Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy

被引：49

作者：

Han, Yuzhu ^{[1
]}

Li, Qingwei ^{[2
]}

机构：

[1] Jilin Univ, Sch Math, Changchun 130012, Jilin, Peoples R China

[2] Dalian Maritime Univ, Dept Math, Dalian 116026, Peoples R China

来源：

COMPUTERS & MATHEMATICS WITH APPLICATIONS | 2018年 / 75卷 / 09期

关键词：

Kirchhoff; Potential well method; Arbitrary initial energy; Global existence; Blow up; NONLINEAR HYPERBOLIC-EQUATIONS; SEMILINEAR PARABOLIC EQUATIONS; POTENTIAL WELLS; WAVE-EQUATIONS; INSTABILITY; LEVEL;

D O I：

10.1016/j.camwa.2018.01.047

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper we will apply the modified potential well method and variational method to the study of the long time behaviors of solutions to a class of parabolic equation of Kirchhoff type. Global existence and blow up in finite time of solutions will be obtained for arbitrary initial energy. To be a little more precise, we will give a threshold result for the solutions to exist globally or to blow up in finite time when the initial energy is subcritical and critical, respectively. The decay rate of the L-2(Omega) norm is also obtained for global solutions in these cases. Moreover, some sufficient conditions for the existence of global and blow-up solutions are also derived when the initial energy is supercritical. (C) 2018 Elsevier Ltd. All rights reserved.

引用

页码：3283 / 3297

页数：15

共 17 条

[1]

Chipot M, 2003, REND SEMIN MAT U PAD, V110, P199

[2] GLOBAL SOLVABILITY FOR THE DEGENERATE KIRCHHOFF EQUATION WITH REAL ANALYTIC DATA [J].

DANCONA, P ;

SPAGNOLO, S .

INVENTIONES MATHEMATICAE, 1992, 108 (02) :247-262

[3]

Gazzola F, 2005, DIFFER INTEGRAL EQU, V18, P961

[4] Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates [J].

Ghisi, Marina ;

Gobbino, Massimo .

JOURNAL OF DIFFERENTIAL EQUATIONS, 2008, 245 (10) :2979-3007

[5] Some remarks on the wave equations with nonlinear damping and source terms [J].

Ikehata, R .

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1996, 27 (10) :1165-1175

[6]

LEVINE HA, 1973, ARCH RATION MECH AN, V51, P371

[7] Global existence blow up and extinction for a class of thin-film equation [J].

Li, Qingwei ;

Gao, Wenjie ;

Han, Yuzhu .

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2016, 147 :96-109

[8]

Lions J.L., 1969, QUELQUES METHODS RES

[9] On potential wells and applications to semilinear hyperbolic equations and parabolic equations [J].

Liu, YC ;

Zhao, JS .

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2006, 64 (12) :2665-2687

[10] On potential wells and vacuum isolating of solutions for semilinear wave equations [J].

Liu, YC .

JOURNAL OF DIFFERENTIAL EQUATIONS, 2003, 192 (01) :155-169

← 1 2 →