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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 ' (R d ) with a positive symmetric tempered measure as a space correlation Γ. The stochastic wave equation 2 t 2 u=Δu(u)(u)𝒲 ˙ with initial conditions u(0,x)=u 0 (x) and the heat equation tu=Δu(u)(u)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 Γ is extended to a generalised function and the Fourier transform of which is not necessarily absolutely continuous with respect to the Lebegue measure λ. Let Condition (H) be: there exists κ such that Γκλ0. Condition (G) is defined by Condition (H) together with

log 1 |y|I |y|1 L(Γ),d=2;1 |y| d-2 I |y|1 L(Γ),d>2·

The main results are: (1) For d3, (G) ensures the existence and uniqueness of the stochastic wave equation on 0t< and when (H) is true and |b(x)|>ε and there exist solutions of the stochastic wave equation of some u 0 (x) and v 0 (x) for 0tT, then (G) holds conversely. (2) The same conclusions as in (1) hold for the heat equation for any dimension d.

60H15Stochastic partial differential equations
60G60Random fields
35K05Heat equation
35L05Wave equation (hyperbolic PDE)