Existence and Uniqueness of Solutions for Non-Linear Stochastic Partial Differential Equations

摘要

We state some results on existence and uniqueness for the solution of non linear stochastic PDEswith deviating arguments. In fact, we consider the equationdx(t) + (A(t; x(t)) + B(t; x(¿ (t))) + f(t)) dt = (C(t; x(½(t))) + g(t)) dwt ;where A(t; :) ; B(t; :) and C(t; :) are suitable families of non linear operators in Hilbert spaces,wt is a Hilbert valued Wiener process, and ¿ ; ½ are functions of delay. If A satisfies a coercivitycondition and a monotonicity hypothesis, and if B ; C are Lipschitz continuous, we prove thatthere exists a unique solution of an initial value problem for the precedent equation. Some examplesof interest for the applications are given to illustrate the results.