Multivalued SPDEs driven by additive space-time white noise and additive white noise (Q2739926)

The authors prove the existence and uniqueness theorem for the multivalued quasi-linear parabolic stochastic partial differential equation NEWLINE\[NEWLINE\partial Y(x,t)/\partial t=\partial^2 Y(x,t)/\partial x^2 + f(x,t,Y(x,t))+\partial^2 W(x,t)/\partial x\partial t+K(x,t),NEWLINE\]NEWLINE NEWLINE\[NEWLINEY(x,0)=Y_0,\qquad Y(0,t) =Y(1,t)=0,NEWLINE\]NEWLINE where \((Y,K)\) is a solution of the considered problem; \(K\) is a \(\sigma\)-finite random measure on \([0,1]\times R^{+}\); \(W(x,t)\) is the Brownian sheet; the function \(f\) satisfies conditions: \(f\in\bigcap_{T>0}L^2([0,1]\times [0,T])\), \(f\) is locally bounded; \(z\to f(x,t,z)\) is continuous and decreasing for any \((x,t)\in [0,1]\times R^{+}\); \(Y_0\) is continuous function which vanishes at the boundary of \([0,1]\). Also the existence and uniqueness result is proved for the stochastic elliptic equation with the Dirichlet boundary condition NEWLINE\[NEWLINE-\Delta U(x)=f(x,U(x))+ \dot W(x)+\eta,\;x\in D,\quad U(x)=0,\;x\in\partial D,NEWLINE\]NEWLINE where \(\dot W(x), x\in D\), is a white noise in \(D\) and \(\eta\) is a random measure satisfying a kind of minimality condition which forces the solution \(U\) to belong to the convex region \(G\).