On a certain stochastic quasilinear hyperbolic equation

D. A. Khrychev

Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 363-388

Résumé

The author considers the first boundary value problem for the equation $$ \frac{\partial^2u(t,x)}{\partial t^2}+k\,\frac{\partial u}{\partial t}-\Delta u+|u|^\rho u=\frac{\partial w(t,x)}{\partial t},\qquad t>0, \quad x\in\mathscr O\Subset\mathbf R^n, $$ where $k\geqslant0$, $\rho>0$, and $w(t)$ is a Wiener process in the space $L^2(\mathscr O)$. The initial values are assumed random and independent of the process $w(t)$. The existence of a space-time statistical solution is proved and (under a certain restriction on $\rho$) the existence of a strong solution. A steady state space-time statistical solution is constructed for $k>0$. Bibliography: 12 titles.