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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.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
Full Text: DOI
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