A Study of Space-time Fractional SPDE in Bounded Domains

Abstract

This dissertation considers space-time fractional stochastic heat equation on a regular bounded domain B in R^d, d ⩾ 1 with Dirichlet boundary condition: ∂βt ut(x) = −(−Δ)α2 ut(x) + I1−βt [λσ(ut(x)) ˙W (t, x)] for x ∈ B and t > 0 ut(x) = 0 for x /∈ B and t > 0, and the initial condition u_0 : B → R+ is a non-random measurable and bounded function that has support with positive measure inside B, where the time fractional differential operators ∂β_t and I_t^(1−β) respectively denote the Caputo derivative and the Riemann-Liouville fractional integral operator. The fractional Laplacian operator −(−Δ)^α/2 , where 0 < α ⩽ 2, is the L2-generator of a symmetric α-stable process XBt killed when exiting B. ˙W denotes a space-time white noise and σ : R → R is a globally Lipschitz function satisfying lσ|x| ⩽ |σ(x)| ⩽ Lσ|x| where lσ and Lσ are positive constants. The positive parameter λ is called the level of the noise. We studied the long time behavior of their solutions with respect to the level λ of the noise and show how the choice of the order β ∈ (0, 1) of the fractional time derivative affects the growth and decay behavior of their solution. Further, we showed the continuity of the solution ut(x) of (0.1) with respect to the fractional parameter β. Our results extend the main results in Foondun [29] to fractional Laplacian as well as higher dimensional cases. We also study the temporal Hölder continuity of mild random field solutions for space-time fractional stochastic heat equation driven by noise colored in space which can be obtained by constructing relevant moment bounds for increments of the stochastic convolution (Y ⊛ σ(u))t(y). Our techniques are based on connecting the space-time fractional Green functions Y,Z to the fundamental solution of ∂tPt+ν/2 (−Δ)^(α/2)Pt = 0, P0 = δ0 via the subordination of Wright-type function W−β, γ(−ρ).