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On a nonlinear stochastic wave equation in the plane: Existence and uniqueness of the solution. (English) Zbl 1017.60072
The authors study the stochastic wave equation with a nonlinearity of polynomial type, \[ {\partial^2u\over\partial t^2} (t,x)-\Delta u(t, x)+|u(t,x)|^p u(t,x= \sigma(u(t,x)) u(t,x), \]
\[ u(0,x)= u_0(x),\qquad{\partial u\over\partial t} (0,x)= v_0(x), \] \(t\in [0,T]\), \(x\in\mathbb{R}^2\), with \(\rho> 0\) and when \(\sigma\) is bounded and \(u_0\) and \(v_0\) have compact suport. \(F\) is a generalized Gaussian noise satisfying certain integrability properties. The authors prove an existence and uniqueness result for this class of stochastic equations. The methods used in the proofs are based on functional analysis and probability tools. The authors use an approximation procedure via regularized versions of the equation. In order to get the results, they have to prove some new regularity properties for the Green function associated with the wave equation.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35L70 Second-order nonlinear hyperbolic equations
35R60 PDEs with randomness, stochastic partial differential equations
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