Antiperiodic Problems for Nonautonomous Parabolic Evolution Equations

Abstract

This work focuses on the antiperiodic problem of nonautonomous semilinear parabolic evolution equation in the form $u\mathrm{\prime }(t)=A(t)u(t)+f(t,u(t))$, $t\in \mathbb{R}$, $u(t+T)=-u(t)$, $t\in \mathbb{R}$, where ${(A\left(t\right))}_{t\in \mathbb{R}}$ (possibly unbounded), depending on time, is a family of closed and densely defined linear operators on a Banach space $X$. Upon making some suitable assumptions such as the Acquistapace and Terreni conditions and exponential dichotomy on ${(A\left(t\right))}_{t\in \mathbb{R}}$, we obtain the existence results of antiperiodic mild solutions to such problem. The antiperiodic problem of nonautonomous semilinear parabolic evolution equation of neutral type is also considered. As sample of application, these results are applied to, at the end of the paper, an antiperiodic problem for partial differential equation, whose operators in the linear part generate an evolution family of exponential stability.