# American Institute of Mathematical Sciences

July 2018, 17(4): 1449-1478. doi: 10.3934/cpaa.2018070

## Local and global existence of solutions to a strongly damped wave equation of the $p$-Laplacian type

 Department of Mathematics, University of Nebraska-Lincoln, 203 Avery Hall, Lincoln, NE 68588-0130, USA

* Corresponding author: Nicholas J. Kass has been partially supported by NSF grant DMS-1211232

Received  May 2017 Revised  December 2017 Published  April 2018

This article focuses on a quasilinear wave equation of
 $p$
-Laplacian type:
 $u_{tt} - Δ_p u -Δ u_t = 0$
in a bounded domain
 $\Omega \subset \mathbb{R}^3$
with a sufficiently smooth boundary
 $\Gamma = \partial \Omega$
subject to a generalized Robin boundary condition featuring boundary damping and a nonlinear source term. The operator
 $Δ_p$
,
 $2 , denotes the classical $ p$-Laplacian. The nonlinear boundary term $ f(u) $is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from $ {W^{1,p}}\left( \Omega \right) $into $ L^2(\Gamma) $. Under suitable assumptions on the parameters we provide a rigorous proof of existence of a local weak solution which can be extended globally in time provided the source term satisfies an appropriate growth condition. Citation: Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the$ p $-Laplacian type. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070 ##### References:  [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] K. Agre and M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations, 19 (2006), 1235-1270. [3] P. Aviles and J. Sandefur, Nonlinear second order equations with applications to partial differential equations, J. Differential Equations, 58 (1985), 404-427. [4] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010. [5] A. Benaissa and S. Mokeddem, Decay estimates for the wave equation of p-Laplacian type with dissipation of m-Laplacian type, Math. Methods Appl. Sci., 30 (2007), 237-247. [6] A. C. Biazutti, On a nonlinear evolution equation and its applications, Nonlinear Anal., 24 (1995), 1221-1234. [7] 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), 281-304. [8] 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), 835-860. [9] L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683. [10] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible NAvierSTokes Equations and Related Models, Springer, 2013. [11] M. M. Cavalcanti and V. N. Domingos, Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236 (2007), 407-459. [12] F. Chen, B. Guo and P. Wang, Long time behavior of strongly damped nonlinear wave equations, J. Differential Equations, 147 (1998), 231-241. [13] I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951. [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), 295-308. [15] J.-M. Ghidaglia and A. Marzocchi, Longtime behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 22 (1991), 879-895. [16] R. T. Glassey, Blow up theorems for nonlinear wave equations, Math. Z., 132 (1973), 183-203. [17] Y. Guo and M. A. Rammaha, Blow-up of solutions to systems of nonlinear wave equations with supercritical sources, Appl. Anal., 92 (2013), 1101-1115. [18] Y. Guo and M. A. Rammaha, Global existence and decay of energy to systems of wave equations with damping and supercritical sources, Z. Angew. Math. Phys., 64 (2013), 621-658. [19] Y. Guo and M. A. Rammaha, Systems of nonlinear wave equations with damping and supercritical boundary and interior sources, Trans. Amer. Math. Soc., 366 (2014), 2265-2325. [20] Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. S. Titi and D. Toundykov, Hadamard wellposedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differential Equations, 257 (2014), 3778-3812. [21] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, In Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), volume 50 of Progr. Nonlinear Differential Equations Appl., pages 197-216. Birkhäuser, Basel, 2002. [22] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = −Au +$ \mathcal{F} $(u), Trans. Amer. Math. Soc., 192 (1974), 1-21. [23] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York, 1972. [24] J.-L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96. [25] M. Nakao and T. Nanbu, Existence of global (bounded) solutions for some nonlinear evolution equations of second order, Math. Rep. College General Ed. Kyushu Univ., 10 (1975), 67-75. [26] P. Pei, M. A. Rammaha and D. Toundykov, Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources, J. Math. Phys., 56 (2015), 081503, 30. [27] P. Radu, Weak solutions to the initial boundary value problem for a semilinear wave equation with damping and source terms, Appl. Math. (Warsaw), 35 (2008), 355-378. [28] M. Rammaha, D. Toundykov and Z. Wilstein, Global existence and decay of energy for a nonlinear wave equation with p-Laplacian damping, Discrete Contin. Dyn. Syst., 32 (2012), 4361-4390. [29] M. A. Rammaha and Z. Wilstein, Hadamard well-posedness for wave equations with pLaplacian damping and supercritical sources, Adv. Differential Equations, 17 (2012), 105-150. [30] 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), 245-275. Workshop on Blow-up and Global Existence of Solutions for Parabolic and Hyperbolic Problems (Trieste, 1999). [31] E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298. [32] E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J., 44 (2002), 375-395. [33] G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math., 32 (1980), 631-643. show all references ##### References:  [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] K. Agre and M. A. Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations, 19 (2006), 1235-1270. [3] P. Aviles and J. Sandefur, Nonlinear second order equations with applications to partial differential equations, J. Differential Equations, 58 (1985), 404-427. [4] V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010. [5] A. Benaissa and S. Mokeddem, Decay estimates for the wave equation of p-Laplacian type with dissipation of m-Laplacian type, Math. Methods Appl. Sci., 30 (2007), 237-247. [6] A. C. Biazutti, On a nonlinear evolution equation and its applications, Nonlinear Anal., 24 (1995), 1221-1234. [7] 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), 281-304. [8] 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), 835-860. [9] L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683. [10] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible NAvierSTokes Equations and Related Models, Springer, 2013. [11] M. M. Cavalcanti and V. N. Domingos, Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236 (2007), 407-459. [12] F. Chen, B. Guo and P. Wang, Long time behavior of strongly damped nonlinear wave equations, J. Differential Equations, 147 (1998), 231-241. [13] I. Chueshov, M. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951. [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), 295-308. [15] J.-M. Ghidaglia and A. Marzocchi, Longtime behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 22 (1991), 879-895. [16] R. T. Glassey, Blow up theorems for nonlinear wave equations, Math. Z., 132 (1973), 183-203. [17] Y. Guo and M. A. Rammaha, Blow-up of solutions to systems of nonlinear wave equations with supercritical sources, Appl. Anal., 92 (2013), 1101-1115. [18] Y. Guo and M. A. Rammaha, Global existence and decay of energy to systems of wave equations with damping and supercritical sources, Z. Angew. Math. Phys., 64 (2013), 621-658. [19] Y. Guo and M. A. Rammaha, Systems of nonlinear wave equations with damping and supercritical boundary and interior sources, Trans. Amer. Math. Soc., 366 (2014), 2265-2325. [20] Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. S. Titi and D. Toundykov, Hadamard wellposedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differential Equations, 257 (2014), 3778-3812. [21] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, In Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), volume 50 of Progr. Nonlinear Differential Equations Appl., pages 197-216. Birkhäuser, Basel, 2002. [22] H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = −Au +$ \mathcal{F} $(u), Trans. Amer. Math. Soc., 192 (1974), 1-21. [23] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York, 1972. [24] J.-L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96. [25] M. Nakao and T. Nanbu, Existence of global (bounded) solutions for some nonlinear evolution equations of second order, Math. Rep. College General Ed. Kyushu Univ., 10 (1975), 67-75. [26] P. Pei, M. A. Rammaha and D. Toundykov, Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources, J. Math. Phys., 56 (2015), 081503, 30. [27] P. Radu, Weak solutions to the initial boundary value problem for a semilinear wave equation with damping and source terms, Appl. Math. (Warsaw), 35 (2008), 355-378. [28] M. Rammaha, D. Toundykov and Z. Wilstein, Global existence and decay of energy for a nonlinear wave equation with p-Laplacian damping, Discrete Contin. Dyn. Syst., 32 (2012), 4361-4390. [29] M. A. Rammaha and Z. Wilstein, Hadamard well-posedness for wave equations with pLaplacian damping and supercritical sources, Adv. Differential Equations, 17 (2012), 105-150. [30] 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), 245-275. Workshop on Blow-up and Global Existence of Solutions for Parabolic and Hyperbolic Problems (Trieste, 1999). [31] E. Vitillaro, Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298. [32] E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J., 44 (2002), 375-395. [33] G. F. Webb, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math., 32 (1980), 631-643.  [1] VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with$ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003 [2] Mohan Mallick, R. 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