## Existenzsätze für semilineare parabolische Systeme mit Funktionalen.(German)Zbl 0535.35038

Fakultät für Mathematik der Universität Karlsruhe. 65 S. (1982).
The author proves global existence results of the solutions of parabolic semilinear differential equations involving functionals, of the general form: $$L_ ku_ k=f_ k(t,x,u(\cdot),u_ x(\cdot))$$ in (0,T]$$\times \Omega$$, $$k=1,...,N$$, $$\Omega$$ open (bounded) domain in $${\mathbb{R}}^ n$$. Here $$L_ k$$ is a (non-autonomous) uniformly parabolic linear operator of the second order; the functional dependence on u, $$u_ x$$ of the functions $$f_ k$$ is, with respect to the time t, of ”delayed” type. The equations are supplemented by boundary conditions of the first or second type, possibly of non-linear type, and appropriate initial conditions. Existence is proved by considering the associated integral equations, making use of fixed point theorems of Banach, Schauder, Tychonoff. The author also outlines a number of models in applied sciences to which its results can be applied. The results extend those existing in the literature, even in the case in which the functional dependence is absent.
Reviewer: P.de Mottoni

### MSC:

 35K55 Nonlinear parabolic equations 35K10 Second-order parabolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions) 47H10 Fixed-point theorems