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Fujita exponent for heat equation with Hörmander vector fields

Marianna Chatzakou, Aidyn Kassymov, Michael Ruzhansky

TL;DR

This work extends Fujita-type theory to nonlinear heat equations governed by Hörmander sum-of-squares operators on R^n, without requiring an ambient group structure. By leveraging heat-kernel estimates obtained through a lifting to stratified groups, the authors identify the critical exponent α_F = 1 + 2/q, where q is the homogeneous dimension of the CC-distance, and prove nonexistence for 1 < α ≤ α_F while establishing global existence for α > α_F under small initial data. They also handle time-dependent nonlinearities φ(t) f(u) by deriving necessary blow-up criteria expressed as the divergence of specific integrals, using a majorant function and a fixed-point iterative scheme. Together, these results generalize the Fujita phenomenon to a broad class of subelliptic heat equations beyond the Euclidean Laplacian.

Abstract

In this paper, we show global existence and non-existence results for the heat equation with some of the squares of smooth vector fields on $\Rn$ satisfying Hörmander's rank condition with a non-linearity of the form $f(u)$, where $f$ is a suitable function and $u$ is the solution. In particular, when $f(u)=u^p$, we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type $\varphi(t)f(u)$.

Fujita exponent for heat equation with Hörmander vector fields

TL;DR

This work extends Fujita-type theory to nonlinear heat equations governed by Hörmander sum-of-squares operators on R^n, without requiring an ambient group structure. By leveraging heat-kernel estimates obtained through a lifting to stratified groups, the authors identify the critical exponent α_F = 1 + 2/q, where q is the homogeneous dimension of the CC-distance, and prove nonexistence for 1 < α ≤ α_F while establishing global existence for α > α_F under small initial data. They also handle time-dependent nonlinearities φ(t) f(u) by deriving necessary blow-up criteria expressed as the divergence of specific integrals, using a majorant function and a fixed-point iterative scheme. Together, these results generalize the Fujita phenomenon to a broad class of subelliptic heat equations beyond the Euclidean Laplacian.

Abstract

In this paper, we show global existence and non-existence results for the heat equation with some of the squares of smooth vector fields on satisfying Hörmander's rank condition with a non-linearity of the form , where is a suitable function and is the solution. In particular, when , we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type .

Paper Structure

This paper contains 7 sections, 9 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Assumptions on the vector fields
  4. Lifting of the vector fields and estimates on the associated CC-balls
  5. Properties of the heat kernel of H
  6. Constant coefficients case
  7. Time-dependent case

Key Result

Theorem 1.1

Let $X$ be a system of $m$ vector fields on $\mathbb{R}^{n}$ satisfying Assumption as1. (i) Let $1<\alpha<\alpha_F=1+\frac{2}{q}$ where $q$ is as in (homdim). Suppose that $f:[0,\infty)\rightarrow \mathbb{R}$ is a locally integrable function such that $f(u)\geq Bu^{\alpha}$ for some $B>0$. Let $0< u does not admit a (non-trivial) distributional solution $u \geq 0$ in $(0,\infty)\times \mathbb{R}^{

Theorems & Definitions (23)

  • Theorem 1.1
  • Definition 1.2: Lifted operator
  • Remark 2.2
  • Example 2.3
  • Theorem 2.4: Theorem 1.2, BB23
  • Remark 2.5
  • Definition 2.6
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 13 more