Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term. (English) Zbl 1030.35125

The author studies the global existence as well as the exponential decay (as \(t\) tends to \(\infty \)) of weak solutions to the problem \[ u_{tt}+ \Delta^2u + \lambda u_t = \sum_{i=1}^N\frac{\partial}{\partial x_i}\sigma_i u_{x_i} \quad \text{in }\Omega \times (0, \infty), \]
\[ u=0, \;\;\frac{\partial u}{\partial n} = 0 \quad \text{on }\partial \Omega \times [0,\infty), \]
\[ u(x,0)=u_0(x), \;\;u_t(x,0)=u_1(x), \quad x\in \Omega, \] where \(\lambda \geq 0\) is a constant. Under different assumptions the solutions blow up in finite time.


35L75 Higher-order nonlinear hyperbolic equations
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI


[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
[2] 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
[3] Ang, D. D.; Dinh, A. P.N., On the strongly damped wave equation \(u_{ tt } \)−Δ \(u\)−Δ \(u_t+f(u)=0\), SIAM J. Math. Anal., 19, 1409-1418 (1988) · Zbl 0685.35071
[4] Aviles, P.; Sandefur, J., Nonlinear second order equations with applications to partial differential equations, J. Differential Equations, 58, 404-427 (1985) · Zbl 0572.34004
[5] Chen, G.-W.; Yang, Z.-J., Existence and non-existence of global solutions for a class of nonlinear wave equations, Math. Meth. Appl. Sci., 23, 615-631 (2000) · Zbl 1007.35046
[6] Ladyzhenskaya, O. A., The Boundary Value Problems of Mathematical Physics (1985), Springer: Springer New York · Zbl 0588.35003
[7] Nakao, M.; Ono, K., Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214, 325-342 (1993) · Zbl 0790.35072
[8] Nishihara, K., Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, J. Differential Equations, 137, 384-395 (1997) · Zbl 0881.35076
[9] Ono, K., Global existence, asymptotic behavior, and global non-existence of solutions for damped non-linear wave equations of Kirchhoff type in the whole space, Math. Meth. Appl. Sci., 23, 535-560 (2000) · Zbl 0953.35099
[10] Zhou, Y.-L.; Fu, H.-Y., Nonlinear hyperbolic systems of higher order generalized Sine-Gordon type, Acta Math. Sinica, 26, 234-249 (1983) · Zbl 0538.35055
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.