# zbMATH — the first resource for mathematics

On semilinear stochastic partial differential equations on the real line. (English) Zbl 0887.60070
The author constructs by the help of the comparison method a solution of the Cauchy problem to a parabolic stochastic partial differential equation of the form ${\partial\over\partial t} u(t,x)= (\Delta u)(t,x)+ f(t,x,u(t,x))+ \sigma(t,x,u(t,x))\dot W(t,x),$ $$t\in(0,T]$$, $$x\in\mathbb{R}$$, $$u(0,x)= v(x)$$, $$x\in\mathbb{R}$$, where $$\Delta$$ denotes the Laplace operator and $$\dot W$$ stands for the space-time Gaussian noise. The “drift” is assumed to be one-sided linearly bounded, continuous and of at most polynomial growth while the “diffusion” is Lipschitzian and linearly bounded.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: