Abstract
Existence of weak solutions of stochastic wave equations with nonlinearities of a critical growth driven by spatially homogeneous Wiener processes is established in local Sobolev spaces and local energy estimates for these solutions are proved. A new method to construct weak solutions is employed.
Highlights
Nonlinear wave equations ut t = u + f (x, u, ut, ∇x u) + g(x, u, ut, ∇x u) W (1.1)subject to random excitations have been thoroughly studied recently under various sets of hypotheses with possible applications in physics in view.The random perturbation has been usually modelled by a spatially homogeneous Wiener process which corresponds to a centered Gaussian random field (W (t, x) : t ≥ 0, x ∈ d ) satisfyingW (t, x)W (s, y) = (t ∧ s)Γ(x − y), t, s ≥ 0, x, y ∈ d for some function or even a distribution Γ called the spatial correlation of W
The Nemytskii operators associated with f and g are globally Lipschitzian and existence of solutions to (1.1) may be proved for rather general spatial correlations Γ
This issue is closely related to the fact that we aim at studying systems of stochastic wave equations (1.1)
Summary
Ad (4) & (5): We are not aware of existence results for stochastic wave equations with non-linearities depending on first derivatives of the solution (the velocity and the spatial gradient) This issue is closely related to the fact that we aim at studying systems of stochastic wave equations (1.1). Such generality is not very substantial for the present paper, the corresponding results are essential in the newly started research in the field of stochastic wave equations in Riemannian manifolds with possible applications in physical theories and models such as harmonic gauges in general relativity, non-linear σ-models in particle systems, electro-vacuum Einstein equations or. The author wishes to thank Jan Seidler and the referee for a kind helping on the redaction of the paper
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