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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
https://doi.org/10.33434/cams.1012330

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 Probl`emes 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
https://doi.org/10.33434/cams.1012330

Abstract

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 Probl`emes 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
Subjects Mathematics
Journal Section Articles
Authors

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

Publication Date September 30, 2022
Submission Date October 20, 2021
Acceptance Date August 2, 2022
Published in Issue Year 2022, Volume 5, Issue 3

Cite

Bibtex @research article { cams1012330, journal = {Communications in Advanced Mathematical Sciences}, issn = {2651-4001}, address = {}, publisher = {Emrah Evren KARA}, year = {2022}, volume = {5}, number = {3}, pages = {137 - 149}, doi = {10.33434/cams.1012330}, title = {Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation}, key = {cite}, author = {Pereira, Ducival and Raposo, Carlos} }
APA Pereira, D. & Raposo, C. (2022). Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation . Communications in Advanced Mathematical Sciences , 5 (3) , 137-149 . DOI: 10.33434/cams.1012330
MLA Pereira, D. , Raposo, C. "Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation" . Communications in Advanced Mathematical Sciences 5 (2022 ): 137-149 <https://dergipark.org.tr/en/pub/cams/issue/72815/1012330>
Chicago Pereira, D. , Raposo, C. "Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation". Communications in Advanced Mathematical Sciences 5 (2022 ): 137-149
RIS TY - JOUR T1 - Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation AU - DucivalPereira, CarlosRaposo Y1 - 2022 PY - 2022 N1 - doi: 10.33434/cams.1012330 DO - 10.33434/cams.1012330 T2 - Communications in Advanced Mathematical Sciences JF - Journal JO - JOR SP - 137 EP - 149 VL - 5 IS - 3 SN - 2651-4001- M3 - doi: 10.33434/cams.1012330 UR - https://doi.org/10.33434/cams.1012330 Y2 - 2022 ER -
EndNote %0 Communications in Advanced Mathematical Sciences Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation %A Ducival Pereira , Carlos Raposo %T Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation %D 2022 %J Communications in Advanced Mathematical Sciences %P 2651-4001- %V 5 %N 3 %R doi: 10.33434/cams.1012330 %U 10.33434/cams.1012330
ISNAD Pereira, Ducival , Raposo, Carlos . "Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation". Communications in Advanced Mathematical Sciences 5 / 3 (September 2022): 137-149 . https://doi.org/10.33434/cams.1012330
AMA Pereira D. , Raposo C. Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation. Communications in Advanced Mathematical Sciences. 2022; 5(3): 137-149.
Vancouver Pereira D. , Raposo C. Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation. Communications in Advanced Mathematical Sciences. 2022; 5(3): 137-149.
IEEE D. Pereira and C. Raposo , "Global Weak Solution, Uniqueness and Exponential Decay for a Class of Degenerate Hyperbolic Equation", Communications in Advanced Mathematical Sciences, vol. 5, no. 3, pp. 137-149, Sep. 2022, doi:10.33434/cams.1012330
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