Decay estimates for wave equations with variable coefficients
HTML articles powered by AMS MathViewer
- by Petronela Radu, Grozdena Todorova and Borislav Yordanov PDF
- Trans. Amer. Math. Soc. 362 (2010), 2279-2299 Request permission
Abstract:
We establish weighted $L^2-$estimates for dissipative wave equations with variable coefficients that exhibit a dissipative term with a space dependent potential. These results yield decay estimates for the energy and the $L^2-$norm of solutions. The proof is based on the multiplier method where multipliers are specially engineered from asymptotic profiles of related parabolic equations.References
- Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste Romรขnia, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843, DOI 10.1007/978-94-010-1537-0
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- Shao Ji Feng and De Xing Feng, Nonlinear internal damping of wave equations with variable coefficients, Acta Math. Sin. (Engl. Ser.) 20 (2004), no.ย 6, 1057โ1072. MR 2130371, DOI 10.1007/s10114-004-0394-3
- Mitsuru Ikawa, Hyperbolic partial differential equations and wave phenomena, Translations of Mathematical Monographs, vol. 189, American Mathematical Society, Providence, RI, 2000. Translated from the 1997 Japanese original by Bohdan I. Kurpita; Iwanami Series in Modern Mathematics. MR 1756774, DOI 10.1090/mmono/189
- Ryo Ikehata, Local energy decay for linear wave equations with variable coefficients, J. Math. Anal. Appl. 306 (2005), no.ย 1, 330โ348. MR 2132904, DOI 10.1016/j.jmaa.2004.12.056
- Han Yang and Albert Milani, On the diffusion phenomenon of quasilinear hyperbolic waves, Bull. Sci. Math. 124 (2000), no.ย 5, 415โ433 (English, with English and French summaries). MR 1781556, DOI 10.1016/S0007-4497(00)00141-X
- Cathleen S. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961), 561โ568. MR 132908, DOI 10.1002/cpa.3160140327
- Takashi Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan 56 (2004), no.ย 2, 585โ626. MR 2048476, DOI 10.2969/jmsj/1191418647
- Michael Reissig, $L_p$-$L_q$ decay estimates for wave equations with time-dependent coefficients, J. Nonlinear Math. Phys. 11 (2004), no.ย 4, 534โ548. MR 2098544, DOI 10.2991/jnmp.2004.11.4.9
- Reissig, M. and Wirth, J., $L^p-L^q$ estimates for wave equations with monotone time-dependent dissipation, Proceedings of the RIMS Symposium on Mathematical Models of Phenomena and Evolution Equations (to appear).
- Todorova, G; Yordanov, B., Weighted $L^2$ Estimates for Dissipative Wave Equations with Variable Coefficients (to appear).
- Grozdena Todorova and Borislav Yordanov, Nonlinear dissipative wave equations with potential, Control methods in PDE-dynamical systems, Contemp. Math., vol. 426, Amer. Math. Soc., Providence, RI, 2007, pp.ย 317โ337. MR 2311533, DOI 10.1090/conm/426/08196
- Jens Wirth, Solution representations for a wave equation with weak dissipation, Math. Methods Appl. Sci. 27 (2004), no.ย 1, 101โ124. MR 2023397, DOI 10.1002/mma.446
- Jens Wirth, Wave equations with time-dependent dissipation. I. Non-effective dissipation, J. Differential Equations 222 (2006), no.ย 2, 487โ514. MR 2208294, DOI 10.1016/j.jde.2005.07.019
- Jens Wirth, Wave equations with time-dependent dissipation. II. Effective dissipation, J. Differential Equations 232 (2007), no.ย 1, 74โ103. MR 2281190, DOI 10.1016/j.jde.2006.06.004
Additional Information
- Petronela Radu
- Affiliation: Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
- Email: pradu@math.unl.edu
- Grozdena Todorova
- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Knoxville, Tennessee 37996
- Email: todorova@math.utk.edu
- Borislav Yordanov
- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Knoxville, Tennessee 37996
- Email: yordanov@math.utk.edu
- Received by editor(s): October 4, 2007
- Published electronically: December 14, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 2279-2299
- MSC (2000): Primary 35L05, 35L15; Secondary 37L15
- DOI: https://doi.org/10.1090/S0002-9947-09-04742-4
- MathSciNet review: 2584601