×

Global existence and blow up of solutions for some hyperbolic systems with damping and source terms. (English) Zbl 1082.35100

The paper deals with the initial boundary value problem to the system \[ u_{tt}-(a+b\| \nabla u\| _2^2+b\| \nabla v\| _2^2)\triangle u+g(u_t)=\mu _1f(u) \]
\[ v_{tt}-(a+b\| \nabla u\| _2^2+b\| \nabla v\| _2^2)\triangle v+g(v_t)=\mu _2h(v). \] In the case of small initial data, the existence (for \(a+b>0\)) and uniqueness (for \(a>0\)) of global regular and weak solutions to the problem are proved. If the function \(g\) is linear and \(f,h\) are of a special form, blow up of a solution is proved and its lifespan is estimated.

MSC:

35L55 Higher-order hyperbolic systems
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
45K05 Integro-partial differential equations

Keywords:

lifespan
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, R. A., Sobolev Space (1975), Academic Press: Academic Press New York
[2] Brito, E. H., Nonlinear initial boundary value problems, Nonlinear Anal. Theory Methods Appl., 11, 1, 125-137 (1987) · Zbl 0613.34013
[3] Cavalcanti, M. M.; Domingos Cavalcanti, V. N., Existence and asymptotic stability for evolution problems on manifolds with damping and source terms, J. Math. Anal. Appl., 291, 109-127 (2004) · Zbl 1073.35168
[4] Georgiev, V.; Todorova, G., Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109, 295-308 (1994) · Zbl 0803.35092
[5] Ikehata, R., On the existence of global solutions for some nonlinear hyperbolic equations with Newman conditions, TRU Math., 24, 1-17 (1988) · Zbl 0707.35094
[6] Kirchhoff, G., Vorlesungen uber Mechanik (1883), Teubner: Teubner Leipzig · JFM 08.0542.01
[7] Levine, H. A., Instability and nonexistence of global solutions to nonlinear wave equations of the form \(Pu_{tt} = - Au +(u)\), Trans. Amer. Math. Soc., 192, 1-21 (1974) · Zbl 0288.35003
[8] Levine, H. A., Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physicsthe method of unbounded Fourier coefficient, Math. Ann., 214, 205-220 (1975) · Zbl 0286.35006
[9] Lions, J. L., Quelques méthode de Résolution des probléme aux Limites Nonlinéaire (1969), Dunod Gauthier-Villars: Dunod Gauthier-Villars Paris · Zbl 0189.40603
[10] Matsuyama, T.; Ikehata, R., On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl., 204, 729-753 (1996) · Zbl 0962.35025
[11] Nishihara, K.; Yamada, Y., On global solutions of some degenerate quasilinear hyperbolic equation with dissipative damping terms, Funkcial. Ekvac., 33, 151-159 (1990) · Zbl 0715.35053
[12] Ono, K., Global existence, decay and blow-up of solutions for some mildly degenerate nonlinear Kirchhoff strings, J. Differential Equations, 137, 273-301 (1997) · Zbl 0879.35110
[13] Park, J. Y.; Bae, J. J., On the existence of solutions of the degenerate wave equations with nonlinear damping terms, J. Korean Math. Soc., 35, 2, 465-489 (1998) · Zbl 0904.35052
[14] Park, J. Y.; Bae, J. J., Variational inequality for quasilinear wave equations with nonlinear damping terms, Nonlinear Anal., 50, 1065-1083 (2002) · Zbl 1004.35100
[15] Payne, L. E.; Sattinger, D. H., Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22, 273-303 (1975) · Zbl 0317.35059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.