×

Global solutions and finite time blow up for damped semilinear wave equations. (English) Zbl 1094.35082

Weak solution of the equation \(u_{tt}-\triangle u -\omega \triangle u_t+\mu u_t=| u| ^{p-2}u\) on a bounded domain \(\Omega\) satisfying zero boundary condition on \(\partial \Omega \) and initial data for \(t=0\) is treated. Uniform boundedness of every global solution is shown. For some constants \(\omega ,\mu \), high energy initial data for which the solution blows up are constructed.

MSC:

35L75 Higher-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063
[2] Ball, J., Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10, 31-52 (2004) · Zbl 1056.37084
[3] Carvalho, A. N.; Cholewa, J. W., Local well posedness for strongly damped wave equations with critical nonlinearities, Bull. Austral. Math. Soc., 66, 443-463 (2002) · Zbl 1020.35059
[4] Cazenave, T., Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60, 36-55 (1985) · Zbl 0568.35068
[5] Esquivel-Avila, J., The dynamics of a nonlinear wave equation, J. Math. Anal. Appl., 279, 135-150 (2003) · Zbl 1015.35072
[6] Esquivel-Avila, J., Qualitative analysis of a nonlinear wave equation, Discrete Contin. Dyn. Syst., 10, 787-804 (2004) · Zbl 1047.35103
[7] Gazzola, F., Finite time blow-up and global solutions for some nonlinear parabolic equations, Differential Integral Equations, 17, 983-1012 (2004) · Zbl 1150.35336
[8] F. Gazzola, T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, in press; F. Gazzola, T. Weth, Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, in press · Zbl 1212.35248
[9] Georgiev, V.; Todorova, G., Existence of a solution of the wave equation with nonlinear damping and source term, J. Differential Equations, 109, 295-308 (1994) · Zbl 0803.35092
[10] Hale, J. K.; Raugel, G., Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys., 43, 63-124 (1992) · Zbl 0751.58033
[11] Haraux, A., Dissipative Dynamical Systems and Applications, Res. Appl. Math., vol. 17 (1991), Masson: Masson Paris, 132 p · Zbl 0726.58001
[12] Haraux, A.; Jendoubi, M. A., Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9, 95-124 (1999) · Zbl 0939.35122
[13] Ikehata, R., Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27, 1165-1175 (1996) · Zbl 0866.35071
[14] Ikehata, R.; Suzuki, T., Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26, 475-491 (1996) · Zbl 0873.35010
[15] Jendoubi, M. A.; Poláčik, P., Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping, Proc. Roy. Soc. Edinburgh Sect. A, 133, 1137-1153 (2003) · Zbl 1046.37045
[16] Levine, H. A., Instability and nonexistence of global solutions to nonlinear wave equations of the form \(P u_{t t} = - A u + F(u)\), Trans. Amer. Math. Soc., 192, 1-21 (1974) · Zbl 0288.35003
[17] Levine, H. A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5, 138-146 (1974) · Zbl 0243.35069
[18] Levine, H. A.; Serrin, J., Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137, 341-361 (1997) · Zbl 0886.35096
[19] Levine, H. A.; Todorova, G., Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy, Proc. Amer. Math. Soc., 129, 793-805 (2001) · Zbl 0956.35087
[20] Nehari, Z., On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95, 101-123 (1960) · Zbl 0097.29501
[21] Ohta, M., Remarks on blowup of solutions for nonlinear evolution equations of second order, Adv. Math. Sci. Appl., 8, 901-910 (1998) · Zbl 0920.35025
[22] 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, 151-177 (1997) · Zbl 0878.35081
[23] Pata, V.; Squassina, M., On the strongly damped wave equation, Comm. Math. Phys., 253, 511-533 (2004) · Zbl 1068.35077
[24] Payne, L. E.; Sattinger, D. H., Saddle points and instability of nonlinear hyperbolic equations, Israel Math. J., 22, 273-303 (1975) · Zbl 0317.35059
[25] Pucci, P.; Serrin, J., Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150, 203-214 (1998) · Zbl 0915.35012
[26] Pucci, P.; Serrin, J., Some new results on global nonexistence for abstract evolution with positive initial energy, Topol. Methods Nonlinear Anal., 10, 241-247 (1997) · Zbl 0911.35035
[27] Sattinger, D. H., On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30, 148-172 (1968) · Zbl 0159.39102
[28] Tsutsumi, M., On solutions of semilinear differential equations in a Hilbert space, Math. Japon., 17, 173-193 (1972) · Zbl 0273.34044
[29] Vitillaro, E., Global existence theorems for a class of evolution equations with dissipation, Arch. Rational Mech. Anal., 149, 155-182 (1999) · Zbl 0934.35101
[30] Webb, G. F., Compactness of bounded trajectories of dynamical systems in infinite-dimensional spaces, Proc. Roy. Soc. Edinburgh Sect. A, 84, 19-33 (1979) · Zbl 0414.34042
[31] Willem, M., Minimax Theorems, Progress Nonlinear Differential Equations Appl., vol. 24 (1996), Birkhäuser Boston: Birkhäuser Boston Boston, MA, 162 p · Zbl 0856.49001
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.