# American Institute of Mathematical Sciences

December  2012, 32(12): 4361-4390. doi: 10.3934/dcds.2012.32.4361

## Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping

 1 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588-0130, United States 2 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588 3 Department of Mathematics & Computer Science, Berry College, Mount Berry, GA 30149-5014, United States

Received  June 2011 Revised  January 2012 Published  August 2012

This paper presents a study of the nonlinear wave equation with $p$-Laplacian damping: $u_{tt} - \Delta u - \Delta _p u_t = f(u)$ evolving in a bounded domain $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions. The nonlinearity $f(u)$ represents a strong source which is allowed to have a supercritical exponent, i.e., the Nemytski operator $f(u)$ is not locally Lipschitz from $H{1\atop 0}(\Omega)$ into $L^2(\Omega)$. The nonlinear term $- \Delta _p u_t$ acts as a strong damping where the $-\Delta _p$ denotes the $p$-Laplacian. Under suitable assumptions on the parameters and with careful analysis involving the Nehari Manifold, we prove the existence of a global solution and estimate the decay rates of the energy.
Citation: Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
##### References:
 [1] K. Agre and M. A. Rammaha, Global solutions to boundary value problems for a nonlinear wave equation in high space dimensions,, Differential Integral Equations, 14 (2001), 1315.   Google Scholar [2] C. O. Alves, M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Rammaha and D. Toundykov, 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. Ser. S, 2 (2009), 583.   Google Scholar [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar [4] D. D. Áng and A. Pham Ngoc Dinh, Mixed problem for some semilinear wave equation with a nonhomogeneous condition,, Nonlinear Anal., 12 (1988), 581.   Google Scholar [5] V. Barbu, "Nonlinear Semigroups and Differential Equations in BAnach Spaces,", Editura Academiei Republicii Socialiste România, (1976).   Google Scholar [6] V. Barbu, I. Lasiecka and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms,, Trans. Amer. Math. Soc., 357 (2005), 2571.   Google Scholar [7] L. Bociu, Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping,, Nonlinear Anal., (2008).   Google Scholar [8] L. Bociu and I. Lasiecka, Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping,, Appl. Math. (Warsaw), 35 (2008), 281.  doi: 10.4064/am35-3-3.  Google Scholar [9] L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping,, Discrete Contin. Dyn. Syst., 22 (2008), 835.  doi: 10.3934/dcds.2008.22.835.  Google Scholar [10] L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping,, J. Differential Equations, 249 (2010), 654.  doi: 10.1016/j.jde.2010.03.009.  Google Scholar [11] L. Bociu and P. Radu, Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping,, Discrete Contin. Dyn. Syst., (2009), 60.   Google Scholar [12] L. Bociu, M. A. Rammaha and D. Toundykov, On a wave equation with supercritical interior and boundary sources and damping terms,, Math. Nach., 284 (2011), 2032.  doi: 10.1002/mana.200910182.  Google Scholar [13] E. Di Benedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827.   Google Scholar [14] 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.   Google Scholar [15] R. T. Glassey, Blow-up theorems for nonlinear wave equations,, Math. Z., 132 (1973), 183.  doi: 10.1007/BF01213863.  Google Scholar [16] A. Haraux, "Nonlinear Evolution Equations - Global Behavior of Solutions,", Lecture Notes in Mathematics, (1981).   Google Scholar [17] W. G. Kelley and A. C. Peterson, "The Theory of Differential Equations. Classical and Qualitative,", 2nd ed., (2010).   Google Scholar [18] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential Integral Equations, 6 (1993), 507.   Google Scholar [19] I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms,, Nonlinear Anal., 64 (2006), 1757.   Google Scholar [20] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt=-Au+ Ţ(u) ,, Trans. Amer. Math. Soc., 192 (1974), 1.   Google Scholar [21] H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation,, Arch. Rational Mech. Anal., 137 (1997), 341.   Google Scholar [22] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273.   Google Scholar [23] D. R. Pitts and M. A. Rammaha, Global existence and non-existence theorems for nonlinear wave equations,, Indiana Univ. Math. J., 51 (2002), 1479.   Google Scholar [24] M. A. Rammaha and Z. Wilstein, Hadamard well-posedness for wave equations with p-Laplacian damping and supercritical sources,, Adv. Differential Equations, 17 (2012), 105.   Google Scholar [25] M. A. Rammaha and T. A. Strei, Global existence and nonexistence for nonlinear wave equations with damping and source terms,, Trans. Amer. Math. Soc., 354 (2002), 3621.   Google Scholar [26] D. Toundykov, Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions,, Nonlinear Anal., 67 (2007), 512.   Google Scholar [27] E. Vitillaro, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy,, Rend. Istit. Mat. Univ.Trieste, 31 (2000), 245.   Google Scholar [28] —, A potential well theory for the wave equation with nonlinear source and boundary damping terms,, Glasg. Math. J., 44 (2002), 375.   Google Scholar [29] —, Global existence for the wave equation with nonlinear boundary damping and source terms,, Journal of Differential Equations, 186 (2002), 259.   Google Scholar [30] G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation,, Canad. J. Math., 32 (1980), 631.   Google Scholar [31] M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, 24 (1996).   Google Scholar

show all references

##### References:
 [1] K. Agre and M. A. Rammaha, Global solutions to boundary value problems for a nonlinear wave equation in high space dimensions,, Differential Integral Equations, 14 (2001), 1315.   Google Scholar [2] C. O. Alves, M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Rammaha and D. Toundykov, 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. Ser. S, 2 (2009), 583.   Google Scholar [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar [4] D. D. Áng and A. Pham Ngoc Dinh, Mixed problem for some semilinear wave equation with a nonhomogeneous condition,, Nonlinear Anal., 12 (1988), 581.   Google Scholar [5] V. Barbu, "Nonlinear Semigroups and Differential Equations in BAnach Spaces,", Editura Academiei Republicii Socialiste România, (1976).   Google Scholar [6] V. Barbu, I. Lasiecka and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms,, Trans. Amer. Math. Soc., 357 (2005), 2571.   Google Scholar [7] L. Bociu, Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping,, Nonlinear Anal., (2008).   Google Scholar [8] L. Bociu and I. Lasiecka, Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping,, Appl. Math. (Warsaw), 35 (2008), 281.  doi: 10.4064/am35-3-3.  Google Scholar [9] L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping,, Discrete Contin. Dyn. Syst., 22 (2008), 835.  doi: 10.3934/dcds.2008.22.835.  Google Scholar [10] L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping,, J. Differential Equations, 249 (2010), 654.  doi: 10.1016/j.jde.2010.03.009.  Google Scholar [11] L. Bociu and P. Radu, Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping,, Discrete Contin. Dyn. Syst., (2009), 60.   Google Scholar [12] L. Bociu, M. A. Rammaha and D. Toundykov, On a wave equation with supercritical interior and boundary sources and damping terms,, Math. Nach., 284 (2011), 2032.  doi: 10.1002/mana.200910182.  Google Scholar [13] E. Di Benedetto, $C^{1+\alpha}$ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827.   Google Scholar [14] 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.   Google Scholar [15] R. T. Glassey, Blow-up theorems for nonlinear wave equations,, Math. Z., 132 (1973), 183.  doi: 10.1007/BF01213863.  Google Scholar [16] A. Haraux, "Nonlinear Evolution Equations - Global Behavior of Solutions,", Lecture Notes in Mathematics, (1981).   Google Scholar [17] W. G. Kelley and A. C. Peterson, "The Theory of Differential Equations. Classical and Qualitative,", 2nd ed., (2010).   Google Scholar [18] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential Integral Equations, 6 (1993), 507.   Google Scholar [19] I. Lasiecka and D. Toundykov, Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms,, Nonlinear Anal., 64 (2006), 1757.   Google Scholar [20] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt=-Au+ Ţ(u) ,, Trans. Amer. Math. Soc., 192 (1974), 1.   Google Scholar [21] H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation,, Arch. Rational Mech. Anal., 137 (1997), 341.   Google Scholar [22] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273.   Google Scholar [23] D. R. Pitts and M. A. Rammaha, Global existence and non-existence theorems for nonlinear wave equations,, Indiana Univ. Math. J., 51 (2002), 1479.   Google Scholar [24] M. A. Rammaha and Z. Wilstein, Hadamard well-posedness for wave equations with p-Laplacian damping and supercritical sources,, Adv. Differential Equations, 17 (2012), 105.   Google Scholar [25] M. A. Rammaha and T. A. Strei, Global existence and nonexistence for nonlinear wave equations with damping and source terms,, Trans. Amer. Math. Soc., 354 (2002), 3621.   Google Scholar [26] D. Toundykov, Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions,, Nonlinear Anal., 67 (2007), 512.   Google Scholar [27] E. Vitillaro, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy,, Rend. Istit. Mat. Univ.Trieste, 31 (2000), 245.   Google Scholar [28] —, A potential well theory for the wave equation with nonlinear source and boundary damping terms,, Glasg. Math. J., 44 (2002), 375.   Google Scholar [29] —, Global existence for the wave equation with nonlinear boundary damping and source terms,, Journal of Differential Equations, 186 (2002), 259.   Google Scholar [30] G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation,, Canad. J. Math., 32 (1980), 631.   Google Scholar [31] M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, 24 (1996).   Google Scholar
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