Doubly nonlinear parabolic equation involving a mixed local-nonlocal operator and a convection term

TL;DR

This paper investigates a complex doubly degenerate parabolic equation combining local and nonlocal operators, analyzing existence, uniqueness, and various long-term behaviors of solutions under different nonlinear conditions.

Contribution

It introduces the concept of weak-mild solutions for a mixed local-nonlocal operator equation and studies their qualitative behaviors including stabilization, extinction, and blow-up.

Findings

Existence and uniqueness of weak-mild solutions established.

Conditions for stabilization, extinction, and blow-up identified.

Behavior depends on nonlinearities and source term properties.

Abstract

In this paper we study a doubly degenerate parabolic equation involving a convection term and the operator $\mathcal{A}_\mu u:=-\Delta_p u +\mu (-\Delta)^s_q u$ which is a linear combination of the $p$-Laplacian and the fractional $q$-Laplacian, and results in a mixed local-nonlocal nonlinear operator. The problem we study is the following, \begin{equation*} \begin{cases} \partial_t \beta(u)+ \mathcal{A}_\mu u= div (\overset{\to}{f}(u))+g(t,x,u) \quad \text{in} \;Q_T:=(0,T)\times \Omega, u=0 \quad \text{in} \; (0,T)\times (\mathbb{R}^d \backslash \Omega), u(0)=u_0 \text{ in } \Omega. \end{cases}\ \end{equation*} We discuss existence, uniqueness and qualitative behavior of, what we call {\it weak-mild} solutions, that is weak solutions of this problem that when interpreted as $v=\beta(u)$ they are a mild solutions. In particular, we investigate stabilization to steady state, extinction and blow up in finite time and show how the occurrence of such behaviors depend on specific conditions on the nonlinearities $\beta$ (typically of porous media type), $\overset{\to}{f}$ and the source term $g$, and on their relation, in terms of certain regularity and growth conditions.