## 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)

### Keywords:

hyperbolic evolution equations; Lyapunov functions
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