Semilinear parabolic equations in $L^1(\Omega)$

Abstract

This paper studies existence, regularity and continuous dependence upon the data of solutions to parabolic semilinear problems of the form: $u'(t)=Au(t) +g[u(t)]$, $u(0)=u_0$. Here, $A:D(A)\to X$ generates an analytic semigroup on a Banach space $X$ and $g:D(g)\to X$. It is assumed that $D(g)$ contains a certain interpolation space of $X$ and $D(A)$; this will allow to treat parabolic partial semilinear problems in the cases where the nonlinear term depends also on the gradient of $u$.