Existence results for linear evolution equations of parabolic type

TL;DR

This paper investigates the existence and regularity of solutions to linear parabolic evolution equations, both deterministic and stochastic, in Banach spaces, with applications to stochastic heat equations.

Contribution

It establishes maximal regularity for deterministic solutions and proves existence and regularity of strict solutions in the stochastic case, applying results to stochastic heat equations.

Findings

Maximal regularity for deterministic solutions

Existence of strict solutions in stochastic case

Application to stochastic heat equations

Abstract

We study both strict and mild solutions to parabolic evolution equations of the form $dX+AXdt=F(t)dt+G(t)dW(t)$ in Banach spaces. First, we explore the deterministic case. The maximal regularity of solutions has been shown. Second, we investigate the stochastic case. We prove existence of strict solutions and show their space-time regularity. Finally, we apply our abstract results to a stochastic heat equation.