On nonlinear Miyadera-Voigt perturbations

Abstract

Let \(A,C,P:D(A)\subset X\rightarrow X\) be linear operators on a Banach space X such that \(-A\) generates a strongly continuous semigroup on X, and \(F:X\rightarrow X\) be a globally Lipschitz function. We study the well-posedness of semilinear equations of the form \({\dot{u}}(t)=G(u(t))\), where \(G:D(A)\rightarrow X\) 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.