Unstable Ground State and Blow Up Result of Nonlocal Klein-Gordon Equations

被引：8

作者：

Carriao, Paulo Cesar ^{[1
]}

Lehrer, Raquel ^{[2
]}

Vicente, Andre ^{[2
]}

机构：

[1] Univ Fed Minas Gerais, Dept Math, ICEX, Belo Horizonte, MG, Brazil

[2] Univ Estadual Oeste Parana, Ctr Exact & Technol Sci, Cascavel, PR, Brazil

来源：

JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS | 2023年 / 35卷 / 03期

关键词：

Hyperbolic equation; Blow up of solution; Fractional laplacian; Orbital stability; instability; SCALAR FIELD-EQUATIONS; UNIFORM DECAY-RATES; WAVE-EQUATION; CAUCHY-PROBLEM; FRACTIONAL LAPLACIAN; GLOBAL-SOLUTIONS; STANDING WAVES; EXISTENCE; NONEXISTENCE; INSTABILITY;

D O I：

10.1007/s10884-023-10281-3

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

In this paper we study the behaviour of solutions for a nonlocal hyperbolic equation. We use the Pohozaev manifold combined with a new technique to explicit two invariant regions in the space of initial data. On the first one the solution blows up (in finite or infinite time) and in the second one the solution exists globally. Additionally, we prove that the ground state solution of the elliptic problem associated to the original problem is unstable. The main goal of this paper is to present a new technique which allows us to consider nonlocal problems and to extend the classical result proved by Shatah (Trans Am Math Soc 290(2):701-710, 1985).

引用

页码：1917 / 1945

页数：29

共 37 条

[1] Optimal results for the fractional heat equation involving the Hardy potential [J].

Abdellaoui, Boumediene ;

Medina, Maria ;

Peral, Ireneo ;

Primo, Ana .

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2016, 140 :166-207

[2] Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations [J].

Alibaud, Nathael ;

Cifani, Simone ;

Jakobsen, Espen R. .

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 2014, 213 (03) :705-762

[3] ON EXISTENCE, UNIFORM DECAY RATES AND BLOW UP FOR SOLUTIONS OF SYSTEMS OF NONLINEAR WAVE EQUATIONS WITH DAMPING AND SOURCE TERMS [J].

Alves, Claudianor O. ;

Cavalcanti, Marcelo M. ;

Domingos Cavalcanti, Valeria N. ;

Rammaha, Mohammad A. ;

Toundykov, Daniel .

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S, 2009, 2 (03) :583-608

[4] On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source [J].

Alves, Claudianor O. ;

Cavalcanti, Marcelo M. .

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 2009, 34 (03) :377-411

[5]

Ambrosetti A., 1973, Journal of Functional Analysis, V14, P349, DOI 10.1016/0022-1236(73)90051-7

[6]

BERESTYCKI H, 1983, ARCH RATION MECH AN, V82, P313

[7]

Brezis H., 1973, OPERATEURS MAXIMAUX

[8]

Carriao PC., 2016, ELECTRON J DIFFER EQ, V2016, P1

[9] Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity [J].

Chang, X. ;

Wang, Z-Q .

NONLINEARITY, 2013, 26 (02) :479-494

[10] Fractional heat equations with subcritical absorption having a measure as initial data [J].

Chen, Huyuan ;

Veron, Laurent ;

Wang, Ying .

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2016, 137 :306-337

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