The existence of solutions of stochastic parabolic equations with power nonlinearities. (English. Ukrainian original) Zbl 0942.60059

Melnik, S. A.

Theory Probab. Math. Stat. 53, 113-118 (1996); translation from Teor. Jmovirn. Mat. Stat. 53, 103-108 (1995).

The author proves existence theorems for the global and the local in time generalized (in Sobolev sense) solutions of the problem \[ \begin{split} du(t,x)=\\ \text{div}(a(x)|u(t,x)|^{\sigma}\nabla u(t,x)) dt+b(t,x)|u(t,x)|^{\beta-1} u^{+}(t,x) dt + c(t,x)|u(t,x)|^{\gamma-1}u^{+}(t,x) dw(t),\end{split} \]
\[ 0\leq t\leq T,\;x\in G\subset R^{n},\;u(0,x)=u_{0}(x),\;u(t,x)|_{x\in\delta G}=0, \] where \(G\) is a bounded region with a piecewise smooth boundary \(\delta G\); \(\sigma,\beta,\gamma>0\); \(u^{+}=\max(u,0)\); \(w(t)\) is a Wiener process; \(a(x), b(t,x), c(t,x)\) are nonrandom bounded functions.

Reviewer: A.D.Borisenko (Kyïv)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35K10	Second-order parabolic equations
35R60	PDEs with randomness, stochastic partial differential equations

Keywords:

nonlinear stochastic partial differential equation; existence theorem