Nonexistence results of global solutions for fractional order integral equations on the Heisenberg group

TL;DR

This paper investigates fractional order integral equations on the Heisenberg group and establishes conditions under which solutions do not exist globally, demonstrating finite-time blow-up behavior.

Contribution

It provides new nonexistence results for solutions of fractional integral equations on the Heisenberg group, extending the understanding of blow-up phenomena in nonlocal fractional PDEs.

Findings

Proves blow-up of solutions under certain conditions.

Establishes nonexistence of global solutions for specific parameter ranges.

Extends fractional PDE analysis to the Heisenberg group setting.

Abstract

We consider the fractional order integral equation with a time nonlocal nonlinearity $$^{c}\mathbf{D}_{0\mid t}^{\beta}\left( u \right) +\left(-\Delta_{\mathbb{H}} \right)^{m} \left( u \right) = \frac{1}{\Gamma(\alpha)}\int_{0}^{t}\left( t-\omega\right)^{\alpha-1}\vert u(\omega)\vert^{p} d\omega,$$ posed in $ (.,t)\in\mathbb{H}\times(0,\infty) $, supplemented with an initial data $u(.,0)=u_{0}(.) $,where $ m>1 \ , \ p>1 \ , \ 0<\beta<1 \ , \ 0<\alpha<1 $, and $^{c}\mathbf{D}_{0\mid t}^{\beta} $ denotes the caputo fractional derivative of order $ \beta $, and $\Delta_{\mathbb{H}}$ is the Laplacian operator on the $ (2N+1) $-dimensional Heisenberg group $\mathbb{H} $.Then, we prove a blow up result for its solutions.