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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 \(u\).

MSC:

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