American Institute of Mathematical Sciences

October  2017, 10(5): 1175-1185. doi: 10.3934/dcdss.2017064

Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term

 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  September 2016 Revised  February 2017 Published  June 2017

Fund Project: This work is partially supported by the the Basic and Advanced Research Project of CQCSTC grant cstc2016jcyjA0018, NSFC grant 11201380, Fundamental Research Funds for the Central Universities grant XDJK2015A16, XDJK2016E120, Project funded by China Postdoctoral Science Foundation grant 2014M550453,2015T80948

This paper deals with a higher-order wave equation with general nonlinear dissipation and source term
 $u''+(-Δ)^mu+g(u')=b|u|^{p-2}u,$
which was studied extensively when
 $m=1, 2$
and the nonlinear dissipative term
 $g(u')$
is a polynomial, i.e.,
 $g(u')=a|u'|^{q-2}u'$
. We obtain the global existence of solutions and show the energy decay estimate when
 $m≥1$
is a positive integer and the nonlinear dissipative term
 $g$
does not necessarily have a polynomial grow near the origin.
Citation: Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064
References:
 [1] M. Aassila, Global existence of solutions to a wave equation with damping and source terms, Diff. Inte. Equations, 14 (2001), 1301-1314.   Google Scholar [2] Q. Gao, F. Li and Y. Wang, Blow up of solution for higher-order Kirchhoff-type equations with nonlinear dissipation, Cent. Euro. J. Math., 9 (2011), 686-698.  doi: 10.2478/s11533-010-0096-2.  Google Scholar [3] V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.  Google Scholar [4] R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475-491.   Google Scholar [5] R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27 (1996), 1165-1175.  doi: 10.1016/0362-546X(95)00119-G.  Google Scholar [6] H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Du_{tt}=Au+f(u)$, Trans. Am. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar [7] H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.  Google Scholar [8] P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM. Cont. Opt. Cal. Var., 4 (1999), 419-444.  doi: 10.1051/cocv:1999116.  Google Scholar [9] S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265 (2002), 296-308.  doi: 10.1006/jmaa.2001.7697.  Google Scholar [10] M. Nako, Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term, J. Math. Anal. Appl., 58 (1977), 336-343.  doi: 10.1016/0022-247X(77)90211-6.  Google Scholar [11] K. Ono, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl., 216 (1997), 321-342.  doi: 10.1006/jmaa.1997.5697.  Google Scholar [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, in: Scattering Theiry, vol Ⅲ, Academic Press, New York, London, 1979.  Google Scholar [13] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.  Google Scholar [14] G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, J. Math. Anal. Appl., 239 (1999), 213-226.  doi: 10.1006/jmaa.1999.6528.  Google Scholar [15] S. T. Wu and L. Y. Tsai, On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwanese J. Math., 13 (2009), 545-558.   Google Scholar [16] Y. Ye, Existence and asymptotic behavior of gobal solutions for aclass of nonlinear higher-order wave equation, J. Ineq. Appl. , 2010 (2010), Art. ID 394859, 14 pp. doi: 10.1155/2010/394859.  Google Scholar [17] E. Zauderer, Partial Differential Equations of Applied Mathematics, in: Pure and Applied Mathematics, second edition, A Wiley-interscience Publication, Johu Wiely & Sons, Inc. , New York, 1989.  Google Scholar [18] J. Zhou, X. R. Wang, X. J. Song and C. L. Mu, Global existence and blowup of solutions for a class of nonlinear higher-order wave equations, Z. Angew. Math. Phys., 63 (2012), 461-473.  doi: 10.1007/s00033-011-0165-9.  Google Scholar

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References:
 [1] M. Aassila, Global existence of solutions to a wave equation with damping and source terms, Diff. Inte. Equations, 14 (2001), 1301-1314.   Google Scholar [2] Q. Gao, F. Li and Y. Wang, Blow up of solution for higher-order Kirchhoff-type equations with nonlinear dissipation, Cent. Euro. J. Math., 9 (2011), 686-698.  doi: 10.2478/s11533-010-0096-2.  Google Scholar [3] V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.  Google Scholar [4] R. Ikehata and T. Suzuki, Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26 (1996), 475-491.   Google Scholar [5] R. Ikehata, Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27 (1996), 1165-1175.  doi: 10.1016/0362-546X(95)00119-G.  Google Scholar [6] H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equation of the form $Du_{tt}=Au+f(u)$, Trans. Am. Math. Soc., 192 (1974), 1-21.  doi: 10.2307/1996814.  Google Scholar [7] H. A. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5 (1974), 138-146.  doi: 10.1137/0505015.  Google Scholar [8] P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM. Cont. Opt. Cal. Var., 4 (1999), 419-444.  doi: 10.1051/cocv:1999116.  Google Scholar [9] S. A. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265 (2002), 296-308.  doi: 10.1006/jmaa.2001.7697.  Google Scholar [10] M. Nako, Asymptotic stability of the bounded or almost periodic solution of the wave equation with nonlinear dissipative term, J. Math. Anal. Appl., 58 (1977), 336-343.  doi: 10.1016/0022-247X(77)90211-6.  Google Scholar [11] K. Ono, On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation, J. Math. Anal. Appl., 216 (1997), 321-342.  doi: 10.1006/jmaa.1997.5697.  Google Scholar [12] M. Reed and B. Simon, Methods of Modern Mathematical Physics, in: Scattering Theiry, vol Ⅲ, Academic Press, New York, London, 1979.  Google Scholar [13] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172.  doi: 10.1007/BF00250942.  Google Scholar [14] G. Todorova, Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms, J. Math. Anal. Appl., 239 (1999), 213-226.  doi: 10.1006/jmaa.1999.6528.  Google Scholar [15] S. T. Wu and L. Y. Tsai, On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwanese J. Math., 13 (2009), 545-558.   Google Scholar [16] Y. Ye, Existence and asymptotic behavior of gobal solutions for aclass of nonlinear higher-order wave equation, J. Ineq. Appl. , 2010 (2010), Art. ID 394859, 14 pp. doi: 10.1155/2010/394859.  Google Scholar [17] E. Zauderer, Partial Differential Equations of Applied Mathematics, in: Pure and Applied Mathematics, second edition, A Wiley-interscience Publication, Johu Wiely & Sons, Inc. , New York, 1989.  Google Scholar [18] J. Zhou, X. R. Wang, X. J. Song and C. L. Mu, Global existence and blowup of solutions for a class of nonlinear higher-order wave equations, Z. Angew. Math. Phys., 63 (2012), 461-473.  doi: 10.1007/s00033-011-0165-9.  Google Scholar
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