Stochastic variational inequalities for a wave equation. (English) Zbl 1349.35250

Kim, Jong Uhn

Differ. Integral Equ. 29, No. 1-2, 93-126 (2016).

The author considers a system of evolution variational inequalities \[ u_{tt}-\Delta u-b(\omega,t,x,u)\dot B\geq 0,\quad u_t\geq 0,\quad u_t(u_{tt}-\Delta u-b(\omega,t,x,u)\dot B)=0 \] on a smooth bounded domain \(G\) in \(\mathbb R^d\) with the homogeneous Dirichlet boundary condition and initial conditions \(u(0)=u_0\), \(u_t(0)=u_1\) where \(u_1\geq 0\). Here \(B\) is a cylindrical Wiener process and \(b\) either does not depend on \(u\) (i.e. the noise in the equation is additive) or \(b\) is a deterministic and time-homogeneous affine operator applied on \(u\) (i.e. the noise in the equation is multiplicative). By transferring the stochastic problem to a random problem \[ v_{tt}-\Delta v-f\geq 0,\quad v_t+M\geq 0,\quad (v_t+M)(v_{tt}-\Delta v-f)=0 \] where \(M\) and \(f\) are random processes, the author proves existence, uniqueness and regularity of the solutions to both the stochastic and the random problems in the state space \((H^1_0(G)\cap H^2(G))\times H^1_0(G)\) for the pair \((u,u_t)\). In case of the multiplicative noise, it is additionally assumed that \(d\leq 3\).

Reviewer: Martin Ondreját (Praha)

MSC:

35L85	Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35R60	PDEs with randomness, stochastic partial differential equations
35L05	Wave equation

Keywords:

stochastic wave equation; penalty method; regularity