×

Fourth order wave equations with nonlinear strain and source terms. (English) Zbl 1113.35113

Summary: We study the initial boundary value problem for fourth order wave equations with nonlinear strain and source terms. First we introduce a family of potential wells and prove the invariance of some sets and vacuum isolating of solutions. Then we obtain a threshold result of global existence and nonexistence. Finally we discuss the global existence of solutions for the problem with critical initial condition \(I(u_{0})\geqslant 0, E(0)=d\). So the Esquivel-Avila’s results are generalized and improved.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] An, L. J.; Peirce, A., A weakly nonlinear analysis of elasto-plastic-microstructure models, SIAM J. Appl. Math., 55, 136-155 (1995) · Zbl 0815.73022
[2] Chen, Guowang; Yang, Zhijian, Existence and non-existence of global solutions for a class of nonlinear wave equations, Math. Methods Appl. Sci., 23, 615-631 (2000) · Zbl 1007.35046
[3] Zhang, Hongwei; Chen, Guowang, Potential well method for a class of nonlinear wave equations of fourth-order, Acta Math. Sci. Ser. A, 23, 6, 758-768 (2003), (in Chinese)
[4] Yang, Zhijian, Global existence,asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187, 520-540 (2003) · Zbl 1030.35125
[5] Esquivel-Avila, J. A., Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal., 63, 5-7, 331-343 (2005) · Zbl 1159.35390
[6] Sattinger, D. H., On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30, 148-172 (1968) · Zbl 0159.39102
[7] Payne, L. E.; Sattinger, D. H., Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22, 273-303 (1975) · Zbl 0317.35059
[8] Tsutsumi, M., On solutions of semilinear differential equations in a Hilbert space, Math. Japan, 17, 173-193 (1972) · Zbl 0273.34044
[9] Tsutsumi, M., Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. RTMS, 8, 211-229 (1972/73) · Zbl 0248.35074
[10] Lions, J. L., Quelques methods de resolution des problem aux limits nonlinears (1969), Dunod: Dunod Paris · Zbl 0189.40603
[11] Ikehata, R., Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27, 1165-1175 (1996) · Zbl 0866.35071
[12] Nakao, M.; Ono, K., Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214, 2, 325-342 (1993) · Zbl 0790.35072
[13] Cavalcanti, M. Marcelo; Valéria, N.; Cavalcanti, Domingos, 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
[14] Cavalcanti, M. Marcelo; Valéria, N.; Cavalcanti, Domingos; Martinez, Patrick, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203, 119-158 (2004) · Zbl 1049.35047
[15] Zhang, J., On the standing wave in coupled non-linear Klein-Gordon equations, Math. Methods Appl. Sci., 26, 1, 11-25 (2003) · Zbl 1034.35080
[16] Vitillaro, E., A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J., 44, 3, 375-395 (2002) · Zbl 1016.35048
[17] Ono, K., On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20, 2, 151-177 (1997) · Zbl 0878.35081
[18] Esquivel-Avila, J. A., Qualitative analysis of a nonlinear wave equation, Discrete Contin. Dyn. Syst., 10, 3, 787-804 (2004) · Zbl 1047.35103
[19] Esquivel-Avila, J. A., The dynamics of a nonlinear wave equation, J. Math. Anal. Appl., 279, 135-150 (2003) · Zbl 1015.35072
[20] Esquivel-Avila, J. A., A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations, Nonlinear Anal., 52, 1111-1127 (2003) · Zbl 1023.35076
[21] Gazzola, Filippo; Squassina, Marco, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Lineairé, 23, 2, 185-207 (2006) · Zbl 1094.35082
[22] Liu, L.; Wang, M., Global solutions and blow-up of solutions for some hyperbolic systems with damping and source terms, Nonlinear Anal., 64, 1, 69-91 (2006) · Zbl 1082.35100
[23] Gan, Zaihui; Zhang, Jian, Instability of standing waves for Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions, J. Math. Anal. Appl., 307, 219-231 (2005) · Zbl 1068.35066
[24] Cavalcanti, M. Marcelo; Valéria, N.; Cavalcanti, Domingos, 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
[25] Liu, Yacheng, On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192, 155-169 (2003) · Zbl 1024.35078
[26] Liu, Yacheng; Zhao, Junsheng, Multidimensional viscoelasticity equations with nonlinear damping and source terms, Nonlinear Anal., 56, 851-865 (2004) · Zbl 1057.74007
[27] Liu, Yacheng; Zhao, Junsheng, Nonlinear parabolic equations with critical initial conditions \(J(u_0) = d\) or \(I(u_0) = 0\), Nonlinear Anal., 58, 873-883 (2004) · Zbl 1059.35064
[28] Ebihara, Y.; Nakao, M.; Nanbu, T., On the existence of global classical solution of initial-boundary value problem for \(u - u^3 = f\), Pacific J. Math., 60, 63-69 (1975) · Zbl 0324.35061
[29] Miranda, M. M.; Medeiros, L. A., On the existence of global solutions of a coupled nonlinear Klein-Gordon equations, Funkcial. Ekvac., 30, 134-145 (1987) · Zbl 0637.35055
[30] Ikehata, R., A note on the global solvability of solutions to some nonlinear wave equations with dissipative terms, Differential Integral Equations, 8, 607-616 (1995) · Zbl 0812.35081
[31] Matsuyama, Tokio; Ikehata, Ryo, 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
[32] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
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.