Date of Award
5-1-2023
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
Schurz, Henri
Abstract
We used indirect Lyapounov method to prove existence, uniqueness, stability, uniformboundedness and Holder continuity of approximate Fourier series solutions ¨ u to damped and undamped, semi-linear stochastic wave equations of the form (in Ito sense) ˆ utt = σ 2△u + B(u, ut) + G(u, ut) ∂W ∂t with cubic nonlinearities B(u, ut) = a1u−a2||u||2u−κut and homogeneous boundary conditions (HBCs) on general 3D cubes D = [0, lx]×[0, ly]×[0, lz] . The driving Q-regular space-time noise W with linear-growth bounded, state-dependent diffusion intensities G(u, ut) is supposed to be general in space (x, y, z) ∈ D. We also studied the expected total energy functional e(t) which depends on many parameters, for instance, the length parameters, the diffusion parameters, the transport coefficient a1, and the damping coefficient κ. We provided some examples to show the applicability of our technique and explain our main results. At the end, we confirm our findings by illustrating with some pictures.
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