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Existence and uniqueness of a solution of a reaction-diffusion equation with polynomial nonlinearity and white noise disturbance. (English) Zbl 0594.60063
This paper deals with the initial boundary value problem to the formal semilinear partial stochastic differential equation \[ (*)\quad \frac{\partial}{\partial t}u(t,x)=a^ 2\frac{\partial^ 2}{\partial x^ 2}u(t,x)+f(u(t,x))+\sigma \xi (t\quad,x),\quad t>0,\quad x\in (0,L). \] Here \(\xi\) is a space-time Gaussian white noise. Using the Green’s function approach and the stochastic integration theory the above mentioned problem can be reformulated in an integral equation possessing a precise mathematical sense.
It is shown that this integral equation is uniquely solvable if \(f(u)=\sum^{n}_{k=0}a_ ku^ k,\) \(a_ n<0\), n odd. Moreover, there is a space-time continuous version of this solution. These results were meanwhile extended by the author to the following situation: Let f be locally Lipschitz-continuous and such that there exist two monotonously nonincreasing functions h and g satisfying \(g\leq f\leq h\). An analogous result was obtained for the initial problem, where (0,L) is replaced by (-\(\infty,\infty)\).

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
35R60 PDEs with randomness, stochastic partial differential equations
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