Semilinear nonautonomous parabolic equations with unbounded coefficients in the linear part

TL;DR

This paper investigates the existence and stability of solutions to semilinear nonautonomous parabolic equations with unbounded coefficients, extending previous results by allowing unboundedness in the linear operator coefficients.

Contribution

It introduces new conditions for existence and stability of solutions in both bounded continuous and L^p spaces, considering unbounded coefficients in the linear part.

Findings

Established existence of solutions under new unbounded coefficient conditions

Provided stability criteria for the null solution

Extended analysis to both C_b and L^p spaces

Abstract

We study the Cauchy problem for the semilinear nonautonomous parabolic equation $u_t=\mathcal{A}(t)u+\psi(t,u)$ in $[s,\tau]\times {{\mathbb R}^d}$, $\tau> s $, in the spaces $C_b([s, \tau]\times{{\mathbb R}^d})$ and in $L^p((s, \tau)\times{{\mathbb R}^d}, \nu)$. Here $\nu$ is a Borel measure defined via a tight evolution system of measures for the evolution operator $G(t,s)$ associated to the family of time depending second order uniformly elliptic operators $\mathcal{A}(t)$. Sufficient conditions for existence in the large and stability of the null solution are also given in both $C_b$ and $L^p$ contexts. The novelty with respect to the literature is that the coefficients of the operators $\mathcal{A}(t)$ are allowed to be unbounded.