On nonlinear Miyadera-Voigt perturbations

2022/04/21 by Mohamed Fkirine, Fkirine, Mohamed, Saïd Hadd +1
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Advanced Mathematical Physics Problems #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2204.09836

Abstract

Let A,C,P:D(A)⊂ X→ X be linear operators on a Banach space X such that -A generates a strongly continuous semigroup on X, and F:X→ X be a globally Lipschitz function. We study the well-posedness of semilinear equations of the form u(t)=G(u(t)), where G:D(A)→ X is a nonlinear map defined by G=-A+C+F∘ 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.

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