Sunday A. Asogwa | Searcharxiv

arXiv2018

Critical parameters for reaction-diffusion equations involving space-time fractional derivatives

We will look at reaction-diffusion type equations of the following type, $$\partial^β_tV(t,x)=-(-Δ)^{α/2} V(t,x)+I^{1-β}_t[V(t,x)^{1+η}].$$ We first study the equation on the whole space by making sense of it via an integral equation. Roughly speaking, we will show that when $0<η\leqη_c$, there is no global solution other than the trivial one while for $η>η_c$, non-trivial global solutions do exist. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the time derivative induces a significant change in the behaviour of the solution.

math.AP↗

arXiv2016

Intermittency fronts for space-time fractional stochastic partial differential equations in $(d+1)$ dimensions

We consider time fractional stochastic heat type equation $$\partial^β_tu_t(x)=-ν(-Δ)^{α/2} u_t(x)+I^{1-β}_t[σ(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $ν>0$, $β\in (0,1)$, $α\in (0,2]$, $d<\min\{2,β^{-1}\}\a$, $\partial^β_t$ is the Caputo fractional derivative, $-(-Δ)^{α/2} $ is the generator of an isotropic stable process, $\stackrel{\cdot}{W}(t,x)$ is space-time white noise, and $σ:\R \to\RR{R}$ is Lipschitz continuous. Mijena and Nane proved in \cite{JebesaAndNane1} that : (i) absolute moments of the solutions of this equation grows exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. The last result was proved under the assumptions $α=2$ and $d=1.$ In this paper we extend this result to the case $α=2$ and $d\in\{1,2,3\}.$

math.PR↗