Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises
The authors investigate the existence and uniqueness of global mild solutions to the abstract second-order stochastic differential equation perturbed by jump noises \[ u_{tt} + A^2 u + \nabla_u \Phi(u) + f(t, u, u_t) = \int_Z g(t,u,u_t, z) \tilde{N}(t, dz), \tag{1} \] where \(u\) is a Hilbert space \(H\)-valued stochastic process, \(A\) is a self-adjoint operator on \(H\) such that \(A \geq \mu I\) for some \(\mu > 0\), \(\Phi\) : \(D(A)\) \(\to\) \([0, \infty)\) is an energy function and \(\tilde{N}\) is a compensated Poisson random measure. Actually, this equation covers both beam equations with nonlocal nonlinear terms and nonlinear wave equations. The proof for the existence of global solutions is technically due to an Itô formula for the local mild solution. Under appropriate conditions on \(f\) and \(g\), they prove the non-explosion and the asymptotic stability of the mild solution. In particular, the unique global mild solution \(w\) \(=\) \((u, u_t)\) satisfies the following estimate \[ \sup_{ t \geq 0} {\mathbb E}[ | w(t) |_{{\mathcal H}}^2 + m( | B^{1/2} u_0 |_H^2 )] < \infty, \tag{2} \] where \({\mathcal H} = D(A) \times H\), \(B\) is the Laplace operator with fixed boundary conditions, and \(m\) : \([0, \infty)\) \(\to\) \([0, \infty)\) is a \(C^1\)-class increasing function such that \(m(0) =0\) and \(\Phi(u)\) \(=\) \(\frac{1}{2} m( | B^{1/2} u |^2 )\) for \(u\) \(\in\) \(D(A)\). For other related works, see, e.g., [the second author et al., Probab. Theory Relat. Fields 132, No. 1, 119--149 (2005; Zbl 1071.60053)] for stochastic nonlinear beam equations, [\textit{S. Albeverio} et al., J. Math. Anal. Appl. 371, No. 1, 309--322 (2010; Zbl 1197.60050)] for invariant measures of SDEs driven by Poisson type noise, and [the second author et al., Nonlinear Anal., Real World Appl. 17, 283--310 (2014; Zbl 1310.60091)] for SPDEs driven by Lévy noise.
scientific article; zbMATH DE number 6661737
scientific article