Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

被引:0
作者
V. A. Galaktionov
S. I. Pohozaev
机构
[1] University of Bath,Department of Math. Sci
[2] Keldysh Institute of Applied Mathematics,undefined
[3] Steklov Mathematical Institute,undefined
来源
Journal of Evolution Equations | 2006年 / 6卷
关键词
35K55; Dual porous medium equation; blow-up; global solutions; critical exponents; thin film equations;
D O I
暂无
中图分类号
学科分类号
摘要
Our first basic model is the fully nonlinear dual porous medium equation with source \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{t} = {\left| {\Delta u} \right|}^{{m - 1}} \Delta u + u^{p} \quad in\;{\mathbb{R}}^{N} \times {\mathbb{R}}_{ + } ,\quad m > 1,\;\; p > 1,$$\end{document} for which we consider the Cauchy problem with given nonnegative bounded initial data u0. For the semilinear case m=1, the critical exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0} = 1 + \frac{2}{N}$$\end{document} was obtained by H. Fujita in 1966. For p ∈(1, p0] any nontrivial solution blows up in finite time, while for p > p0 there exist sufficiently small global solutions. During last thirty years such critical exponents were detected for many semilinear and quasilinear parabolic, hyperbolic and elliptic PDEs and inequalities. Most of efforts were devoted to equations with differential operators in divergent form, where classical techniques associated with weak solutions and integration by parts with a variety of test functions can be applied. Using this fully nonlinear equation, we propose and develop new approaches to calculating critical Fujita exponents in different functional settings.
引用
收藏
页码:45 / 69
页数:24
相关论文
empty
未找到相关数据