Abstract
The goal of this paper is to investigate the finite time blow-up of solutions with supercritical boundary/interior sources and nonlinear boundary/interior damping. First, we prove that if the interior and boundary sources dominate their corresponding damping terms, then every weak solution blows up in finite time with positive initial energy. Second, without any restriction on the boundary source, we prove the finite time blow-up of solutions, provided that the interior sources dominate both interior and boundary damping and the initial energy is nonnegative. A similar result has been shown when the boundary source is absent. Moreover, in the absence of the interior sources, we prove that the solution grows as an exponential function.
Similar content being viewed by others
References
Jörgens K. Das Anfangswertproblem im Grossenfür eine Klasse nichtlinearerWellengleichungen. Math Z. 1961;77:295–308.
Segal IE. Non-linear semigroups. Ann Math. 1963;78:339–64.
Lasiecka I. Mathematical control theory of coupled PDE’s, CBMS-SIAM lecture notes. Philadelphia: SIAM; 2002.
Bociu L. Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping. Nonlinear Anal. 2009;71:e560–75.
Bociu L, Rammaha M, Dundykov DT. Wave equations with super-critical interior and boundary nonlinearities. Math Comput Simul. 2012;82:1017–29.
Levine H, Smith R. A potential well theory for the wave equation with a nonlinear boundary condition. J Reine Angwe Math. 1987;374:1–23.
Vitillaro E. Global existence of the wave equation with nonlinear boundary damping and source terms. J Differ Eqns. 2002;186:259–98.
Bociu L, Lasiecka I. Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping. Discrete Contin Dyn Syst. 2008;22:835–60.
Bociu L, Lasiecka I. Hadamard wellposedness for nonlinear wave equations with supercritical sources and damping. J Diff Eqns. 2010;249:654–83.
Bociu L, Rammaha M, Dundykov DT. On a wave equation with supercritical interior and boundary sources and damping terms. Math Nachr. 2011;284:2032–64.
Reed M. Abstract non-linear wave equations.Springer-Verlag; 1976.
Agre K, Rammaha MA. Systems of nonlinear wave equations with damping and source terms. Differ Integr Equ. 2006;19:1235–70.
Georgiev V, Todorova D. Existence of solutions of the wave equations with nonlinear damping and source terms. J Diff Eqns. 1994;109:295–308.
Alves CO, Cavalcanti MM, Domingos VN, 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. 2009;2:583–608.
Guo Y, Rammaha MA. Systems of nonlinear wave equations with damping and supercritical boundary and interior sources. Trans Amer Math Soc. 2012:67.
Guo Y, Rammaha MA. Blow-up solutions to systems of nonlinear wave equations with supercritical sources. Appl Anal Int J. doi:10.1080/00.036811.2011.649734.
Graber P, Said-Houari B. On the wave equations with semilinear porous accoustic bouindary conditions. J Diff Eqns. 2012;252:4898–941.
Adams RA. Sobolev spaces, pure and applied mathematics, vol. 65. New York: Academic; 1975.
Runst T, Sickel W. Sobolev Spaces of fractional Order, Nemytskij operators and nonlinear partial differential equations, de Gruyter series in nonlinear analysis and applications, vol. 3. Berlin: Walter de Gruyter & Co.; 1996.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wu, ST. Blow-up of Positive Initial Energy Solutions for A System of Nonlinear Wave Equations with Supercritical Sources. J Dyn Control Syst 20, 207–227 (2014). https://doi.org/10.1007/s10883-014-9210-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-014-9210-2