The Fujita exponent for a heat equation with mixed local and nonlocal nonlinearities on the Heisenberg group

TL;DR

This paper investigates the critical Fujita exponent for a heat equation on the Heisenberg group with mixed local and nonlocal nonlinearities, establishing conditions for existence, blow-up, and lifespan of solutions.

Contribution

It introduces the first analysis of the Fujita exponent in the context of the Heisenberg group with mixed nonlinearities, combining geometric and nonlinear analysis techniques.

Findings

Identified the critical Fujita exponent separating global existence and blow-up.

Established local and global existence conditions for solutions.

Derived lifespan estimates in the supercritical regime.

Abstract

This article deals with the problems of local and global solvability for a semilinear heat equation on the Heisenberg group involving a mixed local and nonlocal nonlinearity. The characteristic features of such equations, arising from the interplay between the geometric structure of the Heisenberg group and the combined nonlinearity, are analyzed in detail. The need to distinguish between subcritical and supercritical regimes is identified and justified through rigorous analysis. On the basis of the study, the author suggests precise conditions under which local-in-time mild solutions exist uniquely for regular, nonnegative initial data. It is proved that global existence holds under appropriate growth restrictions on the nonlinear terms. To complement these results, it is shown, by employing the capacity method, that solutions cannot exist globally in time when the nonlinearity exceeds a critical threshold. As a result, the Fujita exponent is formulated and identified as the dividing line between global existence and finite-time blow-up. In addition, lifespan estimates were obtained in the supercritical regime, providing insight into how the size of the initial data influences the time of blow-up.