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

Zineb Sabbagh, Ahmad Z. Fino, Mokhtar Kirane

TL;DR

This work analyzes a semilinear heat equation on the Heisenberg group with a mixed nonlinearity comprising a memory term and a local power nonlinearity. By leveraging the Heisenberg heat semigroup, $L^p$-$L^q$ estimates, and capacity-based (nonlinear) blow-up techniques, the authors establish a Fujita-type dichotomy: global existence for small data when $p_1>p_1^*$ and $p_2>p_2^{**}$, and finite-time blow-up when $p_1\le p_1^*$ or $p_2\le p_2^*$. Lifespan estimates are obtained in the supercritical regime, highlighting how initial-data size controls the blow-up time. The results extend Fujita-type theory to a non-Euclidean, sub-Riemannian setting with mixed local and memory nonlinearities, illustrating the influence of geometry and nonlocal effects on parabolic dynamics.

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.

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

TL;DR

This work analyzes a semilinear heat equation on the Heisenberg group with a mixed nonlinearity comprising a memory term and a local power nonlinearity. By leveraging the Heisenberg heat semigroup, - estimates, and capacity-based (nonlinear) blow-up techniques, the authors establish a Fujita-type dichotomy: global existence for small data when and , and finite-time blow-up when or . Lifespan estimates are obtained in the supercritical regime, highlighting how initial-data size controls the blow-up time. The results extend Fujita-type theory to a non-Euclidean, sub-Riemannian setting with mixed local and memory nonlinearities, illustrating the influence of geometry and nonlocal effects on parabolic dynamics.

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.

Paper Structure

This paper contains 7 sections, 10 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Heisenberg group
  4. Heat kernel
  5. Comparison Principle
  6. Global existence
  7. Finite-time blow up

Key Result

Theorem 1.1

Let $u_0\in C_0(\mathbb{H}^n)$, with $n\geq1$ and $p_i>1$ for $i=1,2$. Then there exists a maximal time $T_{\max}>0$ and a unique mild solution $u\in C([0,T_{\max}),C_0(\mathbb{H}^n))$ to problem 1. Moreover, we have the alternative: In addition, if $u_0\geq0,$$u_0\not\equiv0,$ then the solution remains strictly positive for all $0<t<T_{\max}.$ Furthermore, if $u_0\in L^r(\mathbb{H}^n),$ for some

Theorems & Definitions (15)

  • Definition 1.1: Mild solution
  • Theorem 1.1: Local existence
  • Theorem 1.2: Global existence
  • Remark 1.1
  • Theorem 1.3: Blow-up
  • Remark 1.2
  • Theorem 1.4: Lifespan estimates
  • Theorem 1.5: Lifespan estimates
  • Remark 1.3
  • Proposition 2.1
  • ...and 5 more