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

TL;DR

This paper investigates reaction-diffusion equations with space-time fractional derivatives, identifying critical parameters that determine the existence of non-trivial solutions on both unbounded and bounded domains.

Contribution

It introduces a framework for analyzing such equations via integral equations and reveals how time derivatives influence solution behavior and critical parameters.

Findings

No non-trivial solutions for certain parameter ranges on the whole space.

Existence of non-trivial solutions when parameters exceed critical thresholds.

Time derivatives significantly alter solution dynamics on bounded domains.

Abstract

We will look at reaction-diffusion type equations of the following type, $$\partial^\beta_tV(t,x)=-(-\Delta)^{\alpha/2} V(t,x)+I^{1-\beta}_t[V(t,x)^{1+\eta}].$$ 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 $0<\eta\leq\eta_c$, there is no global solution other than the trivial one while for $\eta>\eta_c$, 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.