On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions

Abstract

In this paper we analyze the porous medium equation \begin{equation}\label{ProblemAbstract} \tag{$◊$} %\begin{cases} u_t=Δu^m + a\io u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad \textrm{in}\quad Ω\times I,%\\ %u_ν-g(u)=0 & \textrm{on}\; \partial Ω, t>0,\\ %u({\bf x},0)=u_0({\bf x})&{\bf x} \in Ω,\\ %\end{cases} \end{equation} where $\Omega$ is a bounded and smooth domain of ${\mathbb{R}}^N$, with $N\ge 1$, and $I=[0,t^{\ast })$ is the maximal interval of existence for $u$. The constants $a,b,c$ are positive, $m,p,q$ proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of $u$. Under some hypothesis on the data, including intrinsic relations between $m,p$ and $q$, and assuming that for some positive and sufficiently regular function $u_0(\nx)$ the Initial Boundary Value Problem (IBVP) associated to \eqref{ProblemAbstract} possesses a positive classical solution $u=u(\nx,t)$ on $\Omega \times I$: \begin{itemize} \item [$▹$] when $p>q$ and in 2- and 3-dimensional domains, we determine a \textit{lower bound of} $t^{\ast }$ for those $u$ becoming unbounded in $L^{m(p-1)}(\Omega )$ at such $t^{\ast }$; \item [$▹$] when $p<q$ and in $N$-dimensional settings, we establish a \textit{global existence criterion} for $u$. \end{itemize}