Semilinear Equations Including the Mixed Operator

TL;DR

This paper investigates the existence, uniqueness, and boundedness of solutions to a semilinear evolution equation involving a mixed local and nonlocal operator, providing conditions for both local and global solutions.

Contribution

It introduces a novel analysis of a semilinear equation with a mixed operator, establishing local and global existence results under specific parameter conditions.

Findings

Existence and uniqueness of local solutions in L^()

Conditions for solutions to remain globally bounded in time

Identification of parameter regimes ensuring global existence

Abstract

We study the local and global existence of solutions to a semilinear evolution equation driven by a mixed local-nonlocal operator of the form \( L = -\Delta + (-\Delta)^{\alpha/2} \), where \( 0 < \alpha < 2 \). The Cauchy problem under consideration is \begin{equation*} \partial_t u + t^\beta L u = -h(t) u^p, \quad x \in \mathbb{R}^N, \quad t > 0, \end{equation*} with nonnegative initial data \( u(x, 0) = u_0(x) \). We establish the existence and uniqueness of local solutions in \( L^\infty(\mathbb{R}^N) \) using a contraction mapping argument. Furthermore, we analyze conditions for global existence, proving that solutions remain globally bounded in time under appropriate assumptions on the parameters \( \beta \), \( p \), and the function \( h(t) \).