数学物理学报 ›› 2022, Vol. 42 ›› Issue (5): 1381-1397.

• 论文 • 上一篇    下一篇

带非线性梯度项的p-Laplacian抛物方程的临界指标

鲁呵倩(),张正策*()   

  1. 西安交通大学数学与统计学院 西安 710049
  • 收稿日期:2021-06-29 出版日期:2022-10-26 发布日期:2022-09-30
  • 通讯作者: 张正策 E-mail:look3114054014@stu.xjtu.edu.cn;zhangzc@mail.xjtu.edu.cn
  • 作者简介:鲁呵倩, E-mail: look3114054014@stu.xjtu.edu.cn
  • 基金资助:
    国家自然科学基金(12071044)

The Critical Exponents for the Evolution p-Laplacian Equation with Nonlinear Gradient Terms

Heqian Lu(),Zhengce Zhang*()   

  1. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049
  • Received:2021-06-29 Online:2022-10-26 Published:2022-09-30
  • Contact: Zhengce Zhang E-mail:look3114054014@stu.xjtu.edu.cn;zhangzc@mail.xjtu.edu.cn
  • Supported by:
    the NSFC(12071044)

摘要:

该文讨论了区域$\mathbb{R} ^{N}\times\mathbb{R} ^{+}$中的齐次$p$-Laplacian抛物方程:$u_{t}-\Delta_{p}u=u^{m}+|\nablau|^{q}, $以及其相应的非齐次抛物方程: $u_{t}-\Delta_{p}u=u^{m}+|\nablau|^{q}+h(x)$初值问题的临界指标. 这里, $N\geq1$, $p$, $m$, $q>1$.对于齐次$p$-Laplacian抛物方程初值问题, 得到一个不连续变化的临界指标结果.这个结果表明, 非线性梯度项对临界指标有重要影响, 伴随着$q$从无穷大减小到$p-N/(N+1)$后, 临界指标从$m=p-1+p/N$改变为$m=\infty$.同时, 研究了非齐次$p$-Laplacian抛物方程初值问题, 得到不同的不连续变化的临界指标结果.

关键词: 临界指标, p-Laplacian方程, 非线性梯度项, 爆破, 全局存在

Abstract:

This article studies the critical exponents for the homogeneous evolution $p$-Laplacian equation $u_{t}-\Delta_{p}u=u^{m}+|\nabla u|^{q}$ and its inhomogeneous version $u_{t}-\Delta_{p}u=u^{m}+|\nabla u|^{q}+h(x)$ in $\mathbb{R} ^{N}\times\mathbb{R} ^{+}$, $u(x, 0)=u_{0}(x)$ in $\mathbb{R} ^{N}$, where $N\geq1$, $p$, $m$, $q>1$. We obtain a discontinuous critical exponent result for the homogeneous evolution $p$-Laplacian equation, which demonstrates the gradient term brings about a significant phenomenon of the critical exponent, changing from $m=p-1+p/N$ to $m=\infty$ as $q$ goes to the value $p-N/(N+1)$ from above. Meanwhile, we also investigate the inhomogeneous evolution $p$-Laplacian equation and get a different discontinuous critical exponent result.

Key words: Critical exponents, p-Laplacian equation, Gradient nonlinearity, Blowup, Global existence

中图分类号: 

  • O175.29