Existence of weak solutions for a class of semilinear stochastic wave equations
The authors prove the existence of weak solutions (in the probabilistic sense) for the following semilinear wave equation: NEWLINE\[NEWLINE \frac{\partial^2 u}{\partial t^2}(x,\,t) - \Delta u(x,\,t) + \beta(u(x,\,t)) = \dot{\eta}(x,\,t),\quad (x,\,t)\in D\times [0,\,T] \subset{\mathbb R}^{d+1}, NEWLINE\]NEWLINE where \(\beta\) is a maximal monotone graph in \({\mathbb R}\times{\mathbb R}\) and \(\dot{\eta}\) is the time derivative of a stochastic integral with respect to a Hilbert space-valued martingale which may depend upon \(u\). Some auxiliary results on mild solutions of stochastic evolution equations with Lipschitz nonlinearities are also established.
scientific article; zbMATH DE number 6070873