Abstract parabolic equations with applications to problems in cylindrical space domains

Abstract

In a Banach space $X$ we consider the partial differential equation $$ (*)\quad D_{t}u(t,x)+(-1)^ma(x)D_{x}^{2m}u(t,x)-A(x)u(t,x)=f(t,x) $$ where $m$ is a positive integer, related to the rectangle $(0,T)\times(0,L)$ and the family of closed linear operators $\{A(x)\}_{x\in[0,L]}$. Under suitable assumptions we uniquely solve certain initial and boundary-value problems associated with $(*)$. Some applications are given when,for each $x,$ $A(x)$ is an explicit linear uniformly elliptic differential operator.