Nonlinear stochastic wave equations in 1D with fractional Laplacian, power-law nonlinearity and additive [formula omitted]-regular noise

Abstract

A qualitative study of nonlinear, 1D stochastic fractional wave equations with dissipative nonlinearities of power-law form is conducted on (t,x)∈[0,+∞)×Dutt+σ2(−uxx)α−a1u+a2‖u‖L2(D)ρu−κut=b0∂W0∂t on (t,x)∈[0,+∞)×D with D=[0,L], where positive fractional α-powers of Laplace operator are allowed, perturbed by additive space–time random noise W0 with fairly general covariance operator Q with finite trace(Q)<+∞ (i.e. Q-regularity). The Q-regular space–time noise W0, which is white in time and generally spatially correlated, is supposed to have a Fourier expansion along the eigenfunctions of the Laplace operator under nonrandom, Dirichlet- and Neumann-type homogeneous boundary conditions. We focus on studying the generalized energy functional V incorporating the fractional diffusive energy part and the energy part which is due to the viscous damping force both in continuous and discrete time. In continuous time, the technique of Fourier expansions and appropriate truncation by finite-dimensional systems leads to the control on the generalized energy functional, and hence we may verify existence, uniqueness and boundedness of moments of Fourier series solutions u. The total mean energy E[E(t)] cannot grow more than linearly in time t (with or without damping). Moreover, under the absence of damping (i.e. κ=0), E[E(t)] is governed by a kind of trace formula in case of additive (state-independent), Q-regular space–time noise W0. For numerical computations and more adequate discretization, we suggest to take nonstandard, partial-implicit midpoint-type methods for the Fourier coefficients. These semi-analytical numerical methods (approximate Fourier series) possess the property of conserving the expected, total energy with random initial data under the absence of nonlinearities. Eventually, we estimate probabilities of large fluctuations of some interesting functionals.