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Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process. (English) Zbl 1054.60068

The author considers a semilinear stochastic hyperbolic equation \[ u_{t,t} = \mathcal{A} u + f(u) + g(u) \dot W, \quad u(0)=u_0, \quad u_t(0)=v_0, \] in a \(d\)-dimensional domain \(D\), driven by an \(n\)-dimensional Wiener process \(W\) and where \(\mathcal{A}\) is a uniformly elliptic second order differential operator. When \(f\) and \(g\) are globally Lipschitz, it is known that this type of equation has a unique global mild solution. The author presents a more general set of sufficient assumptions upon \(f\) and \(g\) that imply that the mild solution to this equation exists globally. Moreover, the author also gives conditions on \(f,g\) and the finite spectral measure of \(W\), implying the existence of a real global strong solution. These results apply to equations with polynomial drift. The sufficient conditions are given in terms of Lyapunov functions for the equation.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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