带非线性梯度项的p-Laplacian抛物方程的临界指标  

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

作　　者：鲁呵倩 张正策[1] Heqian Lu;Zhengce Zhang(School of Mathematics and Statistics,Xi'an Jiaotong University,Xi'an 710049)

机构地区：[1]西安交通大学数学与统计学院,西安710049

出　　处：《数学物理学报（A辑）》2022年第5期1381-1397,共17页Acta Mathematica Scientia

基　　金：国家自然科学基金(12071044)。

摘　　要：该文讨论了区域R^(N)×R^(+)中的齐次p-Laplacian抛物方程:u_(t)−Δp^(u)=u^(m)+|\nablau|^(q),以及其相应的非齐次抛物方程:u_(t)−Δp^(u)=u^(m)+|∇u|^(q)+h(x)初值问题的临界指标.这里,N≥1,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=∞.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.This article studies the critical exponents for the homogeneous evolu_(t)ion p-Laplacian equation u_(t)−Δp^(u)=u^(m)+|∇u|^(q) and its inhomogeneous version u_(t)−Δp^(u)=u^(m)+|∇u|^(q)+h(x)in R^(N)×R^(+),u(x,0)=u0(x)in R^(N),where N≥1,p,m,q>1.We obtain a discontinuous critical exponent result for the homogeneous evolu_(t)ion p-Laplacian equation,which demonstrates the gradient term brings abou_(t) a significant phenomenon of the critical exponent,changing from m=p−1+p/N to m=∞as q goes to the value p−N/(N+1)from above.Meanwhile,we also investigate the inhomogeneous evolu_(t)ion p-Laplacian equation and get a different discontinuous critical exponent result.

关 键 词：临界指标 P-LAPLACIAN方程 非线性梯度项 爆破 全局存在 

分 类 号：O175.29[理学—数学]