A non-Gaussian Hardy-type Equation in Fractional Time

TL;DR

This paper introduces a novel fractional time derivative into a non-Gaussian Hardy equation with Osgood-type non-linearity, establishing existence, blow-up, and critical exponents for solutions.

Contribution

It is the first to incorporate fractional derivatives in non-Gaussian Hardy equations, providing new existence results and detailed properties of solutions.

Findings

Established local and global solution existence.

Derived optimal asymptotic estimates.

Identified conditions for solution blow-up and non-existence.

Abstract

A non-Gaussian Hardy equation is studied with a non-linearity of Osgood-type growth. A fractional derivative in time is incorporated for the first time in an research of this type. Existence of local and global solutions are established by combining properties of the fundamental solutions together with the parameters of the non-Gaussian process, leading to optimal asymptotic estimates. Additional properties of the fundamental solutions and instantaneous blow-up results are found. The Banach contraction mapping principle is particularly exploited. It is also defined a critical exponent for existence and non-existence of solutions together with a judicious choice of the initial data.