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On the Cauchy problem for reaction-diffusion equations with white noise. (English) Zbl 0658.60089
This paper is concerned with the formal Cauchy problem \[ (\partial /\partial t)u(t,x)=A(\delta^ 2/\partial x^ 2)u(t,x)+f(u(t,x))+\sigma \xi (t,x), \] \[ (t,x)\in (0,T)\times R,\quad \sigma \geq 0,\quad u(0,x)=\phi (x),\quad x\in R, \] where \(\xi\) is a space-time Gaussian white noise, f: \(R\to R\) is a locally Lipschitz continuous function, and there exist two nonincreasing functions g and \(h:\quad R\to R\) such that \(g\leq f\leq h\), where \[ | h(u)| \leq c_ h(1+| u|^ m)\quad and\quad | g(u)| \leq c_ g(1+| u|^{\ell}) \] for positive constants \(c_ h, c_ g\) and m, \(\ell \geq 0.\)
The author proves theorems on existence and uniqueness of solutions of the Cauchy problem, and existence of a version of a solution which is continuous in (t,x).
Reviewer: L.G.Gorostiza

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
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