Stability and instability in Periodic Parabolic Problems via Lower and Upper Solutions

This monograph is devoted to a qualitative study of the parabolic boundary value problem \begin{equation}\label{0-1.1}\begin{array}{cl}\partial_t u + \AAA u = f(x,t,u, \nabla_x u), & \mbox{in }{\Omega\times {]0,+\infty[}},\\\BBB u=0,& \mbox{on } \partial\Omega\times {[0,+\infty[},\end{array}\end{equation} in the presence of lower and upper solutions. Here $\Omega$ is a bounded domain in $\RR^N$, $N\ge1$, $\partial_t +\AAA$ is a linear second order uniformly parabolic operator and $\BBB$ is a linear first order mixed boundary operator, namely Dirichlet on one component of $\partial \Omega$ and Robin on the other. The coefficients of $A$ and $B$ are $T-$periodicin $t$, $T$ being a given period. The function $f : \Omega \times {]0,+\infty[} \times \RR \times \RR^N \to \RR$ is $T-$periodic in $t$ and satisfies the $L^p-$Carath\'eo\-dory conditions, for some $p>N+2$, as well as a Nagumo condition, that is, $f(x,t,s,\xi)$ grows at most quadratically with respect to $\xi$. According to these assumptions, solutions of (\ref{0-1.1}) are intended in the strong, i.e., $W^{2,1}_p-$, sense. Assuming that a lower solution $\alpha$ and an upper solution$\beta$ of the ($T-$)periodic problem associated with (\ref{0-1.1}) are given, we face the following two basic questions: \noindent\hangindent8mm\makebox[8mm][l]{$(i)$} existence of periodic solutions of (\ref{0-1.1}) and their localization; \noindent\hangindent8mm\makebox[8mm][l]{$(ii)$} qualitative properties of periodic solutions of (\ref{0-1.1}) with special reference to their stability or instability.