Semilinear parabolic equations in \(L^1(\Omega)\). (English) Zbl 1152.35067

Summary: This paper studies existence, regularity and continuous dependence upon the data of solutions to parabolic semilinear problems of the form: \[ u'(t)= Au(t)+ g[u(t)],\quad u(0)= u_0. \] 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 \(u\).


35K90 Abstract parabolic equations
35K55 Nonlinear parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)