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\markboth{\hfil On forced periodic solutions \hfil EJDE--1998/14}%
{EJDE--1998/14\hfil Jos\'e Luiz Boldrini \& Janete Crema \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~14, pp. 1--18. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
\vspace{\bigskipamount} \\
On forced periodic solutions of superlinear quasi-parabolic problems
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35K55, 35B10.
\hfil\break\indent
{\em Key words and phrases:} quasi-parabolic equations, periodic solutions,
p-Laplacian.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted March 11, 1998. Published May 30, 1998.} }
\date{}
\author{Jos\'e Luiz Boldrini \& Janete Crema}
\maketitle
\begin{abstract}
We study the existence of periodic solutions for a class of
quasi-parabolic equations involving the $p$-Laplacian (or any other
nonlinear operators of similar class) perturbed by nonlinear terms and
forced by rather irregular periodic in time excitations (including what
we call abrupt changes). These equations may model problems for which,
aside from the presence of the kind of nonlinear dissipation associated
to the $p$-Laplacian, other nonlinear and not necessarily dissipative
mechanisms occur. We look for boundedness conditions on these periodic
excitations and nonlinear perturbations sufficient to guarantee the
existence of periodic responses (solutions) of the same period.
\end{abstract}
\newcommand{\dv}{\mathop{\rm div}}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}
\section{Introduction} \label{Introduction}
Monotone operators, in particular the ones that are subdifferentials of
convex functions, like the $p$-Laplacian, appear frequently in equations
modeling the behaviour of viscoelastic materials (see Le Tallec [10] for
instance).
These nonlinear operators may be accompanied by others nonlinear operators
that may complicate the analysis of the equations and the prediction of the
behaviour of the material. This is specially true when these accompanying
operators may, under certain circumstance, impart energy into the system
formed by the interaction between the material and external actions.
Mathematically speaking, this means that the overall operator associated to the
problem is no longer always monotonic and not even always dissipative. Thus,
due to this imbalance between the gain and the dissipation of energy, it is
not clear whether when acted by periodic external forces, the internal
dissipation will be enough for the system respond also in a periodic way.
This raises the problem of knowing under what conditions such periodic
responses exist.
Obviously, this sort of question also appears when the principal part of the
operator modeling to the internal dissipation is linear. But, in this
case, one can resort to a larger collection of mathematical techniques to
study the problem, and, due to our better knowledge of the behavior of linear
equations as compared to the present knowledge of the nonlinear ones, the
answer to the previous question can be more easily obtained. In fact, there
is plenty of articles considering several aspects (and not only the question
of existence) of the problem. For instance, with the help of techniques of
super and subsolution, Hess (\cite{Hess1}, \cite{Hess2}, \cite{Hess3},
\cite{Hess4} \cite{Hess5}) studied the existence of periodic solutions and
its related properties (like the study of principal eigenvalues and
bifurcation), when the principal part of the involved operator is linear.
Other aspects of this problem, always when the principal part of the
involved operator is linear, can be found in papers by Dancer and Hess
(\cite{DancerHess1}, \cite{DancerHess2}, \cite{DancerHess3}), Buonocore
\cite{Buonocore}, Nkashama \cite{Nkashama} and Esteban \cite{Esteban}.
Here, on the other hand, we will be interested in giving a partial answer
only to the the question of existence of forced periodic solutions, but for a
class of nonlinear equations including the $p$-Laplacian as the principal
part of the operator. The analysis of such cases is harder, as it is in
general when the principal part of the operator is also nonlinear. It is also
much less studied.
In fact, few papers consider the question in such situations. For instance,
Mizoguchi \cite{Mizoguchi} considered the problem of existence of positive
periodic solutions for equations in which the principal part of the operator
was related to the operator appearing in porous media equation.
Yamada \cite{Yamada} treated a problem involving the $p$-Laplacian, perturbed
by a maximal monotonic operator, and proved the existence of periodic
solutions when the boundary conditions changed periodically. We observe that
the problem considered in \cite{Yamada} is always dissipative.
In \cite{Otani}, \^Otani proved rather general results concerning the
existence of strong solutions of abstract equations of a certain class ,
which allows the $p$-Laplacian as the principal part of the operator. We will
compare his results with ours later on in the paper.
For simplicity of exposition, we will start by considering in detail a rather
simple model problem. We begin by studying cases where there are no changes
in the basic type of internal dissipation mechanism during the periodic cycle
(later we will consider a case where this change of type can occur). We will
also assume that the periodic excitation is of a special type.
Let $\Omega$ be an open bounded and regular set in $R^N$ and $0 p-1$. This will be done in Section
\ref{ExistenceSimpleModel}
Next, in the following section, we will see that our method of proof can be
easily extended to abstract problems involving operators similar to
$\Delta_p$. In fact, our results will be true for equations of form:
\begin{equation}
\label{AbstractEquation}
\begin{array}{lcl}
& u_t + A(u) = F(u) \, , & \\
& u(0) = u(T) \, , & \\
\end{array}
\end{equation}
with $A$ a strongly monotone, coercive and hemicontinuous operator in a
suitable reflexive separable Banach space and $F$ a nonlinear mapping
satisfying growth conditions similar to the ones for the previously described
special case (\ref{SpecialEquation}). In Section \ref{Abstract Result}, we
will also give examples of application of this abstract result to other
equations.
Then, we will pay attention to situations where at certain specified times
(and in a periodic way) there are sudden (abrupt) changes of the involved
operator.
A problem of this sort is studied in Kawohl and R\"uhl \cite{KawohlRuhl},
where it is proved the existence of periodic solutions for an equation
involving the Laplacian, perturbed by another maximal monotonic operator,
with a certain boundary condition changing abruptly and periodically in time.
We remark again that, as in the case of \cite{Yamada}, the system studied in
\cite{KawohlRuhl} is always dissipative.
The results in \^Otani \cite{Otani} can also be applied to certain situations
of abrupt changes. They do not apply, however, to the sort of situation we
will be interested, and we will comment more on this point later on.
Here, as before, we want to study situations of abrupt changes in cases where
it is possible to impart energy into the system. Moreover, we would like to
allow for abrupt changes in type of the principal part of the operator (which
models the internal dissipation), effectively changing its degree of
dissipation (which mathematically can be gauged, for instance, by its type of
coerciveness).
For this, we will consider a model case given the following problem: being
$p_i\geq 2$, $i=1,2$, $\Omega$ as before a bounded and regular region of
${\mathbb R}^{N}$, $\overline{t} \in (0,T)$ and $I_1 = (0,\overline{t})$, $I_2 =
(\overline{t},T)$, we want to find a suitable solution of
\begin{equation}
\label{Abrupt}
\begin{array}{lcl}
& u_t(t,x) - \Delta_{p_1}u(t,x) = m_1(t,x)g_1(u(t,x))+h_1(t,x)
\quad \mbox{for} \quad (t,x)\in I_1\times\Omega \, , & \\
& u_t(t,x) - \Delta_{p_2}u(t,x) = m_2(t,x)g_2(u(t,x))+h_2(t,x)
\quad \mbox{for} \quad (t,x)\in I_2\times\Omega \, , & \\
& u(t,\cdot)|_{\partial\Omega} = 0, \quad \mbox{for} \quad t\in (0,T),\\
& u(0,x) = u(T,x), \quad \mbox{for} \quad x\in \Omega \, . &
\end{array}
\end{equation}
As in the case of our previous simpler model (\ref{SpecialEquation}),
our goal will be to find conditions so that there are periodic solutions
when $|g_i(v)| \leq a_i (|v|^{s_i} + 1) \quad \forall v \in {\mathbb R}$, $i=1,2$.
We will establish a result of this sort in Section \ref{Abrupt Changes}.
Section \ref{Remarks} finishes the paper with several remarks concerning the
existence of strong solutions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}
\label{Preliminaries}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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For the sake of fixing notations and ease the reading, in this section we
will recall results that we will be using in the paper.
Let $V$ be a Banach space and $V'$ be its topological dual. Let $H$ be a
Hilbert space such that $V \subset H \subset V'$, with continuous and dense
inclusions. We will denote by $|\cdot|_V$, $|\cdot|_{V'}$, \ and \
$|\cdot|_H$ \ respectively the norms of \ $V$, $V'$ \ and \ $H$; by
$\langle \ , \rangle_{V',V}$ \ we denote the duality pairing between \ $V'$
\ and \ $V$, and by $\langle \ , \rangle_{H}$ \, , the inner product of \ $H$.
For $T>0$, we consider the Banach space $L^p(0,T;V) = \{u:(0,T) \rightarrow
V; \int_0^T |u(t)|^p_V \ dt < \infty\}$, with norm $|u|_{L^pV} = ({ \int_0^T}
|u(t)|^p_V \ dt)^{1/p}$. Let $p'$ be defined by $1/p+1/p'=1$ and denote
$u_t={\displaystyle\frac{d u}{dt}}$; we recall that the Banach space $\{ u
\in L^p(0,T;V); \ u_t \in L^{p'}(0,T;V')\}$, with norm given by
$|u|_{L^pV} + |u_t|_{L^{p'}V'}$ \, , is a subspace of $C([0,T];H)$, the class
of continuous functions defined on $[0,T]$ with values in $H$. Moreover, for
$u$ and $v$ in this space, there holds
\begin{equation}
\label{IntegrationByParts}
\int^T_0 \langle u_t(t), v(t) \rangle_{V'V}
+ \langle v_t(t), u(t) \rangle_{V'V} \, dt
= \langle u(T),v(T) \rangle_H - \langle u(0),v(0) \rangle_H.
\end{equation}
For a proof of these results, see Lions \cite{Lions}, pp. 156 and 321.
We will need the following compactness criterion given by the Aubin-Lions
(see Lions \cite{Lions}, p. 58, Theorem 5.1 and Strauss \cite{Strauss}, p.
34, Theorem 2):
\begin{lemma}
\label{Aubin-LionsLemma}
Let $X,Y$ and $Z$ three Banach spaces, $X$ and $Z$ reflexive, such that $X
\subset Y \subset Z$ with continuous inclusions. Moreover, assume that the
inclusion $X\subset Y$ is compact and that $1\leq p \leq \infty$ and $1 < q
\leq\infty$. Then, the space $\{ u \in L^p(0,T; X); \ u_t \in L^q(0,T;Z)\}$
is compactly included in $L^p(0,T; Y)$.
\end{lemma}
In the analysis of the cases with abrupt changes, we will have the opportunity
to use the following result stated in Brezis
\cite{Brezis}, Section 1, Theorem 1.3.
\begin{lemma}
\label{FixedPointContraction}
Let $E$ be an uniformly convex Banach space and let $C$ be a closed convex
subset of $E$. Let {\cal F} be a family of (not necessarily strict)
contractions from $C$ to $C$ with the property that for any $T, T'\in {\cal
F}$ it is true that $TT'=T'T \in {\cal F}$. Suppose also that there is
$\overline x \in C$ such that the set $\{T\overline x, \ \forall T \in {\cal
F} \}$ is bounded. Then, there exists $x_0 \in C$ such that $T x_0 = x_0$ for
any $T
\in {\cal F}$. In particular, if $T: {\cal C} \rightarrow {\cal C}$ is a (not
necessarily strict) contraction, $T$ has a fixed point if and only if there
is $\overline x \in C$ such that $\{ T^n \overline x \}_{n \geq 0}$ is
bounded.
\end{lemma}
The following results will be useful to obtain strong solutions:
\begin{lemma}
\label{Tychonoff-Schauder}
{\rm (Tychonoff-Schauder Fixed Point Theorem}
Let $K$ be a compact convex subset of a locally convex topological vector
space $X$. Let $T$ be an upper semi-continuous (multivalued) mapping from $K$
into $X$ such that for each $x$ in $K$, $T(x)$ is a closed convex subset of
$X$ whose intersection with $K$ is nonempty. Then $T$ has a fixed point in
$K$, i.e., there exists an element $x_0$ in $K$ such that $x_0\in T(x_0)$.
\end{lemma}
\vspace{0.5cm}
\begin{lemma}
\label{Semicontinuity}
Let $K$ and $K_1$ be two compact topological spaces and $T$ be a
multivalued mapping from $K$ into $K_1$ with $T(x)$ closed for each $x\in
K$. Then $T$ is upper semicontinuous if and only if the graph of\,
$T,\, G(T) := \left\{ [u,w]\in K\times K_1\,;\,w\in T(u)\right\}$, is a
closed subset of $K\times K_1$\,.
\end{lemma}
The proof of the last two results can be found in Browder \cite{Browder} (the
first one is Corollary 2 of Theorem 6.3), while the second is Proposition
6.2.)
\vspace{0.5cm}
$L^q(\Omega)$ will denote the usual Banach space of real valued function
defined on $\Omega$, with norm $|u|_q = (\int_\Omega |u(x)|^q dx)^{1/q}$.
When $q =2$,\ $\langle \cdot, \cdot \rangle_2$ denotes the usual inner
product in the Hilbert space $L^2(\Omega)$.
The norm of $u \in W^{1,p}_0(\Omega)$, the usual Sobolev space, will be
denoted
$ |u|_{1,p} = (\int_\Omega |\nabla u(x)|^p \, dx )^{1/p}$.
One can consult Adams \cite{Adams} for general properties of Sobolev spaces.
Here we only recall that if $p\geq 2$, then $W^{1,p}_0 (\Omega) \subset
L^2(\Omega) \subset W^{-1,p'} (\Omega)$, with continuous and dense inclusions.
To rigorously define the $p$-Laplacian, we first consider the functional
$\Phi: W^{1,p}_0 (\Omega) \rightarrow {\mathbb R}$, defined for any $u \in W^{1,p}_0
(\Omega)$ by $\Phi(u)= \frac{1}{p}|u|^p_{1,p}$.
Concerning this functional, it is known that it is of class $C^1$ and that,
for any $u$ and $v$ belonging to $W^{1,p}_0 (\Omega)$ one has
$$\langle \Phi'(u), v \rangle_{W^{-1,p'}, W^{1,p}_0}
= \int_\Omega |\nabla u(x)|^{p-2}
\langle \nabla u(x), \nabla v(x)\rangle \, dx\,$$
When $\Delta_p u \equiv \dv (|\nabla u|^{p-2} \nabla u) \in L^2(\Omega)$, we
have
$\langle \Phi'(u),v \rangle_{W^{-1,p'}, W^{1,p}_0}
= - \int_\Omega \Delta_p u(x) \cdot v(x) \, dx$.
Thus, from now on we are going to denote $\Phi'$ by $-\Delta_p$, and, for
simplicity of notation, the duality map between $W^{-1,p'}(\Omega)$ and
$W^{1,p}_0 (\Omega)$ just by $\langle \cdot, \cdot \rangle$.
It is also known that $-\Delta_p: W^{1,p}_0(\Omega) \rightarrow
W^{-1,p'}(\Omega)$ has the following properties:
\begin{description}
\item{(i)} {\it Strong Monotonicity:} There is $\alpha >0$ such that for any
$u,v \in W^{1,p}_0(\Omega)$ it holds
\begin{equation}
\label{StrongMonotonicity}
\langle (-\Delta_p)u - (-\Delta_p)v, u - v \rangle
\geq
\alpha |u-v|^p_{1,p} \, .
\end{equation}
\item{(ii)} {\it Hemicontinuity:} For any $\lambda \in {\mathbb R}$ and any
$u,v,w \in W^{1,p}_0 (\Omega)$, the following function is continuous:
\begin{equation}
\label{Hemicontinuity}
\lambda \rightarrow
\langle -\Delta_p(u + \lambda w),v \rangle \, ..
\end{equation}
\item{(iii)} {\it Coercivity:} For any $ u \in W^{1,p}_0 (\Omega)$, it holds
that
\begin{equation}
\label{Coercivity}
\langle -\Delta_p u, u \rangle = |u|^p_{1,p} \, .
\end{equation}
\item{(iv)} {\it Boundness:} For any $u \in W^{1,p}_0 (\Omega)$, it is true
\begin{equation}
\label{Boundness}
|- \Delta_p u|_{W^{-1,p'}} \leq |u|^{p-1}_{1,p} \, .
\end{equation}
\end{description}
Properties (\ref{Coercivity}) and (\ref{Boundness}) easily follow from the
definition of $\Delta_p$; (\ref{Hemicontinuity}) follows from the fact that
$\Phi \in C^1(W_0^{1,p}(\Omega))$. (\ref{StrongMonotonicity}) follows from
Tolksdorf \cite{Tolksdorf}, Section 2, Lemma 1.
\vspace{0.5cm}
We also need to clarify the meaning of weak and strong solutions:
\vspace{0.5cm}
\noindent
{\bf Definitions:}
\begin{description}
\item{(i)}
Let $f \in L^{p'}(0,T;W^{-1,p'}(\Omega))$ and $u_0 \in L^2(\Omega)$. We say
that $u$ is a {\em weak solution} of
\begin{equation}
\label{IVAlmostLinearizedEquation}
\begin{array}{lcl}
& u_t-\Delta_p u = f \, , & \\
& u|_{\partial \Omega} = 0 \, , & \\
& u(0) = u_0 \, , &
\end{array}
\end{equation}
when $u \in W = \{w \in L^p(0,T;W^{1,p}_0(\Omega)); w_t \in
L^{p'}(0,T;W^{-1,p'}(\Omega)) \}$, $u$ satisfies
(\ref{IVAlmostLinearizedEquation}) in $L^{p'}(0,T;W^{-1,p'}(\Omega))$ and
$u(0)=u_0$ (which makes sense because $W \subset C([0,T];L^2(\Omega))$).
\item{(ii)}
In the case where $f \in L^2(0,T;L^2(\Omega))$, $u$ is called a {\em strong
solution} of (\ref{IVAlmostLinearizedEquation}) if it is a weak solution ,
$u_t \in L^2(0,T;L^2(\Omega))$, and (\ref{IVAlmostLinearizedEquation}) is
satisfied in $L^2(0,T;L^2(\Omega))$.
\end{description}
\vspace{0.5cm}
We will also use the T-periodic ``almost linearized" problem associated
to the previous initial value problem:
\begin{equation}
\label{PeriodicAlmostLinearizedEquation}
\begin{array}{lcl}
& u_t - \Delta_p u = f(x,t) \quad\quad \mbox{in}\quad \Omega \, , & \\
& u|_{\partial\Omega} = 0 \, , & \\
& u(.,0) = u(.,T) \, , &
\end{array}
\end{equation}
Concerning this problem, the following result that can be found in Lions
\cite{Lions}, Chapter 2, Section 7.4, p. 236:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
\label{PeriodicAlmostLinearizedExistence}
Let $f \in L^{p'}(0,T;W^{-1,p'}(\Omega)), \ p\geq 2$ and
\begin{equation}
\label{W}
W = \{ w \in L^p(0,T;W^{1,p}_0(\Omega)); \ \ w_t \in
L^{p'}(0,T;W^{-1,p}(\Omega)\}.
\end{equation}
Then, there is a unique weak solution $u \in W$ of
(\ref{PeriodicAlmostLinearizedEquation}). In particular,
when $f$ is $T$-periodic, $u$ is also $T$-periodic.
\end{lemma}
This lemma allows the consideration of the Solution Operator associated to
the $T$-periodic ``almost linearized" Problem
\ref{PeriodicAlmostLinearizedEquation}. We will need certain properties
of this solution operator acting in suitable functional spaces. In the
following we will state two of this sort of result; their proofs are quite
classical, and so they will only be sketched.
\begin{proposition}
\label{SolutionOperatorFirstProposition}
Let $p \geq 2$ and $q >1$ such that $1 - N/p > -N/q$. Then, the solution
operator $S$ defined as
\begin{eqnarray}
\label{SolutionOperator1}
& S: L^{p'}(0,T;W^{-1,p'}(\Omega)) \rightarrow L^{p}(0,T;L^{q}(\Omega))&\\
& f \mapsto S(f) = u \, , & \nonumber
\end{eqnarray}
where $u$ is the solution of (\ref{PeriodicAlmostLinearizedEquation}), is
completely continuous. Moreover, there are positive constants $C_1$ and $C_2$
such that
\begin{eqnarray}
\label{BoundnessOfS}
&& |S(f)|_{L^pW^{1,p}_0} \leq |f|_{L^{p'}W^{-1,p'}}^{1/(p-1)} \qquad ,\\
\label{BoundnessOfSt}
&&|\frac{d}{dt} S(f)|_{L^{p'}W^{-1,p'}}
\leq C_1|f|_{L^{p'}W^{-1,p'}} \qquad ,\\
\label{ContinuityOfS}
&& |S(f)-S(g)|_{L^pW^{1,p}_0} \leq
C_2|f-g|_{L^{p'}W^{-1,p'}}^{1/(p-1)}.
\end{eqnarray}
\end{proposition}
\noindent
{\bf Proof:} By taking $v = u$ in (\ref{IntegrationByParts}), working with
classical energy estimates and using (\ref{Boundness}) we obtain
(\ref{BoundnessOfS}) to (\ref{ContinuityOfS}).
The continuous inclusion
$L^p(0,T;W^{1,p}_0(\Omega)) \subset L^p(0,T;L^q(\Omega))$
and (\ref{ContinuityOfS}) now imply that $S$ is continuous.
Moreover, for any bounded subset $B$ of $L^{p'}(0,T;W^{-1,p'}(\Omega))$, by
(\ref{BoundnessOfS}) and (\ref{BoundnessOfSt}), $S(B)$ is a bounded subset of
$W$. But by Lemma 2.1 we have the compact inclusion $W \subset
L^p(0,T;L^q(\Omega))$, then $S(B)$ is relatively compact in
$L^p(0,T;L^q(\Omega))$, and, consequently, $S$ is completely continuous.
\hfill $\Box$
\vspace{0.5cm}
The proof of the following result is similar to the previous one:
\begin{proposition}
\label{SolutionOperatorSecondProposition}
The Solution Operator $S$ associated to
(\ref{PeriodicAlmostLinearizedEquation}) is also well
defined when acting in the following spaces:
\begin{eqnarray}
\label{SolutionOperator2}
& S: L^{p'}(0,T;W^{-1,p'}(\Omega)) \rightarrow L^\infty(0,T;L^2(\Omega)) & \\
& f \mapsto S(f) = u \, .& \nonumber
\end{eqnarray}
Moreover, it is a continuous and bounded operator satisfying:
\begin{equation}
\label{FurtherContinuityOfS}
|S(f)-S(g)|_{L^\infty L^2}^2 \leq C |f - g|^{\frac{2}{p-1}}_{L^{p'}W^{-1/p'}}
+ C |f - g|^{p'}_{L^{p'}W^{-1,p'}}.
\end{equation}
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.5cm}
We remark that when $g=0$ then $S(g)=0$; consequently, this last inequality
furnishes
\begin{equation}
\label{FurtherBoundnessOfS}
|S(f)|_{L^\infty L^2}^2
\leq
C |f|^{\frac{2}{p-1}}_{L^{p'}W^{-1/p'}} + C |f|^{p'}_{L^{p'}W^{-1,p'}}.
\end{equation}
\noindent
with $C=C(\Omega, T, p, \alpha) > 0$.
\vspace{0.5cm}
Finally, we observe that in the derivations of the a priori estimates holding
in this paper, we will follow the usual procedure and denote by $C$ a generic
constant depending only on the problem data. Sometimes we will display the
dependence of these constant on the data, and, in those cases where we want
to stress the role of the constants, we will use other symbols, like $C_i$,
$D_i$, $a_i$, and so on.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Existence of Solutions in the Case of the First Model Equation}
\label{ExistenceSimpleModel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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First of all we want to briefly describe the arguments we will be using to
prove existence of periodic solutions. We will adapt rather classical
techniques: we will consider the Solution Operator associated to the
$T$-periodic ``almost linearized" problem
\ref{PeriodicAlmostLinearizedEquation} acting in suitable functional spaces
and the Nemytskii operator corresponding to the function in the right hand
side of (\ref{SpecialEquation}). Then, we will look for solutions of the
problem as fixed points of the composition of these two operators. As it is
usual in this setting, the main difficulty will be to obtain suitable a
priori estimates to guarantee that we are in the conditions of some fixed
point theorem (in our case, the Schauder Fixed Point Theorem.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.5cm}
We start by investigating the existence of weak solutions of Problem
\ref{SpecialEquation} in the case $0 \leq s < p-1$. We have:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}
\label{SpecialExistence}
Suppose that $p \geq 2$,\ $m \in L^\infty(0,T)$,\ $h \in L^{p'}(0,T;
W^{-1,p'}(\Omega))$, and $g: {\mathbb R} \rightarrow {\mathbb R}$ is a continuous function
such that for some $a > 0$ and $s \in [0,p-1)$ there holds $|g(v)|\leq a
(|v|^{s}+1)$. Then, Problem (\ref{SpecialEquation}) has a weak solution.
\end{theorem}
\noindent
{\bf Proof:} The Nemytskii operator, $N_H$, associated to the function
$H(u,x,t) = m(t) g(u)+h(x,t)$,which is defined by $N_H(u)(x,t)=H(u(x,t),x,t)=
m(t) g(u(x,t))+h(x,t)$, is a continuous and bounded map from
$L^p(0,T;L^{p}(\Omega)$ to $L^{p'}(0,T;W^{-1,p'}(\Omega))$.
On the other hand, because $0 \leq s0$, when $|u|_{L^pL^p} \leq R$ then
also $|S \circ N_H(u)|_{L^pL^p} \leq R$. Fixing such $R$, by restricting now
to a closed ball of radius $R$ in $L^p(0,T;L^p(\Omega))$, we can apply
Schauder Fixed Point Theorem and conclude that there is $u \in W$ satisfying
the abstract equations $u_t - \Delta_p u = N_H(u)$ and $u(0)=u(T)$, that is,
there is a solution of (1.3).
\hfill $\Box$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.5cm}
Now, we will consider equations (\ref{SpecialEquation}) in the case $s=p-1$.
Then, it holds:
\begin{theorem}
\label{Case s=p-1}
Let $p \geq 2$, $m$, $h$ and $g$ as in Theorem \ref{SpecialExistence}, with
$s=p-1$.
Then, if $|m|_{L^\infty}$ is sufficiently small, (\ref{SpecialEquation}) has
a weak solution.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
{\bf Proof:} Following the proof of Theorem \ref{SpecialExistence}, we
observe that in this case we also have that
$S \circ N_H: L^p(0,T;L^p(\Omega)) \rightarrow L^p(0,T;L^p(\Omega))$
is a completely continuous operator.
However, (\ref{BoundnessOfSoNH}) reduces to
$|S \circ N_H(u)|_{L^pL^p} \leq a_3 |u|_{L^pL^p} +a_4$.
If we had $0 < a_3 < 1$, we could take
$R \geq {\displaystyle\frac{a_4}{1-a_3}}$ to obtain $a_3 R + a_4 \leq R$,
and the rest of the argument used in Theorem \ref{SpecialExistence} could be
applied, furnishing the existence of the required solution.
But the condition $0 < a_3 < 1$ is attained when $\|m\|_{L^\infty}$ is small
enough since, from (\ref{a1a2}) and (\ref{a3a4}), $a_3$
approaches zero as $|m|_{L^\infty}$ approaches zero.
\hfill $\Box$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.5cm}
Now we will study the existence of solutions of \ref{SpecialEquation} when
$s > p-1$, \ $p \geq 2$.
This case will require a modified treatment. In fact, let us briefly describe
what would be the trouble if one had proceeded with the argument exactly as
before. Then, one would observe that when $a_3$ were small enough (which as
previously could be attained when $|m|_{L^\infty}$ is sufficiently small), it
would still be possible to find $R >0$ such that $a_3 R^{s/(p-1)} + a_4 \leq
R$ (observe (\ref{BoundnessOfSoNH}) in the case $s > p-1$.) However, in this
case it would not be true that $L^{p/s}(0,T;L^{q/s}(\Omega))
\subset L^{p'}(0,T;W^{-1,p'}(\Omega))$ for $q>1$. Consequently, the
composition $S\circ N_H$ could not be defined in the previous functional
spaces.
Thus, to prove the existence of solutions when $s > p-1$, we will have to
explore the additional properties of the Solution Operator $S$ given by
Proposition \ref{SolutionOperatorSecondProposition} in order to define $S
\circ N_H$ in other suitable spaces. By doing this, we will be able to prove
following result.
\begin{theorem}
\label{SupercriticalExistence}
Let $m$, $h$ and $g$ as in Theorem \ref{SpecialExistence}. Let also $p \geq
2$ and $p-1 < s < p-1 +\Gamma$ with either $\Gamma = 2 p /N$ when $N > p$ or
$\Gamma=2$ when $N \leq p$. Then, if \ \ $|g(v)| \leq a (|v|^{s}+1)$ and \ \
$|m|_{L^\infty}$ is sufficiently small, (\ref{SpecialEquation}) has a weak
solution.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent
{\bf Proof:} The argument is as follows: we will choose suitable $k,r >1$
such that the operator
$S \circ N_H: L^k(0,T;L^r(\Omega)) \rightarrow L^k(0,T;L^r(\Omega))$
is completly continuous. Then, with similar arguments to the ones used in
Theorem \ref{Case s=p-1}, by choosing $|m|_\infty$ small enough, we will
obtain a fixed point for $S \circ N_H$ and, thus, the desired solution.
We start by recalling that, by Propositions
\ref{SolutionOperatorFirstProposition} and
\ref{SolutionOperatorSecondProposition}, the Solution Operator
$S$ associated to problem \ref{PeriodicAlmostLinearizedEquation} is a
continuous and bounded operator from $L^{p'}(0,T;W^{-1,p'}(\Omega))$ to
either
$L^\infty(0,T;L^2(\Omega))$ or
$L^p(0,T,L^q(\Omega))$
(with $q$ satisfying $\displaystyle 1- \frac{N}{p} > -\frac{N}{q}\,$ and to
be chosen later on) as the image space.
By interpolation,
$X=L^\infty(0,T;L^2(\Omega)) \cap L^p(0,T,L^q(\Omega))$ is
continuously imbedded in $L^{k(\theta)}(0,T;L^{r(\theta,q)}(\Omega))$,
for any $\theta \in [0,1]$ and
\begin{equation}
\label{kr}
k(\theta) = p/\theta \quad \mbox{and} \quad
r(\theta,q) = q/ [q(1-\theta) + 2 \theta].
\end{equation}
Moreover, for any $u \in X$ we have
\begin{equation}
\label{InterpolationInequality}
|u|_{L^{k(\theta)}L^{r(\theta,q)}} \leq |u|^\theta_{L^pL^q}
|u|^{1-\theta}_{L^\infty L^2}.
\end{equation}
This implies that, for $\theta \in (0,1]$,
\begin{eqnarray}
\label{SolutionOperatorThird}
& S: L^{p'}(0,T;W^{-1,p'}(\Omega)) \rightarrow
L^{k(\theta)}(0,T;L^{r(\theta,q)}(\Omega)) &\\
& f \mapsto S(f)=u & \nonumber
\end{eqnarray}
is a completely continuous operator. Moreover, by using (\ref{BoundnessOfS}),
(\ref{FurtherBoundnessOfS}) and (\ref{InterpolationInequality}), we obtain:
\begin{equation}
\label{StillFurtherBoundnessOfS}
|S(f)|_{L^{k(\theta)}L^{r(\theta,q)}} \leq
|f|_{L^{p'}W^{-1,p'}}^{1/(p-1)}
[ c_5 + c_6 |f|_{L^{p'}W^{-1,p'}}^{p/2} ]^{(1-\theta)/2}.
\end{equation}
Now, we are going to choose a suitable $\theta_0 \in (0,1]$ such that when
$p-1 < s < p-1 + \Gamma$ we have
$L^{k(\theta)/s}(0,T;L^{r(\theta,q)/s}(\Omega)) \subset
L^{p'}(0,T;W^{-1,p'}(\Omega))$
continuously.
For this, we observe that if we require
\[
s \leq \min \{ f_1(\theta) = (p-1)/ \theta \ , \ f_2(\theta) =
2(q-1)/[q(1-\theta )+2\theta ] \} = s(\theta) ,
\]
then, by (\ref{kr}), we have
\begin{eqnarray*}
& k(\theta)/s = p/(\theta s) \geq p'= p/(p-1) \, , & \\
& r(\theta,q)/s = 2 q / \{ [ q(1-\theta ) + 2 \theta ] s \} \geq q' \, , &
\end{eqnarray*}
and, thus, the required continuous inclusion
$L^{k/s}(0,T,L^{r/s}(\Omega) \subset L^{p'}(0,T;W^{-1,p'}(\Omega)$.
By analyzing the expressions of $f_1(\theta)$ and $f_2(\theta)$ it is
easy to conclude that
\begin{equation}
\label{theta_0}
\theta_0(q) = \frac{q (p+1) -2 q}{q (p+1) -2 p}\in (0,1),
\end{equation}
furnishes the highest possible superior bound for $s(\theta)$ (under the
required restrictions).
That is, we must take
$s \leq s(\theta) = {\displaystyle\frac{p-1}{\theta_0(q)}= p
+ 1 - 2\,\frac{p}{q}} = p + 1 + \Gamma$, with $q$ satisfying
$\displaystyle 1-\frac{N}{p} > \frac{N}{q}$. This gives us
the conditions on $\Gamma$ described in the statement of the theorem. With
$\theta_0(q)$ given by (\ref{theta_0}), we can now take the corresponding
values of $k(\theta_0)$ and $r(\theta_0,q)$ \, given by the expressions in
(\ref{kr}). Thus, $S$ defined as in (\ref{SolutionOperatorThird}) will be
continuous. The Nemytskii operator $N_H: L^k(0,T;L^r(\Omega)) \rightarrow
L^{p'}(0,T;W^{-1,p'}(\Omega))$ will also be continuous and bounded in the
sense that, with $C=C(\Omega,T,k,s,a)>0$,
\begin{equation}
\label{OtherEstimateForNH}
|N_H(u)|_{L^{p'} W^{-1,p'}} \leq C |m|_{L^\infty}
(|u|^s_{L^kL^r} + 1) + |h|_{L^{p'} W^{-1,p'}}.
\end{equation}
We conclude that $S \circ N_H: L^k(0,T;L^r(\Omega)) \rightarrow
L^k(0,T;L^r(\Omega))$ is, as in the previous cases, completely continuous.
Now, from (\ref{StillFurtherBoundnessOfS}) and (\ref{OtherEstimateForNH}),
with the help of Young inequality, we obtain that, when $|u|_{L^kL^r} \leq
R$ for any fixed $R>0$, then
\begin{equation}
\label{SNH}
|S \circ N_H(u)|_{L^kL^r} \leq D_1 R^{s ( \zeta_1 + \zeta_2 )} + D_2,
\end{equation}
where
\[
\begin{array}{l}
\zeta_1 = 1/(p-1),\ \quad \quad
\zeta_2 = (p (1-\theta_0)+ 2\theta_0)/ 2(p-1), \\
\\
D_1= C^{1-\theta_0} (C|m|_{L^\infty})^{\zeta_1}(\zeta_1/\zeta_2)
+ C^{1-\theta_0}(C|m|_{L^\infty})^{\zeta_2}\\
\\
D_2= C^{1-\theta_0}
\{
(C|m|_{L^\infty})^{\zeta_1}
[(\zeta_2 - \zeta_1)/\zeta_2]
+(C(|m|_{L^\infty}
+|h|_{L^{p'}W^{-1,p'}})
\}^{\zeta_1}\\
\qquad + C^{(1-\theta_0)} C^{\zeta_2}
(|m|_{L^\infty} + |h|_{L^{p'}W^{-1,p'}})^{\zeta_2},
\end{array}
\]
again with $C=C(\Omega,p,q,T,a)>0$.
We observe that because $s > p-1$ then $s(\zeta_1 + \zeta_2) > 1$.
So, proceeding as in the final part of the proof of Theorem \ref{Case s=p-1},
we start by fixing $R>0$ large enough such that $D_2 \leq R/2$; then, we
observe that as $|m|_{L^\infty} \rightarrow 0$, we also have $D_1 \rightarrow
0$. Thus, by taking $|m|_{L^\infty}$ small enough, we have
$D_1 R^{s(\zeta_1 + \zeta_2)} \leq R/2$. Under these conditions, (\ref{SNH})
implies that $S \circ N_H(\overline{B_R(0)}) \subset \overline{B_R(0)}$,
where $\overline{B_R(0)}$ is the ball of radius $R$ and center $0$ in
$L^k(0,T;L^r(\Omega)$.
Thus using Schauder Fixed Point Theorem we conclude that $S \circ N_H$ has
a fixed point, and consequently, (\ref{SpecialEquation}) has a weak solution.
\hfill $\Box$
\vspace{0.5cm}
\noindent
{\bf Remark:}
We must stress that the above results are concerned with the existence of
{\it forced periodic solutions}. In particular, if $h$ were identically zero
(recall the observation just after equation (\ref{SpecialEquation},) from the
above results we could not conclude that the constructed solution is
nontrivial (i.e., $u \neq 0$.) This is the reason for the fact that, when
$p=2$, the above stated range of allowed $s$ is larger than the corresponding
range found when one is looking for nontrivial solutions in the case
$g(u)=u^s$ and $h=0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{An Abstract Result}
\label{Abstract Result}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The results stated in the last section hold for more general dissipative
operators sharing with $\Delta_p$ certain properties. We state below the
abstract version of these previous results, omitting their proofs since they
follow exactly as before.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}
\label{AbstractExistence}
Suppose $V \subset H \subset V'$ with continuous and dense inclusions, and
let $A$ be a monotonic and hemicontinuous operator $A: V \rightarrow V'$ for
which there are constants
$\alpha_1 > 0$, $\alpha_2 >0$, $\beta \geq 0$,
$\gamma_1 \geq 0$ and $\gamma_2 \geq 0$
such that for any $u, v \in V$ it holds
$\langle Au - Av, u-v \rangle_{V'V} \geq \alpha_1 |u-v|_V^p$ \ ,
$\langle Au,u \rangle_{V'V} \geq \alpha_2 |u|_V ^p - \beta$ \ and
$|Au|_{V'} \leq \gamma_1|u|_V^{p-1} + \gamma_2$.
\noindent
Let also $X$ be a Banach space such that $V \subset X \subset V'$ with dense
and continuous inclusions and $V \subset X$ compactly. Moreover, let $F:
L^p(0,T;X) \rightarrow L^{p'}(0,T;V')$ be a continuous and bounded map
satisfying
\[
|F(u)|_{L^{p'}V'} \leq k_1 |u|_{L^pX}^s + k_2,
\]
for some non negative constants $k_1$ and $k_2$. Then the problem
\[
\begin{array}{lcl}
& u_t + Au = F(u) \, , & \\
& u(0) = u(T)
\end{array}
\]
has a weak solution when one of the following conditions is satisfied:
\begin{description}
\item{(i)} When $0 \leq s < p-1$.
\item{(ii)} When $s=p-1$ and $k_1$ is sufficiently small.
\item{(iii)} $V = W^{1,p}_0(\Omega)$, $X=L^q(\Omega)$ with
$1-(N/p) > -(N/q)$, \, $k_1$ is small enough and $p-1 < s < p-1 +
\Gamma$, with either $\Gamma = 2 p /N$ when $N > p$ or $\Gamma=2$ when $N
\leq p$.
\end{description}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.5cm}
\noindent
{\bf Examples of problems for which the abstract result applies:}
\begin{description}
\item{(a)} Let $p$ and $m$, $g$, $h$ as in Theorem \ref{SpecialExistence}
and a function $c \in C^1({\mathbb R})$ satisfying $c'(t) \geq 0$ and also $\delta
t^{p-1} \leq c(t^p) t^{p-1} \leq \alpha t^{p-1} + \beta$ for all $t
\in {\mathbb R}$ and certain positive constants $\alpha$, $\beta$, $\delta$. Then,
consider the problem:
\[
\begin{array}{lcl}
& u_t - \dv (c(|\nabla u|^p) |\nabla u|^{p-2} \nabla u) = m(t) g(u) + h(t,x)
\, , & \\
& u|_{\partial \Omega} = 0 \, , & \\
& u(\cdot, 0) = u(\cdot, T) \, .
\end{array}
\]
\item{(b)} Let $m$, $g$, $h$ as in Theorem \ref{SpecialExistence} and $p
> p_2 > p_1 > 2$ and $ \lambda, \lambda_1, \lambda_2 >0$ such that $p=2 p_2 -
p_1$, $\lambda_2^2 \leq 4 \lambda_1 \lambda$ and $[\lambda_2 (p_2 -1)]^2 - 4
\lambda \lambda_1 (p-1)(p_1-1) \geq 0$. Then, consider the problem:
\[
\begin{array}{lcl}
& u_t - \lambda \Delta_p u + \lambda_2 \Delta_{p_2} u
- \lambda_1 \Delta_{p_1} u = m(t) g(u) + h(x,t) \, & \\
& u|_{\partial \Omega} = 0 \, , & \\
& u(\cdot, 0) = u(\cdot, T) \, . &
\end{array}
\]
\end{description}
To prove that the operators corresponding to above two problems are strongly
monotone, one can use Lemma 1 in Section 2 in Tolksdorf \cite{Tolksdorf}. The
other properties required in the last theorem are easily proved. The above
smallness condition on $k_1$, when it applies, corresponds in the examples to
a smallness condition on $|m|_{L^\infty}$, as in the results of the previous
sections. Thus, for the above examples we can obtain the corresponding
existence results of $T$-periodic weak solutions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Existence of Solutions of the Model with Abrupt Changes}
\label{Abrupt Changes}
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Now we will consider the Problem \ref{Abrupt}.
\begin{theorem}
\label{AbruptExistence}
Let
$\overline t \in (0,T)$,
$I_1=[0,\overline{t}]$, and
$I_2=[\overline{t},T]$.
Suppose that for each $i=1,2$,
$m_i \in L^\infty(I_i)$,
$h_i\in L^{p'_i}(I_i\,;\,W^{-1,p'_i}(\Omega))$ and
$g_i$ is a continuous real
function satisfying
$|g_i(v)| \leq a(|v_i|^{s_i}+1)$.
Suppose also that one of the following three conditions is true.
\begin{description}
\item{(i)} $0 \leq s_i < p_i - 1$, \ for \ $i=1,2$.
\item{(ii)} $p_i-1 \leq s_i < p_i + 1$ \ for \ $i=1,2$, \ $N \leq p_i$ and
$\displaystyle\max_{i=1,2}|m_i|_{L^\infty}$ is small enough.
\item{(iii)} $p_i-1 \leq s_i < p_i - 1 + 2 \displaystyle\frac{p_i}{N}$
\ for \ $i=1,2$, \ $N > p_i$ and
$\displaystyle\max_{i=1,2}|m_i|_{L^\infty}$ is small enough.
\end{description}
Then (\ref{Abrupt}) has a weak solution.
\end{theorem}
\noindent
{\bf Proof:} We will just sketch the proof since after constructing a
suitable solution operator it uses arguments similar to those used in the
previous theorems.
One starts by defining the following spaces:
$L_i = L^{p_i}(I_i\,;\,W^{1,p_i}_0(\Omega))$,
$L'_i = L^{p'_i}(I_i\,;\,W^{-1,p_i'}(\Omega))$,
$W_i = \left\{ w \in L_i \,;\, w_t \in L'_i\right\}$.
Then one considers the following problem: given
$(f_1,f_2) \in L_1'\times L_2'$, find $(u_1,u_2) \in W_1 \times W_2$
such that
\begin{equation}
\label{AbruptQuasilinearization}
\begin{array}{lcl}
&(u_1)_t(t,x) - \Delta_{p_1}u_1(t,x) = f_1(t,x),
\quad \mbox{for} \quad (t,x)\in I_1\times\Omega \, , & \\
& u_1(t,\cdot)|_{\partial\Omega} = 0, \quad \mbox{for} \quad t\in I_ 1 \, ,
& \\
& (u_2)_t(t,x) - \Delta_{p_2}u_2(t,x) = f_2(t,x),
\quad \mbox{for} \quad (t,x)\in I_2\times\Omega \, , & \\
& u_2(t,\cdot)|_{\partial\Omega} = 0, \quad \mbox{for} \quad t\in I_2 \, ,
& \\
& u_1(0,x) = u_2(T,x), \quad \mbox{for} \quad x\in \Omega \, , & \\
& u_1(\overline t,x) = u_2(\overline t,x). \quad \mbox{for} \quad
x \in \Omega \, . &
\end{array}
\end{equation}
Observe that the last condition make sense because
\[
\begin{array}{ll}
W_1 \times W_2 & \subset
C(I_1;W^{-1,p_i'}(\Omega) )
\times
C(I_2;W^{-1,p_2'}(\Omega)) \\
& \subset
C(I_1;W^{-1, \min \{ p_1', p_2' \} }(\Omega) )
\times
C(I_2;W^{-1, \min \{ p_1', p_2' \} }(\Omega)).
\end{array}
\]
To prove that Problem \ref{AbruptQuasilinearization} has a unique solution,
one can proceed as follows:
consider the Poincare map ${\cal K}$ associated to
$(u_i)_t - \Delta_{p_i} u = f_i$
for $t \in I_i$, $u_i(0)=u_0 \in L^2(\Omega)$,
$u_2( \overline{t}) = u_1( \overline{t}) $.
By using the monotonicity and coercivity of
$\Delta_{p_i}$, one proves that ${\cal K}$ is a non-strict contraction and
that there is a sufficiently large $R>0$ so that
${\cal K}({\bar B_R(0)}) \subset {\bar B_R(0)}$.
So according to Lemma \ref{FixedPointContraction}, ${\cal K}$ has a fixed
point. Using the monotonicity again, one obtains that is fixed point is
unique.
This defines a solution operator for problem \ref{AbruptQuasilinearization}:
$ S : L'_1 \times L'_2 \rightarrow W_1 \times W_2$, where
$S(h_1,h_2)=(u_1,u_2)$.
Like in the previous theorems, it is then possible to show that $S$ is
completely continuous and, moreover, that for $i=1,2$, with suitable positive
constants $C_i$, $\alpha_i$, $i=1,2$, the following estimates are true:
\begin{eqnarray*}
& \sum _{i=1,2} \alpha_i/ p_i |u_i -v_i |
\leq \sum C_i |f_i - h_i|^{p'_i} \,, & \\
& \sum _{i=1,2} \alpha_i/ p_i |u_i|^p_i
\leq \sum C_i|f_i|^{p'_i} \, , & \\
& \displaystyle |\frac{d}{dt} u_i|
\leq |f_i|_{L'_i} + (p_i/ \alpha_i)^{1/p'_i}
(C_1|f_1|^{p'_1} + C_2|f_2|^{p'_2})^{1/p'_i} \, . &
\end{eqnarray*}
By introducing the natural Nemytskii operators, $N_{H_1}$ and $N_{H_2}$,
corresponding respectively to the second members of (\ref{Abrupt}), with the
help of the above estimates, one proves that $N_{H_1} \times N_{H_2}: W_1
\times W_2 \rightarrow L'_1 \times L'_2$ is continuous.
Now, working with $S \circ (N_{H_1} \times N_{H_2}) $ and repeating the
arguments done in the proofs of the previous theorems, we obtain a fixed
point of $S \circ (N_{H_1} \times N_{H_2}) $ and then the stated result.
\hfill $\Box$
\vspace{0.5cm}
\noindent
{\bf Remarks:}
\begin{description}
\item{(i)} As in Section \ref{Abstract Result}, it is possible to state a
result on the existence of periodic solutions for abstract operators $A_i$
having properties similar to the ones of $\Delta_{p_i}$.
\item{(ii)} Adapting ideas of Kawohl and R\"uhl \cite{KawohlRuhl}, it is
possible to prove existence of periodic solutions of the following problem
involving abrupt changes:
\[
\begin{array}{lcl}
& u_t - \Delta_{p_i} + B_i(t,u) = 0, \quad \mbox{for} \quad
(t,x)\in I_i \times \Omega = Q_i\, , \quad i=1,2 \, , & \\
& -\displaystyle\frac{\partial u}{\partial\eta} \in \beta_i(u),
\quad \mbox{for} \quad (t,x)\in I_i \times\partial\Omega \,, & \\
& u(0,x) = u(T,x) \, , \quad \mbox{for} \quad x \in \Omega \, , &
\end{array}
\]
where the $B_i$\,, $i=1,2,$ are coercive maximal monotonic operators such
that $B_i(t,\cdot)$\,, $i=1,2,$ are locally Lipschitizian in $L_2(\Omega)$.
We remark that the presence of such operators in the above equations prevents
the system to be dissipative all the time. For the proof of this result, see
\cite{CremaBoldrini}.
\end{description}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Commentaries on Strong Solutions}
\label{Remarks}
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With further restrictions on the problem data, it is possible to obtain
strong solutions of the previous problems. Let us comment on this.
When $h \in L^2(0,T;L^2(\Omega))$ instead of $L^{p'}(0,T;W^{-1,p'}(\Omega))$,
one can repeat the arguments done in the proof of Theorem
\ref{SupercriticalExistence} to find exponents $k(\theta)$ and $r(\theta,q)$
\, (as in (\ref{kr})) so that $L^{k/s}(0,T;L^{r/s}(\Omega))\subset
L^2(0,T;L^2(\Omega))$. Then, similarly as before, we could look for a fixed
point of $S\circ N_H : L^k(0,T; L^r(\Omega)) \rightarrow L^k(0,T;
L^r(\Omega))$, with $S(v) = u$.
However, by considering $S\circ N_H$ acting on such $L^k(0,T; L^r(\Omega))$
one does not obtain the best possible result concerning superior bounds for
$s$.
To obtain better results, we proceed as follows:
we observe that, when $h\in L^2(0,T;L^2(\Omega))$, Brezis [2], Chapter 3,
Theorem 3.6 furnishes the existence of a strong solution
of(\ref{PeriodicAlmostLinearizedEquation}) such that $u_t\in
L^2(0,T;L^2(\Omega))$ and, moreover, the mapping $[0,T]\ni
t\mapsto\Phi(u(t)) = \int_\Omega|\nabla u(t,x)|^p \, dx$ is
continuous. In particular, we conclude that for
$ 1- (N/p) > -(N/q)$, the solution belongs to
$ \widetilde{W} = \left\{ w\in L^q (0,T;W^{1,p}(\Omega))\,,\,w_t\in
L^2(0,T;L^2(\Omega))\right\}$.
Hence, for all $s$ so that $0\leq s\leq q/2$, we can consider
$ S\circ N_H : \widetilde{W} \mapsto \widetilde{W}$
>From this argument, we have gained regularity for $u$; but the operator
$S\circ N_H$ is not completely continuous. In fact, it is not even continuous
when one considers the natural norm topology of $\widetilde{W}$. Therefore,
it is not possible proceed exactly as before to obtain a fixed point.
However, as in \^Otani \cite{Otani}, by considering weak topologies and using
Lemma \ref{Semicontinuity}, it is possible to show that $S \circ N_H$ is
semi-continuous and satisfies one can verify that $S o N_H$ does satisfy the
hypotheses of the Tychonoff-Schauder Fixed Point Theorem (Lemma
\ref{Tychonoff-Schauder}) and conclude that:
\begin{theorem}
\label{StrongerResult}
Let $h\in L^2(0,T;L^2(\Omega))$ and suppose that either $0\leq s\leq p/2$ or
$p/2 < s \leq q/2$ with $1- (N/p) > -(N/q)$ and $|m|_{L^\infty}$ is small
enough. Then (\ref{SpecialEquation}) has a strong solution.
\end{theorem}
\noindent
{\bf Remarks:}
\begin{description}
\item{(i)} With the obvious modifications, there holds a result similar to
Theorem \ref{StrongerResult} concerning strong solutions of (\ref{Abrupt}).
\item{(ii)} When $p=2$ (that is, when the principal part of the operator
is linear), $\Omega$ is of class $C^2$, for instance, and $h\in
L^2(0,T;L^2(\Omega))$, it is possible to prove that there are strong periodic
solutions when $1 \leq s \leq \frac {N+2}{N-2}$, in the case $N>2$. This
result, which gives an improved upper bound for $s$ as compared to that given
in Theorem \ref{StrongerResult}, can be obtained by using the same arguments
as above, together with the better regularizing properties the Laplacian
operator (Dautray, Lions \cite{DautrayLions}, for instance) to embed the
solution in higher order Sobolev spaces.
\item{(iii)} We recall that in the proofs of the existence results for
weak solutions presented in the previous sections, we used the usual Schauder
Fixed Point Theorem instead of using the Tychonov-Schauder Fixed Point
Theorem as we did above. Hence, one could wonder whether the use of Lemma
2.2 and weak topologies could give increased upper bounds for $s$ in the
results in the previous sections. This would be indeed the case if we had a
$L^\infty(0,T;W^{1,p}_0(\Omega))$-estimate for $u= S\circ N_H(v)$.
Unfortunately, we were not able to prove such estimate in the case of weak
solutions, and, only with the presented estimates, the use of
Tychonov-Schauder Fixed Point does not improve the values of $s$.
\item{(iv)}
As we commented in the Introduction, \^Otani in \cite{Otani} established a
rather general existence result on existence of strong periodic solutions of
abstract equations. This could be particularized to equation
(\ref{SpecialEquation}), furnishing thus the existence of certain strong
periodic solutions. However, compared to our result, \^Otani's require more
regular conditions on $h$ and, moreover, a certain smallness condition on it.
This implies that the amplitude of periodic solutions found in \cite{Otani}
are also small (in suitable norms). On the other hand, in our results $h$ can
be less regular (in which case we obtain a weak solution), and we do not
impose any smallness condition on $h$ (to be fair, we impose instead a
smallness condition on $|m|_{L^\infty}$, but only in the cases where it is
necessary). Thus, our periodic solutions (both weak and strong) are found in
a large enough ball, and in fact, it is easy to prove that when $h$ is large,
our solutions are also large (in suitable norms). Therefore, our solutions,
even in the case of strong ones, are different from the ones obtained by the
particularization of \^Otani's results.
Another aspect that deserve attention is that since \^Otani's results allow
for certain time-dependent operators, one could try to apply them to Problem
\ref{Abrupt}. However, due to the required hypotheses in \^Otani's theorem,
this is only true in the special case that $h_i\in L^2((0,T)\times\Omega)$,
with small enough norms (thus furnishing a strong periodic solution), and,
moreover, $p1=p2$. Therefore, the results in \cite{Otani} rules out
situations where there are changes in the type of internal dissipation, as in
(\ref{Abrupt}).
\end{description}
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\bigskip
{\sc Jos\'e Luiz Boldrini}\\
UNICAMP-IMECC; CP 6065; 13081-980 Campinas, SP, Brazil.\\
E-mail address: boldrini@ime.unicamp.br \bigskip
{\sc Janete Crema}\\
USP - ICMSC; CP 668, 13560-970 S\~ao Carlos, SP, Brazil
\end{document}