Stochastic wave equations with polynomial nonlinearity
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.
scientific article; zbMATH DE number 1906187
scientific article