Research Article

### Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation

Year 2022, Volume 5, Issue 3, 137 - 149, 30.09.2022

### Abstract

This paper deals with existence, uniqueness and energy decay of solutions to a degenerate hyperbolic equations given by \begin{align*} K(x,t)u'' - M\left(\int_\Omega |\nabla u|^2\,dx \right) \Delta u - \Delta u' = 0, \end{align*} with operator coefficient $K(x,t)$ satisfying suitable properties and $M(\,\cdot \,) \in C^1([0, \infty))$ is a function which greatest lower bound for $M (\,\cdot\,)$ is zero. For global weak solution and uniqueness we use the Faedo-Galerkin method. Exponential decay is proven by using a theorem due to M. Nakao.

### References

• [1] K. Nishihara, Y. Yamada, On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms, Funkcial. Ekvac., 33 (1990), 151-159.
• [2] J. Ferreira, D. C. Pereira, On a nonlinear degenerate evolution equation with strong damping, Internat. J. Math. & Math. Sci., 15(3) (1992), 543-552.
• [3] D. C. Pereira, Mixed problem for a nonlinear vibrations equation, Bol. Soc. Parana. Mat., 9(2) (1988), 31-42.
• [4] G. F. Carrier, On the nonlinear vibration problem of the elastic string, Q. Appl. Math, 3 (1945), 157-165.
• [5] R. Dickey, Free vibration in dynamic buckling of extensible beam, J. Math. Analysis Applic., 29 (1970), 439-451.
• [6] R. Narasihma, Nonlinear vibrations of an elastic string, J. Sound Vibrations, 8 (1968), 134–146.
• [7] S. I. Pohozaev, On a class of quasilinear hyperbolic equations, Math. USSR Sbornik, 25 (1975), 145-158.
• [8] R. W. Dickey, Infinite systems of nonlinear oscillation equations related the string, Amer. Math. Soc., 23 (1969), 459-468.
• [9] J. L. Lions, On Some Questions in Boundary Value Problems of Mathematical Physics, Contemporary Developments in Continuum Mechanics and Partial Differential Equations, edited by G. M. De La Penha and L. A. Medeiros, North-Holland, Amsterdam, 1978.
• [10] R. F. C. Lobato, D. C. Pereira, M. L. Santos, Exponential decay to the degenerate nonlinear coupled beams system with weak damping, Math. Phys., (2012), 1-14, Article ID 659289.
• [11] S. D. B. Menezes, E. A. Oliveira, D. C. Pereira, J. Ferreira, Existence, uniqueness and uniform decay for the nonlinear beam degenerate equation with weak damping, Appl. Math. Comput. 154 (2004), 555-565.
• [12] K. Nishihara, On a global solution of some quasilinear hyperbolic equation, Tokyo J. Math., 7 (1984), 437-459.
• [13] P. H. Rivera Rodriguez, On local strong solutions of a nonlinear partial differential equation, Appl. Anal., 10 (1984), 93-104.
• [14] A. Arosio, S. Spagnolo, Global Solutions to the Cauchy Problem for a Nonlinear Hyperbolic equation, Nonlinear Partial Differential Equations and Their Applications, College de France Seminar Vol. 6, edited by Bresis, H. and Lions, J. L., Pitman, London, 1984.
• [15] Y. Yamada, Some nonlinear degenerate wave equations, Nonlinear Anal., 11 (1987), 1155-1168.
• [16] T. Yamazaki, On local solutions of some quasilinear degenerate hyperbolic equations, Funkcial. Ekvac., 31 (1988), 439-457.
• [17] N. A. Lar’kin, A class of degenerate hyperbolic equations, Dinamika Sploshn. Sredy, 36 (1978), 71–77.
• [18] N. A. Lar’kin, On a class of quasi-linear hyperbolic equations having global solutions, Sov. Math., Dokl. 20 (1979), 28-31.
• [19] F. Colombini, S. Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in C¥, Acta Math., 148 (1982), 243-253.
• [20] D. Lupo, K. R. Payne, N. I. Popivanov, On the degenerate hyperbolic Goursat problem for linear and nonlinear equations of Tricomi type, Nonlinear Anal. Theory Methods Appl., 108 (2014), 29-56.
• [21] R. L. Cruz, J. C. Juajibioy, L. Rendon, Generalized Riemann problem for a totally degenerate hyperbolic system, Electron. J. Differ. Equ., 196 (2018), 1-14.
• [22] N. Kakharman, T. Kal’menov, Mixed Cauchy problem with lateral boundary condition for noncharacteristic degenerate hyperbolic equations, Bound. Value. Probl., 35 (2022), 1-11.
• [23] E. H. Brito, The damped elastic stretched string equation generalized: existence, uniqueness, regularity and stability, Appl. Anal., 13 (1982), 219-233.
• [24] K. Nishihara, Exponential decay of solutions of some quasilinear hyperbolic equations with linear damping, Nonlinear Anal., 8 (1984), 623-636.
• [25] K. Nishihara, Global existence and asymptotic behaviour of the solution of some quasilinear hyperbolic equation with linear damping, Funkcial. Ekvac., 32 (1989), 343-355.
• [26] Y. Yamada, On some quasilinear wave equations with dissipative terms, Nagoya Math. J., 87 (1982), 17-39.
• [27] J. G. Dix, Decay of solutions of a degenerate hyperbolic equation, Electron. J. Differ. Equ., 21 (1998), 1–10.
• [28] E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Inc., New York, 1955.
• [29] J. L. Lions, Quelques M´ethodes de R´esolution des Problemes Aux Limites Non Lin´e aires, Dunod, Paris, 1969.
• [30] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and physics, Springer Verlag, Applied Math. Sci., 68, 1988.
• [31] M. Nakao, Decay of solutions for some nonlinear evolution equations, J. Math. Analysis Appl., 60 (1977), 542-549.
Year 2022, Volume 5, Issue 3, 137 - 149, 30.09.2022

### References

• [1] K. Nishihara, Y. Yamada, On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms, Funkcial. Ekvac., 33 (1990), 151-159.
• [2] J. Ferreira, D. C. Pereira, On a nonlinear degenerate evolution equation with strong damping, Internat. J. Math. & Math. Sci., 15(3) (1992), 543-552.
• [3] D. C. Pereira, Mixed problem for a nonlinear vibrations equation, Bol. Soc. Parana. Mat., 9(2) (1988), 31-42.
• [4] G. F. Carrier, On the nonlinear vibration problem of the elastic string, Q. Appl. Math, 3 (1945), 157-165.
• [5] R. Dickey, Free vibration in dynamic buckling of extensible beam, J. Math. Analysis Applic., 29 (1970), 439-451.
• [6] R. Narasihma, Nonlinear vibrations of an elastic string, J. Sound Vibrations, 8 (1968), 134–146.
• [7] S. I. Pohozaev, On a class of quasilinear hyperbolic equations, Math. USSR Sbornik, 25 (1975), 145-158.
• [8] R. W. Dickey, Infinite systems of nonlinear oscillation equations related the string, Amer. Math. Soc., 23 (1969), 459-468.
• [9] J. L. Lions, On Some Questions in Boundary Value Problems of Mathematical Physics, Contemporary Developments in Continuum Mechanics and Partial Differential Equations, edited by G. M. De La Penha and L. A. Medeiros, North-Holland, Amsterdam, 1978.
• [10] R. F. C. Lobato, D. C. Pereira, M. L. Santos, Exponential decay to the degenerate nonlinear coupled beams system with weak damping, Math. Phys., (2012), 1-14, Article ID 659289.
• [11] S. D. B. Menezes, E. A. Oliveira, D. C. Pereira, J. Ferreira, Existence, uniqueness and uniform decay for the nonlinear beam degenerate equation with weak damping, Appl. Math. Comput. 154 (2004), 555-565.
• [12] K. Nishihara, On a global solution of some quasilinear hyperbolic equation, Tokyo J. Math., 7 (1984), 437-459.
• [13] P. H. Rivera Rodriguez, On local strong solutions of a nonlinear partial differential equation, Appl. Anal., 10 (1984), 93-104.
• [14] A. Arosio, S. Spagnolo, Global Solutions to the Cauchy Problem for a Nonlinear Hyperbolic equation, Nonlinear Partial Differential Equations and Their Applications, College de France Seminar Vol. 6, edited by Bresis, H. and Lions, J. L., Pitman, London, 1984.
• [15] Y. Yamada, Some nonlinear degenerate wave equations, Nonlinear Anal., 11 (1987), 1155-1168.
• [16] T. Yamazaki, On local solutions of some quasilinear degenerate hyperbolic equations, Funkcial. Ekvac., 31 (1988), 439-457.
• [17] N. A. Lar’kin, A class of degenerate hyperbolic equations, Dinamika Sploshn. Sredy, 36 (1978), 71–77.
• [18] N. A. Lar’kin, On a class of quasi-linear hyperbolic equations having global solutions, Sov. Math., Dokl. 20 (1979), 28-31.
• [19] F. Colombini, S. Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in C¥, Acta Math., 148 (1982), 243-253.
• [20] D. Lupo, K. R. Payne, N. I. Popivanov, On the degenerate hyperbolic Goursat problem for linear and nonlinear equations of Tricomi type, Nonlinear Anal. Theory Methods Appl., 108 (2014), 29-56.
• [21] R. L. Cruz, J. C. Juajibioy, L. Rendon, Generalized Riemann problem for a totally degenerate hyperbolic system, Electron. J. Differ. Equ., 196 (2018), 1-14.
• [22] N. Kakharman, T. Kal’menov, Mixed Cauchy problem with lateral boundary condition for noncharacteristic degenerate hyperbolic equations, Bound. Value. Probl., 35 (2022), 1-11.
• [23] E. H. Brito, The damped elastic stretched string equation generalized: existence, uniqueness, regularity and stability, Appl. Anal., 13 (1982), 219-233.
• [24] K. Nishihara, Exponential decay of solutions of some quasilinear hyperbolic equations with linear damping, Nonlinear Anal., 8 (1984), 623-636.
• [25] K. Nishihara, Global existence and asymptotic behaviour of the solution of some quasilinear hyperbolic equation with linear damping, Funkcial. Ekvac., 32 (1989), 343-355.
• [26] Y. Yamada, On some quasilinear wave equations with dissipative terms, Nagoya Math. J., 87 (1982), 17-39.
• [27] J. G. Dix, Decay of solutions of a degenerate hyperbolic equation, Electron. J. Differ. Equ., 21 (1998), 1–10.
• [28] E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Inc., New York, 1955.
• [29] J. L. Lions, Quelques M´ethodes de R´esolution des Problemes Aux Limites Non Lin´e aires, Dunod, Paris, 1969.
• [30] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and physics, Springer Verlag, Applied Math. Sci., 68, 1988.
• [31] M. Nakao, Decay of solutions for some nonlinear evolution equations, J. Math. Analysis Appl., 60 (1977), 542-549.

### Details

Primary Language English Mathematics Articles Ducival PEREİRA This is me State University of Pará 0000-0003-4511-0185 Brazil Carlos RAPOSO> (Primary Author) Federal University of São João del-Rei 0000-0001-8014-7499 Brazil September 30, 2022 October 20, 2021 August 2, 2022 Year 2022, Volume 5, Issue 3