
Previous Article
Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps
 DCDS Home
 This Issue

Next Article
Entropy formulas for dynamical systems with mistakes
Global existence and decay of energy for a nonlinear wave equation with $p$Laplacian damping
1.  Department of Mathematics, University of NebraskaLincoln, Lincoln, NE, 685880130, United States 
2.  Department of Mathematics, University of NebraskaLincoln, Lincoln, NE 68588 
3.  Department of Mathematics & Computer Science, Berry College, Mount Berry, GA 301495014, United States 
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. 
[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. 
[3] 
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349. doi: 10.1016/00221236(73)900517. 
[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. 
[5] 
V. Barbu, "Nonlinear Semigroups and Differential Equations in BAnach Spaces,", Editura Academiei Republicii Socialiste România, (1976). 
[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. 
[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). 
[8] 
L. Bociu and I. Lasiecka, Blowup 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/am3533. 
[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. 
[10] 
L. Bociu and I. Lasiecka, Local Hadamard wellposedness for nonlinear wave equations with supercritical sources and damping,, J. Differential Equations, 249 (2010), 654. doi: 10.1016/j.jde.2010.03.009. 
[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. 
[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. 
[13] 
E. Di Benedetto, $C^{1+\alpha} $ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827. 
[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. 
[15] 
R. T. Glassey, Blowup theorems for nonlinear wave equations,, Math. Z., 132 (1973), 183. doi: 10.1007/BF01213863. 
[16] 
A. Haraux, "Nonlinear Evolution Equations  Global Behavior of Solutions,", Lecture Notes in Mathematics, (1981). 
[17] 
W. G. Kelley and A. C. Peterson, "The Theory of Differential Equations. Classical and Qualitative,", 2nd ed., (2010). 
[18] 
I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential Integral Equations, 6 (1993), 507. 
[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. 
[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. 
[21] 
H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation,, Arch. Rational Mech. Anal., 137 (1997), 341. 
[22] 
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273. 
[23] 
D. R. Pitts and M. A. Rammaha, Global existence and nonexistence theorems for nonlinear wave equations,, Indiana Univ. Math. J., 51 (2002), 1479. 
[24] 
M. A. Rammaha and Z. Wilstein, Hadamard wellposedness for wave equations with pLaplacian damping and supercritical sources,, Adv. Differential Equations, 17 (2012), 105. 
[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. 
[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. 
[27] 
E. Vitillaro, Some new results on global nonexistence and blowup for evolution problems with positive initial energy,, Rend. Istit. Mat. Univ.Trieste, 31 (2000), 245. 
[28] 
—, A potential well theory for the wave equation with nonlinear source and boundary damping terms,, Glasg. Math. J., 44 (2002), 375. 
[29] 
—, Global existence for the wave equation with nonlinear boundary damping and source terms,, Journal of Differential Equations, 186 (2002), 259. 
[30] 
G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation,, Canad. J. Math., 32 (1980), 631. 
[31] 
M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, 24 (1996). 
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. 
[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. 
[3] 
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349. doi: 10.1016/00221236(73)900517. 
[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. 
[5] 
V. Barbu, "Nonlinear Semigroups and Differential Equations in BAnach Spaces,", Editura Academiei Republicii Socialiste România, (1976). 
[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. 
[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). 
[8] 
L. Bociu and I. Lasiecka, Blowup 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/am3533. 
[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. 
[10] 
L. Bociu and I. Lasiecka, Local Hadamard wellposedness for nonlinear wave equations with supercritical sources and damping,, J. Differential Equations, 249 (2010), 654. doi: 10.1016/j.jde.2010.03.009. 
[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. 
[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. 
[13] 
E. Di Benedetto, $C^{1+\alpha} $ local regularity of weak solutions of degenerate elliptic equations,, Nonlinear Anal., 7 (1983), 827. 
[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. 
[15] 
R. T. Glassey, Blowup theorems for nonlinear wave equations,, Math. Z., 132 (1973), 183. doi: 10.1007/BF01213863. 
[16] 
A. Haraux, "Nonlinear Evolution Equations  Global Behavior of Solutions,", Lecture Notes in Mathematics, (1981). 
[17] 
W. G. Kelley and A. C. Peterson, "The Theory of Differential Equations. Classical and Qualitative,", 2nd ed., (2010). 
[18] 
I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping,, Differential Integral Equations, 6 (1993), 507. 
[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. 
[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. 
[21] 
H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation,, Arch. Rational Mech. Anal., 137 (1997), 341. 
[22] 
L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations,, Israel J. Math., 22 (1975), 273. 
[23] 
D. R. Pitts and M. A. Rammaha, Global existence and nonexistence theorems for nonlinear wave equations,, Indiana Univ. Math. J., 51 (2002), 1479. 
[24] 
M. A. Rammaha and Z. Wilstein, Hadamard wellposedness for wave equations with pLaplacian damping and supercritical sources,, Adv. Differential Equations, 17 (2012), 105. 
[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. 
[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. 
[27] 
E. Vitillaro, Some new results on global nonexistence and blowup for evolution problems with positive initial energy,, Rend. Istit. Mat. Univ.Trieste, 31 (2000), 245. 
[28] 
—, A potential well theory for the wave equation with nonlinear source and boundary damping terms,, Glasg. Math. J., 44 (2002), 375. 
[29] 
—, Global existence for the wave equation with nonlinear boundary damping and source terms,, Journal of Differential Equations, 186 (2002), 259. 
[30] 
G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation,, Canad. J. Math., 32 (1980), 631. 
[31] 
M. Willem, "Minimax Theorems,", Progress in Nonlinear Differential Equations and their Applications, 24 (1996). 
[1] 
Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 3759. doi: 10.3934/eect.2016.5.37 
[2] 
Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of pLaplacian type equation with critical growth and potential vanishing at infinity. Discrete & Continuous Dynamical Systems  A, 2016, 36 (2) : 683699. doi: 10.3934/dcds.2016.36.683 
[3] 
Dimitri Mugnai. Bounce on a pLaplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371379. doi: 10.3934/cpaa.2003.2.371 
[4] 
Robert Stegliński. On homoclinic solutions for a second order difference equation with pLaplacian. Discrete & Continuous Dynamical Systems  B, 2018, 23 (1) : 487492. doi: 10.3934/dcdsb.2018033 
[5] 
Tomás Caraballo, Marta HerreraCobos, Pedro MarínRubio. Global attractor for a nonlocal pLaplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems  B, 2017, 22 (5) : 18011816. doi: 10.3934/dcdsb.2017107 
[6] 
Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higherorder wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems  S, 2017, 10 (5) : 11751185. doi: 10.3934/dcdss.2017064 
[7] 
Bernd Kawohl, Jiří Horák. On the geometry of the pLaplacian operator. Discrete & Continuous Dynamical Systems  S, 2017, 10 (4) : 799813. doi: 10.3934/dcdss.2017040 
[8] 
Leszek Gasiński. Positive solutions for resonant boundary value problems with the scalar pLaplacian and nonsmooth potential. Discrete & Continuous Dynamical Systems  A, 2007, 17 (1) : 143158. doi: 10.3934/dcds.2007.17.143 
[9] 
Nikolaos S. Papageorgiou, Vicenţiu D. Rǎdulescu, Dušan D. Repovš. Nodal solutions for the Robin pLaplacian plus an indefinite potential and a general reaction term. Communications on Pure & Applied Analysis, 2018, 17 (1) : 231241. doi: 10.3934/cpaa.2018014 
[10] 
Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations & Control Theory, 2018, 7 (3) : 335351. doi: 10.3934/eect.2018017 
[11] 
Tae Gab Ha. On viscoelastic wave equation with nonlinear boundary damping and source term. Communications on Pure & Applied Analysis, 2010, 9 (6) : 15431576. doi: 10.3934/cpaa.2010.9.1543 
[12] 
Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems  S, 2009, 2 (3) : 583608. doi: 10.3934/dcdss.2009.2.583 
[13] 
Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251265. doi: 10.3934/mcrf.2011.1.251 
[14] 
Yansheng Zhong, Yongqing Li. On a pLaplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227236. doi: 10.3934/cpaa.2019012 
[15] 
Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the pLaplacian with gradient dependence. Discrete & Continuous Dynamical Systems  S, 2019, 12 (2) : 287295. doi: 10.3934/dcdss.2019020 
[16] 
Francesca Colasuonno, Benedetta Noris. A pLaplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems  A, 2017, 37 (6) : 30253057. doi: 10.3934/dcds.2017130 
[17] 
Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L^{2}, L^{q}) pullback attractors for a stochastic plaplacian equation. Communications on Pure & Applied Analysis, 2017, 6 (2) : 443474. doi: 10.3934/cpaa.2017023 
[18] 
Wen Tan. The regularity of pullback attractor for a nonautonomous pLaplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems  B, 2019, 24 (2) : 529546. doi: 10.3934/dcdsb.2018194 
[19] 
Ronghua Jiang, Jun Zhou. Blowup and global existence of solutions to a parabolic equation associated with the fraction pLaplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 12051226. doi: 10.3934/cpaa.2019058 
[20] 
Louis Tebou. Energy decay estimates for some weakly coupled EulerBernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 4560. doi: 10.3934/mcrf.2012.2.45 
2017 Impact Factor: 1.179
Tools
Metrics
Other articles
by authors
[Back to Top]