Blow-up for a non-linear stable non-Gaussian process in fractional time

TL;DR

This paper investigates the behavior of solutions to a non-linear diffusion problem driven by a stable non-Gaussian process in fractional time, establishing blow-up conditions, critical exponents, and properties of solutions.

Contribution

It introduces a new integral representation of solutions using subordination and proves a Fujita type blow-up result specific to stable non-Gaussian processes in fractional time.

Findings

Established Fujita's critical exponent for the process

Proved blow-up conditions for solutions

Derived properties of solutions such as continuity and non-negativeness

Abstract

The behaviour of solutions for a non-linear diffusion problem is studied. A subordination principle is applied to obtain the variation of parameters formula in the sense of Volterra equations, which leads to the integral representation of a solution in terms of the fundamental solutions. This representation, the so-called mild solution, is used to investigate some properties about continuity and non-negativeness of solutions as well as to prove a Fujita type blow-up result. Fujita's critical exponent is established in terms of the parameters of the stable non-Gaussian process and a result for global solutions is given.