Finite time blow-up for a heat equation in [formula omitted]

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Applied Mathematics Letters Pub Date : 2024-12-26 DOI:10.1016/j.aml.2024.109441

Kaiqiang Zhang

引用次数: 0

Abstract

We consider the semilinear heat equation ut−Δu=|u|p−1u+λu,onRnwhere p>1, and λ∈R is a parameter. When λ=0, the equation reduces to the classical heat equation. We reveal that the parameter λ in the linear term plays an important role in the blow-up conditions. Although the solution may blow up in finite time due to the cumulative effect of the nonlinearities, interestingly, we find that for λ>n2, all non-negative solutions blow up in finite time, which shows that the Fujita exponent is equal to +∞. Our result extends the Theorem 17.1 in Quittner and Souplet (2007). In addition, for λ<0, we provide a new sufficient condition for the finite time blow-up solution.

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[公式略]中热方程的有限时间爆破

考虑半线性热方程ut−Δu=|u|p−1u+λu, onrwhere p>1， λ∈R为参数。当λ=0时，方程化为经典热方程。我们发现线性项中的参数λ在爆破条件中起着重要的作用。虽然由于非线性的累积效应，解可能在有限时间内爆炸，但有趣的是，我们发现对于λ>；n2，所有非负解在有限时间内爆炸，这表明Fujita指数等于+∞。我们的结果推广了Quittner and Souplet（2007）中的定理17.1。此外，对于λ<；0，给出了有限时间爆破解的一个新的充分条件。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applied Mathematics Letters 数学-应用数学

CiteScore

7.70

自引率

5.40%

发文量

347

审稿时长

10 days

期刊介绍： The purpose of Applied Mathematics Letters is to provide a means of rapid publication for important but brief applied mathematical papers. The brief descriptions of any work involving a novel application or utilization of mathematics, or a development in the methodology of applied mathematics is a potential contribution for this journal. This journal''s focus is on applied mathematics topics based on differential equations and linear algebra. Priority will be given to submissions that are likely to appeal to a wide audience.