Global Solution and Asymptotic Behaviour for a Wave Equation Type p-Laplacian with p-Laplacian Damping


  • Ducival Pereira State University of Pará
  • Carlos Raposo Universidade Federal de São João del-Rei
  • Celsa Maranhão Federal University of Pará


Global solution, Potential well, p-Laplacian damping, Asymptotic behaviour, Nakao method


In this work we study the global solution, uniqueness and asymptotic behaviour of the nonlinear equation

utt − ∆pu − ∆put =|u|r−1u

where ∆pu is the nonlinear p-Laplacian operator, 2 ≤ p  < ∞. The global solutions are constructed by means of the Faedo-Galerkin approximations and the asymptotic behavior is obtained by Nakao method. Keywords: p-Laplacian, global solution, asymptotic behaviour, p-Laplacian damping.


Download data is not yet available.

Author Biography

Carlos Raposo, Universidade Federal de São João del-Rei

Mathematics Department


R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Functional Analysis, 14 (1973) 349-381.43

M. S. Alves, C. A. Raposo, J. E. Muñoz Rivera, M. Sepúlveda, O.V. Villagrán, Uniform Stabilization for the Transmission Problem of the Timoshenko System with Memory. Journal of Mathematical Analysis and Applications,369, (2010) 323-345.

D.D. Ang, A.P.N. Dinh, Strong solutions of a quasilinear wave equation with nonlinear damping. SIAM J. Math.Anal., 19 (1988) 337–347.

A. Benaissa, S. Mokeddem, Decay estimates for the wave equation of p-Laplacian type with dissipation of m-Laplaciantype. Math. Methods Appl. Sci., 30 (2007) 237–247.

A.C. Biazutti, On a nonlinear evolution equation and its applications. Nonlinear Analysis: Theory, Methods & Applications, 24 (1995) 1221–1234.

L. Bociu, 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.

I. Chueshov, I. Lasiecka, Existence, uniqueness of weak solution and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete Contin. Dyn. Syst., 15 (2006) 777-809.

E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill Inc., New York, 1955.

C.V. Coffman, Ljusternik-Schnirelman theory and eigenvalue problems for monotone potential operators. J. Funct. Anal., 14 (1973) 237–252.

P. D’Ancona, S. Spagnolo, On the life span of the analytic solutions to quasilinear weakly hyperbolic equations. Indiana Univ. Math. J., 40 (1991) 71–99.

E. Di Benedetto, C1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal., 7 (1983) 827-850.

M. Dreher, The wave equation for the p-Laplacian. Hokkaido Mathematical Journal, 36 (2007) 21–52.

H. Gao, T.F. Ma, Global solutions for a nonlinear wave equation with the p–Laplacian operator. Electronic J.Qualitative Theory Differ. Equ., 11 (1999) 1–13.

J. M. Greenberg, R. C. MacCamy, V. J. Vizel, On the existence, uniqueness, and stability of solution of the equation σ′(ux)uxx+λuxtx=ρ0utt. J. Math. Mech., 17 (1968) 707-728.

J.K. Hale, Ordinary Differential Equations (2nd ed.). Dover Publications, INC, 1997.

J.L. Lions, Quelques méthodes de résolution des problems aux limits non linéaire. Dunod-Gauthier Villars,1969.

T.F. Ma, J. A. Soriano, On weak solutions for an evolution equation with exponential nonlinearities. NonlinearAnal., 37 (1999) 1029–1038.

P. Martinez, A new method to decay rate estimates for dissipative systems. ESAIM Control Optim. Calc. Var., 4(1999) 419-444.

M. Nakao, H. Kuwahara, Decay estimates for some semilinear wave equations with degenerate dissipative terms. Funkcialaj Ekvacioj, 30 (1987) 135-145.44

M. Rammaha, Pei Pei, D. Toundykov, Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources. Journal of Mathematical Physics, 56 (2015) 081503.

M. Rammaha, D. Toundykov, Z. Wilstein, Global existence and decay of energy for a nonlinear wave equation with p-Laplacian damping. Discrete and Continuous Dynamical Systems, 32 (2012) 4361-4390

M. I. Visik, O.A. Ladyzhenskaya, Boundary - value problems for partial differential equations and certain classes of operator equations. Uspekhi Matem. Nauk, 6 (1956), 41-97; English transl., Amer. Math. Soc. Transl., 10 (1958),223-281.

M. Willem, “Minimax Theorems”, Progress in Nonlinear Differential Equations and their Applications. 24, Birkhöuser Boston Inc., Boston, MA, 1996.

Y. Ye, Global Existence and Asymptotic Behavior of Solutions for a Class of Nonlinear Degenerate Wave Equations.Differential Equations and Nonlinear Mechanics, (2007) 19685.

E. Zeidler, Nonlinear Functional Analysis and its Applications: II/B: Nonlinear Monotone Operators. SpringerScience & Business Media LCC, 1990.

Y. Zhijian, Global existence, asymptotic behavior and blow up of solutions for a class of nonlinear wave equations with dissipative term. Journal of Differential Equations, 187 (2003) 520-540.



How to Cite

Pereira, D., Carlos Raposo, & Celsa Maranhão. (2020). Global Solution and Asymptotic Behaviour for a Wave Equation Type p-Laplacian with p-Laplacian Damping. MathLAB Journal, 5, 35-45. Retrieved from



Research Articles

Most read articles by the same author(s)