### Sobolev regularity for solutions of parabolic equations by extrapolation methods

Gabriella Di Blasio

#### Abstract

Let $A$ be the generator of an analytic semigroup on a Banach space $X$. We study the Sobolev regularity of the solutions $u$ of the problem $u'(t)=Au(t)+f(t)$, for almost every $t\in (0,T)$, $u(0)=u_0$. Using extrapolation theory we can investigate optimal conditions for the data $f$ and $u_0$ guaranteeing $u\in W^{\alpha,p}(0,T;X)$ and obtain new existence results for generalized as well as differentiable solutions for parabolic equations.

#### Article information

Source
Adv. Differential Equations Volume 6, Number 4 (2001), 481-512.

Dates
First available in Project Euclid: 2 January 2013