On the solutions of reaction-diffusion equations with white noise. (English) Zbl 0591.35029

Manthey, Ralf

Forschungsergeb., Friedrich-Schiller-Univ. Jena N/85/24, 27 p. (1985).

The paper treats a one-dimensional semi-linear parabolic inhomogeneous initial and boundary value problem subjected to a distributed whitenoise input. A solution is defined by formally applying Duhamel’s principle utilizing a boundedness property of the Green’s function, valid only in the one-dimensional case. By a truncation and prolongation technique the author extends existence theorems for globally Lipschitz-continuous nonlinearities to deriving terms being locally Lipschitz, not necessarily satisfying a linear growth law. So the theory makes accessible some of the important reaction-diffusion systems with stochastic fluctuations recently under consideration. Without the stochastic aspect the theory is given by e.g. D. Henry [”Geometric theory of semilinear parabolic equations”, Lect. Notes Math. 840, Ch. 3 (1981; Zbl 0456.35001)].

Reviewer: G.Leugering

MSC:

35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
60K99	Special processes

Keywords:

semi-linear evolution equation; white-noise fluctuations; sample; continuity; Duhamel’s principle; boundedness property; reaction-diffusion systems; stochastic fluctuations