eta^{\frac{q_M +r}{r}}}\,.
\]
\end{remark}

The following lemmas show the estimates we have in $W_0^{1,p}(0,1)$
norm. Note that, even if the initial data are taken into $L^2(0,1)$,
since the flow is governed by a subdifferential (so it has good
smoothing properties), we can ensure strong estimates in
$W_0^{1,p}(0,1)$ from any positive time elapsed from the origin.

\begin{lemma} \label{lem2.4}
 Given $\delta >0 $ there exists $\tilde{K}_2>0$ such that
 $\|u_{pq}(t)\|_{W_0^{p,q}(0,1)} \leq \tilde{K}_2$ for
$t \geq \delta$ and for all initial data $u_0$
in $L^2(0,1)$
\end{lemma}

\begin{remark} \label{rmk2.5} \rm
The above lemma is a direct consequence of \cite[Lemma 2.1]{TY} and
the first assertion of Lemma \ref{estunifh}. The constant
$\tilde{K}_2$ carries the same dependence of $\tilde{K _1}$, that
means, it is uniform on $ (p, q) \in R$, completely independent of
the initial data and uniform on bounded sets with respect to the
parameter $r$.
\end{remark}

However, if we are interested in estimates since the beginning of
evolution, so we naturally find upper bounds dependent on the
initial data. The demonstration is exactly the same as \cite[Lemma 2.2]{BGP1}.

\begin{lemma} \label{estunifwup}
Let $u_{pq}$ be  a solution of \eqref{eqTY} with
$u_{pq}(0)=u_{0}\in W_0^{1,p}(0,1)$. Given $M>0$ there exists a
positive constant $K_{2}>0$ such that
$\|u_{pq}(t)\|_{_{W_0^{1,p}(0,1)}} < K_2$ for  $t\geq 0$ and
$(p,q)\in R$. Furthermore, the positive constant $K_2$ can be
uniformly chosen for  $(p,q)\in R$, and $\|u_{0}\|_{W_0^{1,p}(0,1)}
\leq M$.
 \end{lemma}

 Finally, from the above lemma we conclude our set of uniform estimates 
of $\{u_{pq}\}$, giving bounds to
the solutions of the problem \eqref{eqTY} in $L^\infty(0,1)$.

\begin{lemma}\label{normaLinfinito} 
Let $u_{pq}$ be a solution of \eqref{eqTY} with
$u_{pq}(0)=u_{0}\in W_0^{1,p}(0,1)$ and 
$\|u_{0}\|_{W_0^{1,p}(0,1)} \leq M$. From Lemma \ref{estunifwup} we obtain
 $$
\|u_{pq}(t)\|_{\infty}\leq{K_{3}(M)},\quad t\geq 0.
$$
 \end{lemma}

\begin{remark} \label{rmk2.8} \rm
If the initial data are in $L^2(0,1)- W_0^{1,p}(0,1)$, for each
$\delta>0$ we find $\tilde{K}_3$ depending on $\delta$ and $p$, with
$$
\|u_{pq}(t)\|_{\infty}\leq{\tilde{K}_{3}(\delta,p)},\quad t\geq
\delta.
$$
 \end{remark}

The existence of the global attractor in $L^2(0,1)$ is a simple
consequence of Lemma \ref{estunifh} and Lemma \ref{estunifwup}, as
it is claimed in \cite[Corollary 2.3]{BGP1}. It is also very
simple to obtain the existence
 of global attractor to the restriction of the semigroup to the space $W_0^{1,p}(0,1)$. In fact, for each $(p,q) \in R$, let us denote by $\{S_{pq}(t)\}$ the semigroup
associated with problem \eqref{eqTY} in $W_0^{1,p}(0,1)$. We prove
below that $\{S_{pq}(t)\}$ is a continuous semigroup of compact
operators. The following result will be necessary.

\begin{lemma}\label{estimaut}
 Let $B \subset W_0^{1,p}(0,1)$ be a bounded set and let $T>0$, $q_M>2$.
There is a constant $K_4$ such
$\| \frac{\partial}{\partial t}u_{pq}(t) \|_{L^2(0,1)} \leq K_4$
for any $p> 2$, $2<q<q_M$, $t\in[0,T]$ and $u_0 \in B$.
\end{lemma}

\begin{proof}
 Multiplying the equation in \eqref{eqTY} by
$\frac{\partial}{\partial t}u_{pq}(t)$ and integrating from $0$ to
$T$ we obtain
\begin{equation}\label{estima_int_de_u_t}
\begin{aligned}
\int_0^T\| \frac{\partial}{\partial t}u_{pq}(s)\|_{L^2(0,1)}^2 ds
 + \varphi^1_{pq}(u_{pq}(T))
&\leq \varphi^2_{q}(u_{pq}(T)) + \varphi^1_{pq}(u_{pq}(0)) \\
&\leq c_0\varphi^1_{pq}(u_{pq}(T)) + \varphi^1_{pq}(u_{pq}(0)) + c
\end{aligned}
\end{equation}
where $c_0 < 1$ and $c$ is the same of Remark
\ref{diferencadesubdif}. On the other hand, if we denote
$f_q(s)\doteq |s|^{q-2} s (1-|s|^r)$ then
\begin{align*}
\frac{1}{2} \frac{d}{dt}\|u_{pq}(t+h) - u_{pq}(t)\|^2_{L^2(0,1)}
&=\langle \frac{\partial}{\partial t}u_{pq}(t+h) -
\frac{\partial}{\partial t}u_{pq}(t), u_{pq}(t+h) - u_{pq}(t)
\rangle\\
&\leq
\langle  f_qu_{pq}(t+h) -  f_q u_{pq}(t), u_{pq}(t+h) - u_{pq}(t) \rangle \\
&\leq  C \|u_{pq}(t+h) - u_{pq}(t)\|^2_{L^2(0,1)},
\end{align*}
where $C\doteq (q_M-1)^{\frac{q_M+r-2}{r}}$. So by Gronwall we obtain
$$
\|u_{pq}(t+h) - u_{pq}(t)\|^2_{L^2(0,1)}
\leq \|u_{pq}(s+h) - u_{pq}(s)\|^2_{L^2(0,1)} e^{2CT}.
$$
Therefore,
$$T \| \frac{\partial}{\partial t}u_{pq}(t)\|^2_{L^2(0,1)} \leq \int_0^T  \|\frac{\partial}{\partial t}u_{pq}(s)\|^2_{L^2(0,1)}ds\:e^{2CT} $$
and from \eqref{estima_int_de_u_t} we conclude that
$$\|\frac{\partial}{\partial t}u_{pq}(t)\|_{L^2(0,1)} \leq K_4.
$$
\end{proof}

\begin{theorem}\label{continuidade}
 For each $t>0$ the mapping $S_{pq}(t):W_0^{1,p}(0,1) \to W_0^{1,p}(0,1)$
is continuous and compact.
\end{theorem}

\begin{proof}
Let $T>0$, $0<t<T$ and $\{u_{0_n}\} \subset W_0^{1,p}(0,1)$ a
sequence converging to $u_0$ in $W_0^{1,p}(0,1)$.
Then $u_{0_n} \to u_0$ in $L^2(0,1)$ and, from \cite[Lemma 2.1]{TY},
$S_{pq}(\cdot) u_{0_n} \to S_{pq}(\cdot)u_0$ in $L^p(0,T; W_0^{1,p}(0,1))$.
Therefore we can conclude that exists a subsequence denoted by
$\{u_n(t)\} \subset \{S_{pq}(t) u_{0_n}\}$ which converges to
$u(t)\doteq S_{pq}(t)u_0$ a.e. in $[0,T]$. Let $A \subset [0,T]$ the
set where $\| u_n(\cdot) - u(\cdot)\|_{W_0^{1,p}(0,1)} \to 0$. Given
an arbitrary $t \in (0,T]$ we claim that $\|
u_n(t)\|_{W_0^{1,p}(0,1)} \to \| u(t)\|_{W_0^{1,p}(0,1)}$. In fact,
for each $\theta \in A$
\begin{align*}
|\varphi^1_{pq} (u_n(t)) -  \varphi^1_{pq} (u(t))|
&\leq |\varphi^1_{pq} (u_n(t)) -  \varphi^1_{pq} (u_n(\theta))|
 +|\varphi^1_{pq} (u_n(\theta)) -  \varphi^1_{pq} (u(\theta))|\\
&\quad  +|\varphi^1_{pq} (u(\theta)) -  \varphi^1_{pq} (u(t))|
\end{align*}
and
\begin{align*}
|\varphi^1_{pq} (u_n(t)) -  \varphi^1_{pq} (u_n(\theta))|
&\leq \int_\theta^t \left|\langle \partial\varphi^1_{pq} (u_n(s)),
\frac{\partial}{\partial s}  u_n(s)
 \rangle\right|ds\\
 &\leq  \frac{3}{2}\int_\theta^t \|\frac{\partial}{\partial s}
u_n(s)\|^2_{L^2(0,1)}ds +
\frac{1}{2}\int_\theta^t \|f_q(u_n(s))\|^2_{L^2(0,1)}ds ,
\end{align*}
where $f_q(s)\doteq |s|^{q-2} s (1-|s|^r)$. We can obtain the same
result changing $u_n$ by $u$ in the above inequality. So, it follows
from Lemma \ref{normaLinfinito} and Lemma \ref{estimaut} that,
 given $\eta>0$, we can choose $\theta \in A$
close to $t$ and $n$ large enough to obtain $|\varphi^1_{pq} (u_n(t)) -
\varphi^1_{pq} (u(t))| \leq \eta$. Therefore we conclude that
$u_n(t)\to u(t)$ in $W_0^{1,p}(0,1)$.

We observe that, in fact, this proof shows that $S_{pq}(t)$ is
continuous from $L^2(0,1)$ to $ W_0^{1,p}(0,1)$.

To prove the second statement, let $B \subset
W_0^{1,p}(0,1)$ a bounded subset. Let us prove that $S_{pq}(t)B$ is
relatively compact in $W_0^{1,p}(0,1)$. As $W_0^{1,p}(0,1)$ is
compactly immersed in $L^2(0,1)$, given any sequence
$\{u_{0_n}\}\subset B$, there is $u_0$ such that $u_{0_n} \to u_0
\in L^2(0,1)$ and so $S_{pq}(t)u_{0_n} \to S_{pq}(t)u_{0}$ in $
W_0^{1,p}(0,1)$, which concludes the proof.
\end{proof}

The existence of a global attractor $\mathcal{A}_{pq}$ for $S_{pq}(t)$
in $ W_0^{1,p}(0,1)$ is a consequence of Lemma \ref{estunifwup},
Theorem \ref{continuidade}, and \cite[Theorem 2.2]{LD}.

\begin{proposition} \label{prop2.11}
 Given $(p,q) \in R$, let $S_{pq}(t):W_0^{1,p}(0,1) \to W_0^{1,p}(0,1)$
the semigroup determined by problem \eqref{eqTY}.
Then $\{S_{pq}(t)\}$ has a global attractor, which is compact and invariant.
\end{proposition}


 \section{Continuity of flows and upper semicontinuity of the attractors}

In this section we  proof that, given $T>0$ and  $(p_0,q_0) \in R$,
the solutions $\{u_{pq}\}$ of \eqref{eqTY} go to the solution
$u_{p_0q_0}$ of \eqref{eqTY} in $C([0,T];L^2(0,1))$ , when $p \to
p_0$ and $q \to q_0$. After that, we will obtain the upper
semicontinuity  of the family of global attractors
 $$\{\mathcal{A}_{pq}\subset W_0^{1,p}(0,1);\
(p,q) \in R \}$$
 of \eqref{eqTY} at $(p_0,q_0)$ in the topologies of $L^2(0,1)$  and
 $C([0,1])$. Furthermore, when $p=p_0$ we will prove the upper
 semicontinuity in  $W_0^{1,p_0}(0,1)$.

First of all we observe that from Section 2, there exists a positive
constant $M$, independent of $t\geq 0$ and $(p,q) \in R$, such that
 $$ \|u_{pq}(t) \|_{W_0^{1,p}(0,1)} \leq M$$
 for all $t\geq 0 $ and $(p,q) \in R$. Following exactly the same steps
in Section 3 of \cite{BGP1} we obtain an adapted version of
Baras'Theorem, \cite{V}, as we state bellow.

\begin{lemma}\label{lema3.1}
 Given $T>0$, the set
\begin{align*}
M^{pq} :=\big\{& u_{pq}\subset W_0^{1,p}(0,1) : (p,q) \in R,\
u_{pq} \text{ is a solution of  \eqref{eqTY} with }  \\
&u_{pq}(0)= u_{0\,pq} \in W_0^{1,p}(0,1),\;
  u_{0\,pq}\to u_0\text{ as } (p,q) \to
(p_0,q_0) \text{ in } L^2(0,1)\\
&\text{and } \|u_{0\,pq}\|_{W_0^{1,p}(0,1)}\leq M, \;
 \forall (p,q) \in R\big\},
 \end{align*}
is relatively compact in $C([0,T];L^2(0,1))$.
\end{lemma}


 \begin{theorem}  \label{teo3.1}
For each $(p,q)\in R$, let $\{u_{pq}\}\subset W_0^{1,p}(0,1)$ be a
solution of
\begin{gather*}
\frac{d}{dt} u_{pq}(t)  - \lambda
(|(u_{pq}(t))_x|^{p-2} (u_{pq}(t))_ x)_x =
|u_{pq}(t)|^{q-2}u_{pq}(t)(1+|u(t)|_{pq}^r), \quad
t>0\\
u_{pq}(0) = u_{0\,pq} \in W_0^{1,p}(0,1).
 \end{gather*}
Suppose that $\|u_{0\,pq}\|_{W_0^{1,p}(0,1)}\leq M$ for every
$(p,q) \in R$ and $u_{0\,pq}\to u_0$ as $ (p,q) \to (p_0,q_0)$ in $
L^2(0,1)$. Then, for each $T>0$, $u_{pq}\to u$ in
$C([0,T];L^2(0,1))$ as $(p,q)\to (p_0,q_0)$, where $u$ is a solution
of
\begin{gather*}
\frac{d}{dt} u(t)  - \lambda (|u_x(t)|^{p_0-2} u_x(t)
)_x =|u(t)|^{q_0-2}u(t)(1+|u(t)|^r),\quad  t>0\\
u(0) = u_0 \in L^{2}(0,1).
 \end{gather*}
\end{theorem}

\begin{proof}
Throughout this proof we denote
  $$
f_q( v (t))=|v(t)|^{q-2}v(t)(1+|v(t)|^r).
$$
 Since
  $$
\|u_{pq}(t) \|_{W_0^{1,p}(0,1)} \leq M
$$
 for all $t>0 $, $(p,q) \in R$ with $M$ independent of $t\geq 0$ and $(p,q) \in R$,
   we obtain that
$\{u_{pq}(t)\}$ is uniformly bounded in $L^\infty (0,1)$ for
$(p,q) \in R$ and $t\in [0,T]$. Furthermore, from Lemma \ref{lema3.1},
$\{u_{pq}\}$ converges in $C([0,T]; L^2(0,1))$ to a function
$u: [0,T] \to L^2(0,1)$, when $p\to p_0$ and $q \to q_0$. Since
$f_q(u_{pq})$ is uniformly integrable in $ L^1([0,T]; L^2(0,1))$ and
\begin{align*}
   \|f_q( u_{pq} (t)) - f_{q_0}(v(t))\|
&\leq   \|f_q( u_{pq} (t)) - f_{q_0}(u_{pq}(t))\|
+ \|f_{q_0}( u_{pq} (t)) - f_{q_0}(u(t))\| \\
&\leq  \bar{K}|q-q_0| + \tilde{K}|u_{pq}(t)-u(t)|,
\end{align*}
we obtain
 $f_q( u_{pq} (t)) \to f_{q_0}(u(t))$ in $L^2(0,1)$ for each $t>0$
when $p \to p_0$ and $q \to q_0$. Now,
 with  the same arguments used in \cite{BGP1}
 we obtain that  $u$ is a weak solution of
 \begin{align*}
\frac{d}{dt} u(t)  - \lambda (|u_x(t)|^{p_0-2} u_x(t)
)_x =|u(t)|^{q_0-2}u(t)(1+|u(t)|^r),\quad t>0\\
u(0) = u_0 \in L^{2}(0,1).
 \end{align*}
 and we obtain the desired result.
\end{proof}

\begin{corollary}\label{coro3.1}
 The family of global attractors
$\{\mathcal{A}_{pq}\subset W_0^{1,p}(0,1)): (p,q) \in R\}$
 of  problem \eqref{eqTY} is upper semicontinuous at $(p_0,q_0)$
in the  $L^2(0,1)$ topology.
 \end{corollary}

 \begin{proof}
The results in Section 2 imply that there exists a bounded set
$\mathcal{B}\subset L^2(0,1)$
such that $\mathcal{A}_{pq}\subset \mathcal{B}$, for every
$(p,q) \in R$.  Since $\mathcal{A}_{p_0,q_0}$  attracts bounded sets of
$L^2(0,1)$, for every $\delta>0$, there is $T_{1}>0$ in such way
that
$$
\sup_{\psi_{pq}\in \mathcal{A}_{pq},\, (p,q) \in R}\operatorname{dist}_{L^2(\Omega)}
(u_{p_0,q_0}(T_{1};\psi_{pq}),\mathcal{A}_{p_0,q_0}) \leq
\frac{\delta}{2},
$$
 where $u_{p_0q_0}(t;\psi_{pq})$ is a solution of
problem \eqref{eqTY} when $p=p_0$ and $q=q_0$ with initial condition
$\psi_{pq}$.

Now, the previous results in this section imply that there exist
$\delta_0>0$ and $\epsilon>0$ such that
$$
\|u_{pq}(t;\psi_{pq})-u_{p_0q_0}(t;\psi_{pq})\|_{L^2(0,1)}
<\frac{\delta}{2},
$$
 for  $|p-p_0|<\delta_0$, $|q-q_0|<\epsilon$ and $T \geq t\geq
T_{1}$.

Thus, for  $|p-p_0|<\delta_0$, we obtain
\begin{align*}
&\operatorname{dist}_{ L^2(0,1)}(u_{pq}(T_{1};\psi_{pq}),\mathcal{A}_{p_0q_0}) \\
& \leq \|u_{pq}(T_{1},\psi_{pq})-u_{p_0q_0}(T_{1};\psi_{pq})\|_{L^2(0,1)}
+ \operatorname{dist}_{ L^2(0,1)}(u_{p_0q_0}(T_{1},\psi_{pq}),\mathcal{A}_{p_0q_0})
< \delta.
\end{align*}
On the other hand, it follows from the invariance of the attractors
that
$$
\operatorname{dist}_{L^2(0,1)}(\mathcal{A}_{pq},\mathcal{A}_{p_0q_0})\leq \delta,
$$
for every $|p-p_0|<\delta_0$ and $q$ such that $|q-q_0|\leq
\epsilon$ showing the upper semicontinuity desired.
\end{proof}

\begin{remark} \label{rmk3.4} \rm
It follows from Theorem \ref{estunifwup}, Corollary \ref{coro3.1},
\cite[Lemma 1.1]{piskarev} and the compact immersion of $W_0^{1,2}(0,1)$
in $C([0,1]$ that the family
$\{\mathcal{A}_{pq}\}$ is upper
semicontinuous at $(p_0,q_0)$ in the topology of $C([0,1])$.
\end{remark}

Now we are interested in obtaining the upper semicontinuity
of global attractors of \eqref{eqTY} in a stronger topology.
To do that, we consider $p$ fixed, and $q \to q_0$.

\begin{theorem} \label{teo3.2}
 For each $(p_0,q)\in R$, let $\{u_{p_0q}\}\subset W_0^{1,p_0}(0,1)$ be a solution
of
 \begin{gather*}
\begin{aligned}
&\frac{d}{dt} u_{p_0q}(t)  - \lambda
(|(u_{p_0q}(t))_x|^{p_0-2} (u_{p_0q}(t))_ x)_x \\
&=|u_{p_0q}(t)|^{q-2}u_{p_0q}(t)(1+|u(t)_{p_0q}|^r), \quad t>0
\end{aligned}\\
u_{p_0q}(0) = u_{0\,p_0q} \in W_0^{1,p_0}(0,1).
 \end{gather*}
Suppose that $\|u_{0\,p_0q}\|_{W_0^{1,p_0}(0,1)}\leq M$ for every
$(p_0,q) \in R$ and $u_{0\,p_0q}\to u_0$ in $ L^2(0,1)$ as
$ q \to q_0$. Then, for each $T>0$, $u_{p_0q}\to u$ in
$C([0,T];W_0^{1,p_0}(0,1))$ as $q\to q_0$, where $u$ is a solution
of
\begin{gather*}
\frac{d}{dt} u(t)  - \lambda (|(u(t))_x|^{p_0-2}(u(t))_ x)_x
=|u(t)|^{q_0-2}u(t)(1+|u(t)|^r),\quad t>0\\
u(0) = u_0 \in W_0^{1,p_0}(0,1).
 \end{gather*}
\end{theorem}

The above theorem is a simple consequence of Tartar's Inequality,
Lemma \ref{normaLinfinito}
and Lemma \ref{estimaut}.

With the same argument as in the proof of  Corollary \ref{coro3.1}
we can prove the next result.

\begin{corollary}\label{coro3.2}
 The family of global attractors
$\{\mathcal{A}_{pq}\subset W_0^{1,p_0}(0,1)): (p_0,q) \in R\}$
 of problem \eqref{eqTY} is upper
semicontinuous at $(p_0,q_0)$ in the topology of $W_0^{1,p_0}(0,1)$.
 \end{corollary}



\section{Continuity of equilibrium sets }
 \label{continuidadedosequilibrios}

In this section, considering $p$ fixed, we  prove the continuity of the family of
equilibrium points of the equation \eqref{eqTY} when $q$ goes to $p$.
To analyze the continuity of the equilibrium sets it is interesting
to remember how the stationary solutions are obtained in \cite{TY}.

Let $\phi_{\alpha q}$ be a solution  of
\begin{equation}\label{auxiliarequilibriumequation}
\begin{gathered}
\lambda (\psi)_x + f_q(\phi_{\alpha q}) =0 ,\quad \text{in } (0,\infty)\\
\phi_{\alpha q}(0)=0, \\
\psi(0)=\alpha
\end{gathered}
\end{equation}
where $\alpha $ is a parameter, $\psi=|(\phi_{\alpha q})_x|^{p-2}(\phi_{\alpha q})_x$
and $f_q(\phi) =|\phi|^{q-2}\phi(1-|\phi|^r)$. We observe that, in
order to a solution of \eqref{auxiliarequilibriumequation} be an
equilibrium point of \eqref{eqTY}, $\alpha$ must be such that
$\phi_{\alpha q}(1)=0$.

We denote by $X(\alpha,p,q)$ the function that measure the
$x$-time that the solution $\phi_{\alpha q}$ of
\eqref{auxiliarequilibriumequation} takes to reach the first maximum
point. Because of the symmetry we have that
$\phi(2X(\alpha,p,q))=0$ or, more generally,
$\phi_{\alpha,q}(2kX(\alpha, p, q))=0$, $k=1,2,\ldots$.
Also, we have that $2n X(\alpha,p,q)=1$ is a  sufficient condition to
$\phi_{\alpha q}$ be an equilibrium point of \eqref{eqTY} with ${n-1}$
zeros in $(0,1) \subset {\mathbb R}$. Due to the symmetry
of the problem, we can only consider $\alpha >0$.

The function $X$ is
$$
X(\alpha,p,q) =
\Big(\frac{\lambda(p-1)}{p}\Big)^{1/p} I(p,q,\tilde{\phi}_{\alpha,q}),
$$
where $\tilde{\phi}_{\alpha,q}$ is the maximum value of $\phi_{\alpha,q}$ and
$$
I(p,q,a)= \int_0^a (F_q(a)-F_q(\phi))^{-1/p}d\phi,
$$
with $F_q(\phi)=F(\phi,q) = \int_0^\phi f_q(s)ds
={ \frac{\phi^q}{q} -\frac{\phi^{q+r}}{q+r}} \in C^1((0,\infty)\times(2,\infty))$.

In \cite{TY}, the authors studied the behavior of the function $Y(p,q)$,
which describes the distance between two consecutive zeros of an equilibrium
 and, analyzing their graphs for $p>q$, $p=q$ and $p<q$, they obtain that
if $p>q$ there exists a decreasing sequence $\lambda_{n}(p,q)$,
$\lambda_{n}(p,q) \to 0$ when $n \to \infty$
such that the equilibrium set $E=\{0\} \cup \cup_{i=0}^{\infty} E_i^\pm$
where $E_i^\pm$ denote the
equilibrium sets within the equilibria with $i$ zeros in $(0,1)$ and  if
$\lambda < \lambda_{n}(p,q)$, the set $E_i^\pm$ is diffeomorphic
to $[0,1]^i$, for $1\leq i \leq n$.  We observe that in this case there are
equilibrium points with any amount of zeros in $(0,1)$.

If $p\leq q$ there exist decreasing sequences $\lambda_n(p,q)\to 0$ and
$\lambda^*_{n}(p,q)\to 0$ such that  $\lambda^*_n(p,q) > \lambda_{n}(p,q)$.
If $p=q$, for $\lambda_{M+1}\leq \lambda <  \lambda_M$,  the equilibrium
set is given by $E=\{0\} \cup \cup_{i=0}^{M} E_i^\pm$.
If $p<q$, for $\lambda_{M+1}< \lambda \leq \lambda_M$ the equilibrium set
is given by $E=\{0\} \cup \cup_{i=0}^{M} (E_i^\pm \cup \{F_i^\pm\})$,
where $E_i^\pm$ denote the equilibrium sets containing equilibria with $i$
zeros in $(0,1)$ and $F_i^\pm = \{\psi_i^\pm\}$ also is equilibrium with
$i$ zeros in $(0,1)$.
Furthermore, if $p\leq q$ and  $\lambda < \lambda_{n}(p,q)$, the set
$E_i^\pm$ is diffeomorphic to $[0,1]^i$, for $1\leq i \leq n$.
In any case,  $E_0^\pm=\{\phi_0^\pm\}$ for $\lambda < \lambda_0(p,q)$.

About the stability of the equilibria, in
\cite[Theorems 4.2, 4.3]{TY}, they obtain that $0$ is asymptotically
stable if $p=q$ and $\lambda \geq \lambda_0$ or if $p<q$, $0$ is
unstable for $p>q$ or $p=q$ and $\lambda< \lambda_0$.
The equilibrium $\phi_0^+$ is asymptotically stable if
$\lambda> \lambda_0^*$ and attractive for $\lambda \leq \lambda_0^*$,
and if $q>p$, $\psi_0$ is unstable for $\lambda \leq \lambda_0$.

Since we deal with the dependence on the
parameter $q$ and there are qualitative changes in the equilibrium sets
depending on the relation between $p$ and $q$, if necessary,
we will exhibit explicitly the parameters
$p$ and $q$.

To prove the continuity of the equilibrium set, we take a sequence of equilibria
 in $E_i^\pm$ with a fixed number of zeros and, analyzing the initial slopes
 of such stationary solutions,
we conclude through the continuity properties of
 problem \eqref{auxiliarequilibriumequation},
that this sequence must converge to an equilibrium point of the limit
problem with the same amount of zeros in $ (0,1)$ or,
when it is not possible, the sequence converges to the null stationary solution.
 We also prove that any sequence of equilibria taken in
 $\{\psi_i\}\subset F_i^\pm$ converges to zero.

 We start with the analysis of the dependence of $\tilde{\phi}_{\alpha, q}$
on $q$ and $\alpha$. We know that $\tilde{\phi}_{\alpha, q}$ is strictly
increasing and $C^1$ in $\alpha$, $\alpha \in [0, \alpha_0)$ (see \cite{BGP1}).
With respect to $q$, since $\tilde{\phi}_{\alpha,q}$ is the maximum value of
$\phi_{\alpha,q}$, then $\tilde{\phi}_{\alpha,q}$ satisfies
$$
F(\tilde{\phi}_{\alpha,q},q) = \lambda \frac{(p-1)}{p} |\alpha|^{\frac{p}{p-1}}.
$$
Calculating
\[
\frac{\partial}{\partial q}F(\phi,q)
={\frac{\phi^q(q\ln \phi -1)}{q^2} -\frac{\phi^{q+r}((q+r)\ln \phi -1)}{(q+r)^2}}
 =\beta(q) -\beta(q+r),
\]
 where $\beta(\theta) = \frac{\phi^\theta(\theta \ln \phi -1)}{\theta^2}$, for
$\theta \geq 2$. As $\beta' (\theta) > 0$,
 thus $\frac{\partial}{\partial q}F(\phi,q)  < 0$.

Also  $\frac{\partial}{\partial \phi}F(\phi,q) =  \phi^{q-1}-\phi^{q+r-1} > 0$,
if $\phi \in (0,1)$.
Using the Implicit Function Theorem, we obtain that the map
$\tilde{\phi}_{\alpha q}$ is $C^1$ on
 $(\alpha,q)$ . Also,
$$
\frac{\partial}{\partial q}\tilde{\phi}_{\alpha q}
= - \frac{\frac{\partial}{\partial q} F(\tilde{\phi}_{\alpha q},q)}
{\frac{\partial}{\partial \phi} F(\tilde{\phi}_{\alpha q},q)} >
 0,
$$
then $\tilde{\phi}_{\alpha q}$ is  strictly increasing on $q$.


Now we analyze  the function $I(p,q,a)$. In \cite{TY}, the authors rewrite
$I(p,q,a)$ as
$$
I=I(p,q,a)=\int_{0}^{a} (F_q(a)-F_q(\phi))^{-1/p}d\phi=
a^{1-q/p} \int_0^1\Phi_q(s,a)^{-1/p}ds,
$$
where
$\Phi_q(s,a)=\frac{1-s^q}{q}-\frac{1-s^{q+r}}{q+r}a^r$. Then
 we obtain $I(p,q,a)$ is $C^2$ on $(2,\infty)\times[2,\infty)\times(0,1]$.
For each $p$ fixed, we analyze the behavior of $I(p,q,a)$ with respect the
parameter $q$. We study the behavior of $I(p,q,a)$ with respect to $q$ for
$a$ close to zero because  $I(p,q,a)$ is $C^2$ on
$(2,\infty)\times[2,\infty)\times(0,1]$ and the major difference in the
cases occurs close to zero. We prove that $I(p,q, a)$ is increasing
with respect to $q$ for $a$ near to zero.

\begin{lemma} \label{lem4.1}
For $0 \leq a <e^{-1/2}$ fixed,
$\frac{\partial}{\partial q} I(p,q,a) > 0$,  for
$(p,q) \in (2,\infty) \times [2,\infty)$.
\end{lemma}

\begin{proof}
In fact, since  $I(p,q,a)= \int_0^a (F_q(a)-F_q(\phi))^{-1/p}d\phi$, it follows that
\begin{align*}
 \frac{\partial}{\partial q} I(p,q,a) 
&=  \int_0^a \frac{\partial}{\partial q}(F_q(a)-F_q(\phi))^{-1/p}d\phi\\ 
& = \int_0^a -\frac{1}{p}  (F_q(a)-F_q(\phi))^{-1/p -1} 
 \frac{\partial}{\partial q} (F_q(a)-F_q(\phi)) d\phi
\end{align*}

Since  $ (F_q(a)-F_q(\phi))^{-1/p -1}>0$, we only consider
\begin{align*}
\frac{\partial}{\partial q} (F_q(a)-F_q(\phi))
&=  \frac{a^q\ln(a)}{q} + \frac{-a^q}{q^2} - \frac{a^{q+r}\ln(a)}{(q+r)}
 + \frac{a^{q+r}}{(q+r)^2} \\
&\quad - \big[ \frac{\phi^q\ln(\phi)}{q} + \frac{-\phi^q}{q^2}
  - \frac{\phi^{q+r}\ln(\phi)}{(q+r)} + \frac{\phi^{q+r}}{(q+r)^2}\big]
\end{align*}

Now we define
${ \varphi(\theta) = \frac{a^\theta}{\theta^2}-\frac{\phi^\theta}{\theta^2}}$.
Then
$\varphi'(\theta) \leq 0$, thus $\varphi(q+r) -\varphi(q) < 0$.
Define also
${ \psi(\theta)= \frac{\theta^q \ln(\theta)}{q}-\frac{\theta^{q+r}
\ln (\theta)}{(q+r)}}$. Then, for $\theta < e^{-1/2}$
$$
\psi'(\theta) \leq [\theta^{q-1} -\theta^{q+r-1}]
\Big(\ln \theta + \frac{1}{q+r}\Big) < 0,
$$
thus $\psi(a)-\psi(\phi) < 0$ for  $0< \phi< a< e^{-1/2}$.
Therefore,
\begin{equation}
\frac{\partial}{\partial q} (F_q(a)-F_q(\phi))
 = \varphi(q+r) -\varphi(q) + \psi(a)-\psi(\phi) < 0
\end{equation}
Finally, we obtain
\begin{equation}
 \frac{\partial I}{\partial q} (p,q,a)
= \int_0^a -\frac{1}{p}  (F_q(a)-F_q(\phi))^{-1/p -1}
\frac{\partial}{\partial q} (F_q(a)-F_q(\phi)) d\phi > 0.
\end{equation}
\end{proof}

Now we consider $p>q$.
In \cite{TY}, the authors show that $\frac{\partial I}{\partial a}(p,q,a) >0$
for $q<p$, then
$$
\frac{\partial X}{\partial q}(\alpha,p,q)
=\big(\frac{\lambda(p-1)}{p}\big)^{1/p}
\Big(\frac{\partial I}{ \partial q}(p,q,\tilde{\phi}_{\alpha q}) +
\frac{\partial I}{\partial a}(p,q,\tilde{\phi}_{\alpha q})
\frac{\partial\tilde{\phi}_{\alpha q}}{\partial q}\Big)> 0, \quad p > q.
$$
Fixed $p$ and $n$, for each $q<p$, we consider $\alpha_q^n$ the initial
condition such that the $x$-time
$X(\alpha^n_q,p,q)$ is kept constant and equal to $1/2n$. We have that
\begin{align*}
0&= \frac{d X}{d q}(\alpha^n_ q,p,q)
 = \frac{\partial X}{\partial \alpha}(\alpha^n_ q,p,q)\frac{d\alpha}{dq} +
\frac{\partial X}{\partial q}(\alpha^n_ q,p,q)   \\
& =
\big(\frac{\lambda(p-1)}{p}\big)^{1/p}\big[
\frac{\partial I}{\partial a}(p,q,\tilde{\phi}_{\alpha q})
\frac{\partial\tilde{\phi}_{\alpha q}}{\partial \alpha}
\frac{d\alpha}{dq}  + \frac{\partial I}{ \partial
q}(p,q,\tilde{\phi}_{\alpha q}])
+ \frac{\partial I}{\partial a}(p,q,\tilde{\phi}_{\alpha q})
 \frac{\partial \tilde{\phi}_{\alpha q}}{\partial q}\big]\\
 & =  \big(\frac{\lambda(p-1)}{p}\big)^{1/p}
 \big[ \frac{\partial I}{\partial a}(p,q,\tilde{\phi}_{\alpha q})
 ( \frac{\partial \tilde{\phi}_{\alpha q}}{\partial \alpha} \frac{d\alpha}{dq}+
\frac{\partial\tilde{\phi}_{\alpha q}}{\partial q})
 + \frac{\partial I}{ \partial q}(p,q,\tilde{\phi}_{\alpha q}) \big] .
\end{align*}
Since ${ \frac{d }{d q}(\tilde{\phi}_{\alpha q})
=\frac{\partial \tilde{\phi}_{\alpha q}}{\partial \alpha} \frac{d\alpha}{dq}+
\frac{\partial\tilde{\phi}_{\alpha q}}{\partial q}}$,
$\frac{\partial I}{ \partial q}>0$,  $\frac{\partial I}{\partial a} >0 $
for $q<p$, and ${}\frac{\partial \tilde{\phi}_{\alpha q}}{\partial \alpha}>0$,
$\frac{\partial\tilde{\phi}_{\alpha q}}{\partial q}>0$
then we conclude that $\frac{d\alpha}{dq}<0$.
We summarize the previous results in the following lemma.

\begin{lemma} \label{lem4.2}
If  $p>q$, let  $\alpha(q)$  be such that  $X(\alpha(q),p,q)$ remains constant.
Then $\alpha(q)$ is decreasing with respect to $q$.
\end{lemma}

Now we can prove the following result.

\begin{theorem}\label{teo4.3}
 Suppose $p>2$ fixed. Let $M$ be the maximum number of
zeros of an equilibrium when $q=p$. Let $\phi_n(q) \in E_n^\pm $ for
$p> q$. If $n\leq M$, then $\phi_n(q)$ converges to another
stationary solution, with the same amount of zeros when $q\to p^-$.
If $n$ is greater than $M$, then $\|\phi_n(q)\|_{C^1(0,1)}$ goes to
zero when $q \to p^-$.
\end{theorem}

\begin{proof}
We rewrite \eqref{auxiliarequilibriumequation} in the  form
\begin{equation} \label{sistemasimples}
\dot{z} = h(z,q),
\end{equation}
where $z=[\phi,\psi]$ and
$h((\phi,\psi),q)=(sign(\psi)|\psi|^{1/(p-1)},-f_q(\phi)/\lambda)$.
We have that the map $h$ depends continuously on $q$ and its local
Lipschitz constant with respect to $z$ is independent of $q$ for
$q\in (q_0,p]$, where $q_0$ is close enough to $p$.
As it is done in \cite{BGP1}, if $\alpha^n_q$ is such that
$X(\alpha^n_q,p,q)=\frac{1}{2n}$, there is an open set
$U \subset {\mathbb R}^2$ such that $(\alpha,q) \in U$ and $\alpha$
is a $C^1$ function of $q$. Then, once we have that the solution $z_q$
of \eqref{sistemasimples} depends continuously on $q$ and on
$(\phi(0),\psi(0))= (0,\alpha_{q})$, (see \cite{Hale2}), $z_q$ converges
to $z_p$ when $q \to p$.
If $n> M$, we obtain $\alpha^n_q \to 0$ when $q \to p^-$. In fact, since
$\alpha^n_q $ is decreasing and bounded, given a sequence $q_j$,
$q_j \to p^-$, and $\alpha^n_j = \alpha^n(q_j)$, there exists $\alpha^n$ such that
$\alpha^n_j \to \alpha^n$. If $\alpha^n >0$, by continuity, we obtain that
when $p=q$ there exists an equilibrium point of \eqref{eqTY}
with $n$ zeros in $(0,1)$. Since $n >M$, it is not possible, then $\alpha^n =0$.
Therefore, from the continuous dependence of initial data and parameters,
 we obtain $\|\phi_j(q)\|_{C^1(0,1)}$ goes to zero when
$q \to p^-$.
\end{proof}

Regarding the case $q>p$, since $\frac{\partial I}{\partial a}(p,q,a) <0$
when $q>p$  and $a$ is close to zero it is not possible analyze the sign
of $\frac{\partial X}{\partial q}$.
In \cite{TY}, it was proved that for each $q$, $q>p$ there is only one
$a^*(q)$ such that $a^*(q)$ is the minimum point of $I(p,q,a)$,
 that means, $\frac{\partial I}{\partial a}(q,a^*(q))=0$ and
$\frac{\partial ^2 I}{\partial a^2}(q,a^*(q))>0$.
We will prove that $a^*(q)$ goes to zero when $q$ goes to $p^+$.


First of all, using the Implicit Function Theorem for
$\frac{\partial I}{\partial a}(q,a)=0$, we obtain that $a^*(q)$ is a $C^1$
function. Then we have the following theorem.

\begin{theorem} \label{teo4.4}
Suppose $p>2$ fixed. Let $\phi_i(q) \in E_i^\pm $ for
$q > p$. Then $\phi_i(q)$ converges to another stationary solution,
with the same amount of zeros when $q\to p^+$. If $\psi_i(q) \in
F_i^\pm $, then $\|\psi_i(q)\|_{C^1(0,1)}$ goes to zero when $q \to
p^+$.
\end{theorem}

\begin{proof}
The first part of the statement follows as in the previous theorem.

Let $q_n$ be a sequence that $q_n \to p^+$ and $a^*_n=a^*(q_n)$.
Since $a^*_n$ is a bounded sequence it contains a convergent subsequence $a^*_{n_k}$.
Suppose that $a^*_{n_k} \to a^* >0$. Then
$I(p,p,a^*) = \lim_{k\to \infty} I(p,q_{n_k}, a^*_{n_k}) $ and
$\frac{\partial I}{\partial a }(p,p,a^*)
= \lim_{k \to \infty} \frac{\partial I}{\partial a }(p,q_{n_k},a^*_{n_k}) =0$,
that means, $a^*$ is a critical point of $I(p,p,a)$.

But, in \cite{TY} the authors have proved that $I(p,p,a)$ is strictly increasing
in $[0,1)$. Then, it is only possible $a^* =0$ for any sequence $a^*_n$.
Thus, we conclude that $a^*(q_n)$ goes to $0$ when $q_n \to p^+$.
Therefore, since that each equilibrium point $\psi_i(q)$ of \eqref{eqTY}
is a solution of \eqref{auxiliarequilibriumequation} with initial date $\phi(0)=0$
and $\psi(0) = {\tilde{\alpha}}_{nq}$,  where ${\tilde{\alpha}}_{nq}$ is the
$\alpha$ such that ${\tilde{\alpha}}_{nq} < a^*(q)$, from the  continuous
dependence with respect initial data and
parameter $q$, we have that $\psi_i(q_n) $ converges to zero when $q_n \to p^+$
in $C^1[0,1]$.
\end{proof}

Now we join  some results about $I(p,q,a)$ for  $q>p$ in the following lemma.

\begin{lemma}  \label{I_min_q>p}
If $q>p$, then \begin{itemize}
\item[(i)] $a^*(q) \to 0$ when $q \to p^+$,
\item[(ii)] $\tilde{I}(q) = I(p,q, a^*(q))$ is increasing with respect to $q$,
\item[(iii)] $\tilde{I}(q) \to I_0=I(p,p,0)$, when $q \to p^+$.
\end{itemize}
\end{lemma}

\begin{proof}
Item (i) follows from the prior discussion.

(ii) $\tilde{I}(q) = I(p,q, a^*(q))$, with $p$ fixed. We obtain
\[
\frac{d\tilde{I}}{dq}(q) = \frac{\partial I}{\partial a }(q,a^*(q)) 
 \frac{d a^*}{dq}(q) + \frac{\partial I}{\partial q }(q,a^*(q)) 
= \frac{\partial I}{\partial q }(q,a^*(q))>0,
\]
which means that the minimum value of $I$ is increasing with $q$.

(iii) It follows by using (ii) and the continuity of $I(p,p,a)$ in 
$a=0$ and $I(p,q,a)$ for $a>0$.
\end{proof}

Since the sequences $\lambda_n(p,q)$ and $\lambda_n^*(p,q)$ depends on $(p,q)$, 
even if  $\lambda $ is fixed  it is possible
to occur changes in the relation between $\lambda$ and $\lambda^*_n(p,q)$ and 
$\lambda_n(p,q)$ when $q \to p$. Then,  before proving the continuity of 
equilibrium sets $E(p,q)$ in $q=p$ we analyze that the possibilities among 
$\lambda$,  $\lambda_n(p,q)$ and $\lambda_n^*(p,q)$.

Let $\{\lambda_n\}$, $\{\lambda_n^*\}$, $\{\lambda_{n}(p,q)\}$ 
and $\{\lambda^*_{n}(p,q)\}$ be defined as follows:
\begin{gather*}
 \lambda^*_n  \doteq \frac{p}{p-1}(2(n+1)I_0)^{-p},\\
\lambda^*_{n}(p,q) \doteq  \frac{p}{p-1}(2(n+1)I^*(q))^{-p}, q>p,
\end{gather*}
where $I^*(q)= I(p,q,a^*(q))$ denotes the minimum value of $I(p,q,a)$ 
with relation to $a$, $I_0=lim_{a\to 0^+} I(p,p,a)$, and
\begin{gather*}
\lambda_n \doteq \lambda_n(p,p)= \frac{p}{p-1}(2(n+1)I(p,p,1))^{-p};
\lambda_n(p,q)= \frac{p}{p-1}(2(n+1)I(p,q,1))^{-p}.
\end{gather*}
Here $\{\lambda^*_n\}$, $\{\lambda^*_{n}(p,q)\}$ are the sequence that 
determine the number of zeros allowed to a stationary solution of \eqref{eqTY}
when $p=q$ and $p<q$ respectively, and $\{\lambda_n(p,q)\}$ determines 
the existence of continuum components in $E_\lambda(p,q)$. All details can
be found in \cite{TY}.

Now we fix $p$ and $\lambda$. We observe that there are only four possibilities
\begin{enumerate}
\item $\lambda \neq \lambda_i$ and $\lambda \neq \lambda_j^*$ for any 
 $i$ and any $j$;
\item $\lambda \neq \lambda_i$ and $\lambda = \lambda_j^*$ for any $i$ 
 and for some $j$;
\item $\lambda = \lambda_i$ and $\lambda \neq \lambda_j^*$ for some $i$ 
 and for any $j$;
\item $\lambda = \lambda_i$ and $\lambda = \lambda_j^*$ for some $i$ 
 and some $j$, $i>j$.
\end{enumerate}
 We have the following:
\smallskip

\noindent\textbf{Case 1.}
Let $j_0$ be the least index such that $\lambda > \lambda_{j_0}(p,p)$. 
Since $I(p,q,1)$ behaves continuously on $q$, if $q$ is close enough to $p$, than
$\lambda > \lambda_{j_0}(p,q)$.
By Lemma \ref{I_min_q>p}, we also have that  $I^*(q) = \min I(p,q,a)$ is 
increasing with $q$ if $q>p$ and $I^*(q) \to I_0$  when $q \downarrow p$.
So, if $\lambda > \lambda^*_{i_0}$ for some given ${i_0}$, then 
$\lambda > \lambda^*_{i_0}(p,q)$, if $q$ is close enough to $p$. 
Therefore, if $ p < q $, $q$
can be chosen in a neighborhood of $p$ in such way that the maximum number 
of zeros of any equilibrium in $E_{\lambda}(p,q)$ is $M$ and, in both case $p>q$ or
$p<q$, components having equilibrium with the same amount of zeros, namely $k$, 
are discrete or continuous according with the cardinality of $E_{\lambda}^k(p,p)$.
Thus, in this case there is no additional qualitative differences between the 
sets of equilibrium beyond those which we deal in the prior discussion.
\smallskip

\noindent\textbf{Case 2.}
To analyze this case, let us consider the variation of $\lambda^*_j(p,q)$ 
with respect to $q$, $q>p$. By Lemma \ref{I_min_q>p}, $I^*(q)$
is increasing with $q$ if $q>p$ and $I^*(q) \to I_0$ when $q \downarrow p$, 
then $\lambda^*_j(p,q)< \lambda^*_j$. Thus, $\lambda = \lambda^*_j$
implies $\lambda > \lambda^*_j(p,q)$. This allows us to conclude that if there
 exist stationary solutions in $E_\lambda(p,q)$ having $n$ zeros in $(0,1)$,
then there is also solutions in $E_\lambda(p,p)$ having $n$ zeros in $(0,1)$.
In other words, once $\lambda = \lambda_j^*$ there is no solution with $j$ 
zeros in $(0,1)$ and, as $\lambda_j^*(p,q) < \lambda_j^* = \lambda$
there is no solution with $j$ zeros in $(0,1)$ for $q>p$. 
Finally, if $\lambda = \lambda_j^*$ then $\lambda < \lambda_k^*$, 
for $0\leq k\leq j-1$ the analysis follows
the Case 1, for solutions with $k$ zeros in $(0,1)$.
\smallskip

\noindent\textbf{Case 3.}
 Once $\lambda_i(p,q)= \frac{p}{p-1}(2(i+1)I(p,q,1))^{-p}$, using the continuity 
of $I(p,q,1)$  we obtain $\lambda_i(p,q)  \to \lambda_i$ when $q \to p$. 
If $I(p,q,1) < I(p,p,1)$ there exists a continuum of solutions with $i$ zeros 
for $(p,q)$. In despite of this, we know that, if $X_j(q)$ is the ``$x$-time'' 
that an equilibrium $\phi_j(q) \in  E_\lambda^j(p,q)$ needs to reach its first 
maximum, then  $X_j(q) \to \frac{1}{2(j-1)}$ as $q \to p$. 
So we obtain that all sequence of stationary solutions in the continuum sets 
$E_\lambda^j(p,q)$ converges to the same equilibrium in $E_\lambda^j(p,p)$, 
when $q \to p$. If $I(p,q,1) > I(p,p,1)$ the solutions with $i$ zeros do not reach
the maximum value equal 1 for $(p,q)$. In this case, by the continuity of $I$  
the maximum value goes to 1.
\smallskip

\noindent\textbf{Case 4.} This case follows from Cases 2 and 3.


\begin{remark} \label{rmk4.6} \rm
Regarding the equilibria $\pm\psi_n$ that appear when $q>p$ it is known that 
with respect to parameter $\lambda$ they arise as spontaneous bifurcations, 
\cite{TY},  but our analysis shows that with respect to $q$, $\pm\psi_n$ 
bifurcate from trivial solution.
\end{remark}

Now we are ready to state our main result concerning to the continuity on 
$q$ of the equilibrium sets
$E(p,q)$. The upper semicontinuity in $L^2 (0,1)$ and
$W_0^{1,p}(0,1)$ follows easily from Theorem \ref{teo3.1} and Corollaries
\ref{coro3.1} and \ref{coro3.2}. From the prior analysis presented in
this section we can conclude the upper and lower semicontinuity in $C^1[0,1]$.

\begin{theorem} \label{teo4.7}
 The family $E(p,q)$ is upper and lower semicontinuous on $q$ as $q$ goes
to $p$ in $C^1[0,1]$.
\end{theorem}

\begin{proof}
If $q\downarrow p$, given any sequence $\{\varphi_q\}$, $\varphi_q \in E(p,q)$ 
for each $q$, there is a subsequence of $\{\varphi_q\}$ containing
only equilibria with the same amount of zeros in $(0,1)$. 
Then we know from Theorem \ref{teo4.4} that this subsequence converges to an
equilibrium in $E(p,p)$.

If $q\uparrow p$, given a sequence $\{\phi_q\}$, $\phi_q \in E(p,q)$ for each $q$, 
which contains a subsequence with the same amount of zeros,
then we know from Theorem \ref{teo4.3} that this subsequence converges to an
equilibrium in $E(p,p)$. But in this case it is also possible to find a sequence 
$\phi_q \in E(p,q)$ in such way that the number of zeros of $\phi_q$ goes to 
infinity with $q$. In this case, we observe that this sequence goes to the 
null solution.

So we conclude from \cite[Lemma 1.1]{piskarev} that $E(p,q)$ is upper 
semicontinuous at $q=p$.

To prove the lower semicontinuity, let $\phi_p \in E(p,p)$. We have three possible 
situations. If the maximum value of $\phi_p$ is
less than 1 and $n$ is the amount of zeros of $\phi_p$ in $(0,1)$, the 
sequence $\phi_q \in E(p,q)$ containing only
equilibria with $n$ zeros converges to $\phi_p$ according with Theorems \ref{teo4.3}
 and \ref{teo4.4}. If $\phi_p$ achieves 1
but does not have flat cores we can repeat the prior argument 
(observe that it is possible only if $\lambda =\lambda_n$ and this situation 
was discussed in the Case 3). When $\phi_p$ presents flat cores, then 
$\lambda < \lambda_n$ and, from the continuity of $\lambda_n(p,q)$ on $q$, 
we conclude that equilibria with $n$ zeros in $E(p,q)$ present flat cores as 
well (we have used an analogous argument in Case 1). In this case,
we construct the approaching sequence. Let $f_i$ be the length of the $i$-th 
flat core, for $i=1, \ldots, n+1$. For $q$ close to $p$, let $X(p,q)$
the $x$-time spent to an equilibrium in $E_n(p,q)$ achieve the maximum value 
equals to 1. If $X(p,q) > X(p,p)$ we pick in $E(p,q)$ an equilibrium $\phi_q$
with $n$ zeros in $(0,1)$ such that the length of $i$-th flat core is 
$f_i -2(X(p,q)-X(p,p))$. If $X(p,q) < X(p,p)$ we choose an equilibrium $\phi_q$ with
$n$ zeros in $(0,1)$ such that the length of $i$-th flat core is 
$f_i +2(X(p,p)-X(p,q))$. In any case $\phi_q \to \phi_p$ as $q \to p$.

The lower semicontinuity follows from  \cite[Lemma 1.1]{piskarev}.
\end{proof}

\subsection*{Acknowledgments}
S. M. Bruschi was  supported by the FEMAT - Funda\c{c}\~ao de Estudos
em Ci\^encias Matem\'aticas.

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\end{document}