RESEARCH ARTICLE

Global solutions and blow-up for a class of strongly damped wave equations systems

  • Yaojun YE ,
  • Lanlan LI
Expand
  • Department of Mathematics and Statistics, Zhejiang University of Science and Technology, Hangzhou 310023, China

Published date: 15 Oct 2022

Copyright

2022 Higher Education Press 2022

Abstract

The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied. The existence of global solutions for this problem is proved by using potential well method, and the exponential decay of global solutions is given through introducing an appropriate Lyapunov function. Meanwhile, blow-up of solutions in the unstable set is also obtained.

Cite this article

Yaojun YE , Lanlan LI . Global solutions and blow-up for a class of strongly damped wave equations systems[J]. Frontiers of Mathematics in China, 2022 , 17(5) : 767 -782 . DOI: 10.1007/s11464-022-1025-y

Acknowledgements

This research was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LY17A010009).
1
Agre K, Rammaha M A. Systems of nonlinear wave equations with damping and source terms. Differential Integral Equations 2006; 19: 1235–1270

2
Alves C O, Cavalcanti M M, Domingos V N, Rammaha M, Toundykov D. On existence uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete Contin Dyn Syst-S 2009; 2: 583–608

3
Gao H J, Ma T F. Global solutions for a nonlinear wave equation with the Laplacian operator. Electron J Qual Theory Differ Equ 1999; 11: 11–13

4
Gazzola F, Squassina M. Global solutions and finite time blow up for damped semilinear wave equations. Ann I H Poincaré-AN 2006; 23: 185–207

5
Gerbi S, Said-Houari B. Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions. Adv Differential Equations 2008; 13: 1051–1074

6
Gerbi S, Said-Houari B. Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions. Nonlinear Anal TMA 2011; 74: 7137–7150

7
Gerbi S, Said-Houari B. Exponential decay for solutions to semilinear damped wave equation. Discrete Contin Dyn Syst, Series S 2012; 5: 559–566

8
Kalantarov V K, Ladyzhenskaya O A. The occurrence of collapse for quasi-linear equation of parabolic and hyperbolic typers. J Soviet Math 1978; 10: 53–70

9
Li G, Sun Y N, Liu W J. Global existence, uniform decay and blow-up of solutions for a system of Petrovsky equations. Nonlinear Anal TMA 2011; 74: 1523–1538

10
Messaoudi S A. Global existence and nonexistence in a system of Petrovsky. J Math Anal Appl 2002; 265: 296–308

11
Messaoudi S A, Said-Houari B. Global nonexistence of positive initial energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. J Math Anal Appl 2010; 365: 277–287

12
Nehari Z. On a class of nonlinear second-order differential equations. Trans Amer Math Soc 1960; 95: 101–123

13
Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Israel J Math 1975; 22: 273–303

14
Rammaha M A, Sakuntasathien S. Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms. Nonlinear Anal TMA 2010; 72: 2658–2683

15
ReedM. Abstract Nonlinear Wave Equations. New York: Springer-Verlag, 1976

16
Said-Houari B. Global existence and decay of solutions of a nonlinear system of wave equations. Appl Anal 2012; 91: 475–489

17
Sattinger D H. On global solutions for nonlinear hyperbolic Equations. Arch Ration Mech Anal 1968; 30: 148–172

18
WillemM. Minimax Theorems. Progress Nonlinear Differential Equations Appl, Vol 24. Boston: Birkhauser, 1996

19
Yang Z J, Chen G W. Global existence of solutions for quasilinear wave equations with viscous damping. J Math Anal Appl 2003; 285: 604–618

Outlines

/