On nonlinear Miyadera-Voigt perturbations

TL;DR

This paper investigates the well-posedness of certain semilinear equations involving nonlinear Miyadera-Voigt perturbations using maximal regularity and fixed point methods, with applications to nonlinear heat equations.

Contribution

It introduces a framework for analyzing semilinear equations with nonlinear Miyadera-Voigt perturbations, establishing existence and uniqueness of solutions.

Findings

Proved well-posedness for a class of semilinear equations with nonlinear perturbations.

Applied the theoretical results to nonlinear heat equations with boundary conditions.

Demonstrated the approach's effectiveness for nonlocal unbounded nonlinear perturbations.

Abstract

Let $A,C,P:D(A)\subset X\to X$ be linear operators on a Banach space $X$ such that $-A$ generates a strongly continuous semigroup on $X$, and $F:X\to X$ be a globally Lipschitz function. We study the well-posedness of semilinear equations of the form $\dot{u}(t)=G(u(t))$, where $G:D(A)\to X$ is a nonlinear map defined by $G=-A+C+F\circ P$. In fact, using the concept of maximal $L^p$-regularity and a fixed point theorem, we establish the existence and uniqueness of a strong solution for the above-mentioned semilinear equation. We illustrate our results by applications to nonlinear heat equations with respect to Dirichlet and Neumann boundary conditions, and a nonlocal unbounded nonlinear perturbation.