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Nonlinear stochastic wave and heat equations. (English) Zbl 0959.60044
This is a continuation of the authors’ paper [Stochastic Processes Appl. 72, No. 2, 187-204 (1997; Zbl 0943.60048)]. Let $$\mathcal{W}$$ be a homogeneous Wiener process valued in $$\mathcal{S}'(R^d)$$ with a positive symmetric tempered measure as a space correlation $$\Gamma$$. The stochastic wave equation $$\frac {\partial ^2}{\partial t^2}u=\Delta u(u)(u)\dot{\mathcal{W}}$$ with initial conditions $$u(0,x)=u_0(x)$$ and the heat equation $$\frac \partial {\partial t}u=\Delta u(u)(u)\dot{\mathcal{W}}$$ with initial condition $$u(0,x)=v_0(x)$$ are considered on $$R^d$$ with Lipschitz $$f$$ and $$b$$. Comparing the paper cited above, in the present paper $$\Gamma$$ is extended to a generalised function and the Fourier transform of which is not necessarily absolutely continuous with respect to the Lebegue measure $$\lambda$$. Let Condition (H) be: there exists $$\kappa$$ such that $$\Gamma \kappa \lambda \geq 0$$. Condition (G) is defined by Condition (H) together with $\Biggl(\log \frac {1}{|y|}\Biggr) I_{|y|\leq 1} \in L( \Gamma),\;d=2; \qquad \frac {1}{|y|^{d-2}}I_{|y|\leq 1} \in L( \Gamma),\;d>2 .$ The main results are: (1) For $$d\leq 3$$, (G) ensures the existence and uniqueness of the stochastic wave equation on $$0 \leq t < \infty$$ and when (H) is true and $$|b(x)|> \varepsilon$$ and there exist solutions of the stochastic wave equation of some $$u_0(x)$$ and $$v_0(x)$$ for $$0 \leq t \leq T$$, then (G) holds conversely. (2) The same conclusions as in (1) hold for the heat equation for any dimension $$d$$.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60G60 Random fields 35K05 Heat equation 35L05 Wave equation
##### Keywords:
stochastic wave; heat equation; homogeneous Wiener process
Zbl 0943.60048
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