Non-existence of solutions for a non-Gaussian equation in fractional time with Osgood type nonlinearity

TL;DR

This paper investigates the conditions under which solutions to a non-Gaussian fractional diffusion equation with Osgood nonlinearity do not exist, identifying critical exponents and phenomena like blow-up and global existence.

Contribution

It introduces new results on non-existence and existence of solutions for a class of fractional non-Gaussian equations with Osgood nonlinearity, highlighting the role of fractional derivatives and diffusion types.

Findings

Critical exponent for non-existence depends on fractional derivative and diffusion type.

Instantaneous blow-up occurs under certain conditions.

Global solutions exist when parameters satisfy specific criteria.

Abstract

Osgood functions in the source term are used to produce results for non-existence of local solutions into the framework of non-Gaussian diffusion equations. The critical exponent for non-existence of local solutions is found to depend on the fractional derivative, the non-Gaussian diffusion and the non-linear term. The instantaneous blow-up phenomenon is studied by exploiting estimates of the fundamental solutions. Nevertheless, theory of super-solutions and fixed points are combined for showing existence of global solutions. In this case, the critical exponent for existence of global solutions depends only on the last two parameters above.