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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 $𝒲$ be a homogeneous Wiener process valued in ${𝒮}^{\text{'}}\left({R}^{d}\right)$ with a positive symmetric tempered measure as a space correlation ${\Gamma }$. The stochastic wave equation $\frac{{\partial }^{2}}{\partial {t}^{2}}u={\Delta }u\left(u\right)\left(u\right)\stackrel{˙}{𝒲}$ with initial conditions $u\left(0,x\right)={u}_{0}\left(x\right)$ and the heat equation $\frac{\partial }{\partial t}u={\Delta }u\left(u\right)\left(u\right)\stackrel{˙}{𝒲}$ with initial condition $u\left(0,x\right)={v}_{0}\left(x\right)$ 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 ${\Gamma }\kappa \lambda \ge 0$. Condition (G) is defined by Condition (H) together with

$\left(log\frac{1}{|y|}\right){I}_{|y|\le 1}\in L\left({\Gamma }\right),\phantom{\rule{4pt}{0ex}}d=2;\phantom{\rule{2.em}{0ex}}\frac{1}{{|y|}^{d-2}}{I}_{|y|\le 1}\in L\left({\Gamma }\right),\phantom{\rule{4pt}{0ex}}d>2·$

The main results are: (1) For $d\le 3$, (G) ensures the existence and uniqueness of the stochastic wave equation on $0\le t<\infty$ and when (H) is true and $|b\left(x\right)|>\epsilon$ and there exist solutions of the stochastic wave equation of some ${u}_{0}\left(x\right)$ and ${v}_{0}\left(x\right)$ for $0\le t\le T$, 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 60G60 Random fields 35K05 Heat equation 35L05 Wave equation (hyperbolic PDE)
##### Keywords:
stochastic wave; heat equation; homogeneous Wiener process