Strong solutions for semilinear problems with almost sectorial operators

Abstract

In this paper we study a semilinear parabolic problem ut + Au = f(t, u), t > τ ; u(τ ) = u0 ∈ X, in a Banach space X, where A : D(A) ⊂ X → X is an almost sectorial operator. This problem is locally well-posed in the sense of mild solutions. By exploring properties of the semigroup of growth 1 − α generated by −A, we prove that the local mild solution is actually strong solution for the equation. This is done without requiring any extra regularity for the initial condition u0 ∈ X and under suitable assumptions on the nonlinearity f. We apply the results for a reaction-diffusion equation in a domain with handle where the nonlinearity f satisfies a polynomial growth |f(t, u) − f(t, v)| ≤ C|u − v|(1 + |u| ρ−1 + |v| ρ−1 ) and we establish values of ρ for which the problem still have strong solution. Mathematical Subject Classification 2020: 35A01, 35D30, 35D35, 35K58 Key words and phrases: Almost sectorial operators, semigroups of growth 1 − α, semilinear problems, strong solutions.