About Decay Of Solution Of The Wave Equation With Dissipation
DOI:
https://doi.org/10.4067/S0716-09172007000100003Abstract
In this work, we consider the problem of existence of global solutions for a scalar wave equation with dissipation. We also study the asymptotic behaviour in time of the solutions. The method used here is based in nonlinear techniques. Key words: wave equation, evolution model, decay of solution, asymptotic behaviour.
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References
[1] F. Conrad and B. Rao - Decay of solutions of wave equations in a star-shaped domain with nonlinear boundary feedback. Asympt. Anal. 7, pp. 159—177, (1993).
[2] L. Cortés-Vega.- A note on resonant frequencies for a system of elastic wave equations, Int. J. Math. Math. Sci. 64, pp. 3485—3498, (2004).
[3] L. Cortés-Vega and Y. Santiago-Ayala, Decaimiento de la ecuación de onda con disipación. PESQUIMAT Revista de la Fac. CC. MM. de la UNMSM. Vol IX, Nro. 1, pp. 39—62, (2006).
[4] L. Hörmander - Linear Partial Differential Operators, Springer-Verlag, New York, 116, (1976).
[5] M. Ikawa - Mixed problems for hyperbolic equations of second order. J. Math. Soc. Japan 20, pp. 580—608, (1968).
[6] S. Kesavan - Topics in Functional Analysis and applications. John Wiley & Sons, (1989).
[7] J. Lagnese, Deacay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50, pp. 163—182, (1983).
[8] J. Lagnese and J. L. Lions, Modelling Analysis and Control of thin
Plates, Masson, Paris, (1989).
[9] I. Lasiecka and R. Triggiani, Control Problems for Systems Described by Partial Differential Equations and Applications, Springer Verlag Lect. Not., 97 (1987).
[10] I. Lasiecka and G. Avalos, Uniform decay rates of solutions to a structural acoustic model with nonlinear dissipation, Appl. Math Comp. Sci, 8, No. 2, pp. 101—127, (1998).
[11] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Diff. Int. Eq, 6, pp. 507—533, (1993).
[12] I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Diffetential Equations, 79, pp. 340—381, (1989).
[13] V. Komornik- Exact controllability and stabilization. John Wiley & Sons, (1994).
[14] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. 69, pp. 163-182, (1990).
[15] P. Martinez. - Decay of solutions of the wave equation with a local highly degenerate dissipation. Asymptotic Analysis 19, pp. 1—17, (1999).
[16] M. Nakao - Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. of Maths 95, pp. 25—42, (1996).
[17] A. Pazy - Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, (1983).
[18] Y. Santiago A. - Una aplicación del Lema de Nakao. PESQUIMAT Revista de la Fac. CC. MM. de la UNMSM, Nro. 2, pp. 23—36, (2006).
[19] Y. Santiago A. - Decaimiento exponencial de la solución débil de una ecuación de onda no lineal. PESQUIMAT Revista de la Fac. CC. MM. de la UNMSM. Vol VIII Nro. 2, pp. 29—43, (2005).
[20] Y. Santiago A. and J. Rivera - Global existence and exponential decay to the wave equation with localized frictional damping. PESQUIMAT Revista de la Fac. CC. MM. de la UNMSM. Vol V. Nro. 2, pp. 1—19, (2002).
[21] D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim. 3, pp. 257—295, (1995).
[22] E. Zuazua - Exponential decay for the semi-linear wave equation with locally distributed damping. Comm. Partial Differential Equations. 15, pp. 205—235, (1990).
[2] L. Cortés-Vega.- A note on resonant frequencies for a system of elastic wave equations, Int. J. Math. Math. Sci. 64, pp. 3485—3498, (2004).
[3] L. Cortés-Vega and Y. Santiago-Ayala, Decaimiento de la ecuación de onda con disipación. PESQUIMAT Revista de la Fac. CC. MM. de la UNMSM. Vol IX, Nro. 1, pp. 39—62, (2006).
[4] L. Hörmander - Linear Partial Differential Operators, Springer-Verlag, New York, 116, (1976).
[5] M. Ikawa - Mixed problems for hyperbolic equations of second order. J. Math. Soc. Japan 20, pp. 580—608, (1968).
[6] S. Kesavan - Topics in Functional Analysis and applications. John Wiley & Sons, (1989).
[7] J. Lagnese, Deacay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50, pp. 163—182, (1983).
[8] J. Lagnese and J. L. Lions, Modelling Analysis and Control of thin
Plates, Masson, Paris, (1989).
[9] I. Lasiecka and R. Triggiani, Control Problems for Systems Described by Partial Differential Equations and Applications, Springer Verlag Lect. Not., 97 (1987).
[10] I. Lasiecka and G. Avalos, Uniform decay rates of solutions to a structural acoustic model with nonlinear dissipation, Appl. Math Comp. Sci, 8, No. 2, pp. 101—127, (1998).
[11] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Diff. Int. Eq, 6, pp. 507—533, (1993).
[12] I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Diffetential Equations, 79, pp. 340—381, (1989).
[13] V. Komornik- Exact controllability and stabilization. John Wiley & Sons, (1994).
[14] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl. 69, pp. 163-182, (1990).
[15] P. Martinez. - Decay of solutions of the wave equation with a local highly degenerate dissipation. Asymptotic Analysis 19, pp. 1—17, (1999).
[16] M. Nakao - Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. of Maths 95, pp. 25—42, (1996).
[17] A. Pazy - Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, (1983).
[18] Y. Santiago A. - Una aplicación del Lema de Nakao. PESQUIMAT Revista de la Fac. CC. MM. de la UNMSM, Nro. 2, pp. 23—36, (2006).
[19] Y. Santiago A. - Decaimiento exponencial de la solución débil de una ecuación de onda no lineal. PESQUIMAT Revista de la Fac. CC. MM. de la UNMSM. Vol VIII Nro. 2, pp. 29—43, (2005).
[20] Y. Santiago A. and J. Rivera - Global existence and exponential decay to the wave equation with localized frictional damping. PESQUIMAT Revista de la Fac. CC. MM. de la UNMSM. Vol V. Nro. 2, pp. 1—19, (2002).
[21] D. Tataru, Boundary controllability for conservative PDEs, Appl. Math. Optim. 3, pp. 257—295, (1995).
[22] E. Zuazua - Exponential decay for the semi-linear wave equation with locally distributed damping. Comm. Partial Differential Equations. 15, pp. 205—235, (1990).
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2017-04-18
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How to Cite
[1]
“About Decay Of Solution Of The Wave Equation With Dissipation”, Proyecciones (Antofagasta, On line), vol. 26, no. 1, pp. 37–71, Apr. 2017, doi: 10.4067/S0716-09172007000100003.
