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

Abstract

In this paper we study a doubly degenerate parabolic equation involving a convection term and the operator $\mathcal{A}_μu:=-Δ_p u +μ(-Δ)^s_q u$ 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 β(u)+ \mathcal{A}_μu= div (\overset{\to}{f}(u))+g(t,x,u) \quad \text{in} \;Q_T:=(0,T)\times Ω, u=0 \quad \text{in} \; (0,T)\times (\mathbb{R}^d \backslash Ω), u(0)=u_0 \text{ in } Ω. \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 $v=β(u)$ 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 $β$ (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.