Nonlinear stochastic wave and heat equations
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 \(d\).
scientific article; zbMATH DE number 1454838