Nonlinear stochastic wave and heat equations

Abstract

We study nonlinear wave and heat equations on ‫ޒ‬ d driven by a spatially homogeneous Wiener process. For the wave equation we consider the cases of d = 1, 2, 3. The heat equation is considered on an arbitrary ‫ޒ‬ d -space. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance kernel of the noise. IntroductionThe paper is concerned with the following stochastic wave equationand heat equationIn (0.1) and (0.2), u 0 and v 0 are given functions, f, b : ‫ޒ‬ → ‫,ޒ‬ and W is a spatially homogeneous Wiener process defined on a filtered probability space ᑯ = ( , ᑠ, (ᑠ t ) t≥0 , ‫.)ސ‬ For the wave equation we consider the cases of d = 1, 2, 3. The heat equation is considered on an arbitrary ‫ޒ‬ d -space. Process W takes values in the space of tempered distributions S ‫ޒ(‬ d ), and has the covariance of the following formwhere ϕ (s) (x) = ϕ(−x), x ∈ ‫ޒ‬ d , and is a positive-definite tempered distribution. Then has to be the Fourier transform of a positive symmetric tempered measure µ on ‫ޒ‬ d . We call and µ the space correlation and spectral measure of W. In Section

Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations

“…As far as we know, there are no publications on the dynamics of coupled parabolichyperbolic stochastic partial differential equations, although stochastic parabolic and wave equations have been widely studied by many authors (see, e.g, the monographs Cerrai (2001), Da Prato andZabczyk (1996) and the references therein for the parabolic case and the papers Barbu and Da Prato (2002), Carmona and Nualart (1993), Dalang and Frangos (1998), Da Prato and Zabczyk (1992, Millet and Morien (2001), Millet and Sanz-Solé (2000), Peszat and Zabczyk (2000), Quer-Sardanyons and Sanz-Solé (2004) for the wave case).…”

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

“…Moreover, under various conditions on σ, (1.2) is necessary for the existence of a solution [10,26].…”

On Convergence of Population Processes in Random Environments to the Stochastic Heat Equation with Colored Noise

“…Not surprisingly, pathwise uniqueness -and thus convergence of the approximations-holds if the coefficients satisfy Lipschitz conditions (see Peszat and Zabczyk [32]). But it can also be shown that pathwise uniqueness holds for the lattice systems if the drift coefficients are Lipschitz continuous and σ satisfies the conditions of Yamada and Watanabe [45].…”

“…The case of a linear noise coefficient has been treated by Dawson and Salehi [10] and Noble [28]. Amongst others, Kotelenez [23], Peszat and Zabczyk [31,32], Brzeźniak and Peszat [2], Dalang [5], Manthey and Mittmann [25], Tindel and Viens [41] and Sanz-Solé and Sarrà [37] investigate solutions with Lipschitz coefficients. For some results on equations with non-Lipschitz coefficient see Viot [42], DaPrato and Zabczyk [34], Krylov [24] and Kallianpur and Sundar [20].…”