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Stochastic wave equations with polynomial nonlinearity. (English) Zbl 1017.60071
The author studies the stochastic wave equation with polynomial nonlinearity, $\partial^2_t u= \nabla^2 u+ f(u)+ \sigma(u)\partial_\tau W(t,x),x\in\mathbb{R}^d,\;t> 0,\quad u(0,x)= g(x),\quad \partial_t u(0,x)= h(x),$ $$d\leq 3$$, where $$W(t,.)$$ is a Wiener random field and the nonlinear terms $$f(u)$$ and $$\sigma(u)$$ are assumed to be polynomials in $$u$$. The author proves an existence and uniqueness result of local and global solutions in Sobolev space $$H_1$$ for a class of these stochastic equations. The proofs are based on an energy inequality for a linear stochastic wave equation. Nevertheless, in order to show that in general the global solution does not exist, the author shows a cubically nonlinear stochastic wave equation with an explosive solution.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations
##### Keywords:
stochastic wave equation; polynomial nonlinearity
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##### References:
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