an:04076231
Zbl 0658.60089
Manthey, Ralf
On the Cauchy problem for reaction-diffusion equations with white noise
EN
Math. Nachr. 136, 209-228 (1988).
00151851
1988
j
60H15 60H20
reaction-diffusion equation; formal Cauchy problem; space-time Gaussian white noise
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).
L.G.Gorostiza