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Global solution and blow-up of critical heat equation with nonlocal interaction

  • Minbo Yang and Jian Zhang ORCID logo EMAIL logo
Published/Copyright: February 10, 2025

Abstract

We consider the Cauchy problem of the critical nonlocal heat equation

u t - Δ u = ( | x | - μ | u | 2 μ ) | u | 2 μ - 2 u in  N × ( 0 , T max ) ,

where N 3 , 0 < μ < min { 4 , N } and 2 μ = 2 N - μ N - 2 is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. The aim of this paper is to study the behaviour of the solutions in time. More precisely, we prove the existence and decay of global solutions and blow-up in finite time. Furthermore, the global uniform bound of the global solutions in H ˙ 1 ( N ) and the asymptotic behaviour of them are given.

MSC 2020: 35B11

Communicated by Christopher D. Sogge


Award Identifier / Grant number: 12471114

Award Identifier / Grant number: LZ22A010001

Funding statement: Minbo Yang was partially supported by National Natural Science Foundation of China (12471114) and Natural Science Foundation of Zhejiang Province (LZ22A010001).

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Received: 2024-11-22
Revised: 2024-12-31
Published Online: 2025-02-10
Published in Print: 2025-06-01

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