Decay estimates for wave equations with variable coefficients
Authors:
Petronela Radu, Grozdena Todorova and Borislav Yordanov
Journal:
Trans. Amer. Math. Soc. 362 (2010), 22792299
MSC (2000):
Primary 35L05, 35L15; Secondary 37L15
Published electronically:
December 14, 2009
MathSciNet review:
2584601
Fulltext PDF
Abstract 
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Additional Information
Abstract: We establish weighted 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 norm of solutions. The proof is based on the multiplier method where multipliers are specially engineered from asymptotic profiles of related parabolic equations.
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 Evans, L.C., Partial Differential Equations, Graduate Studies in Mathematics, AMS, vol. 19 (1998). MR 1625845 (99e:35001)
 3.
 Feng, Shao Ji; Feng, De Xing, Nonlinear internal damping of wave equations with variable coefficients, Acta Math. Sin. (Engl. Ser.) 20 (2004), no. 6, 10571072. MR 2130371 (2005m:35190)
 4.
 Ikawa, Mitsuru, Hyperbolic partial differential equations and wave phenomena, American Mathematical Society, Providence R.I. (2000). MR 1756774 (2001j:35176)
 5.
 Ikehata, R., Local energy decay for linear wave equations with variable coefficients, J. Math. Anal. Appl. 306 (2005), no. 1, 330348. MR 2132904 (2006a:35187)
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 Milani, Albert; Yang, Han On the diffusion phenomenon of quasilinear hyperbolic waves in low space dimensions, Bull. Sci. Math. 124 (2000), 415433. MR 1781556 (2001f:35271)
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 Morawetz, C., The decay of solutions of the exterior initialboundary problem for the wave equation, Comm. Pure Appl. Math. Sci. 14 (1961), 561568. MR 0132908 (24:A2744)
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 Narazaki, T.  estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004), 585626. MR 2048476 (2005a:35206)
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 Reissig, M., decay estimates for wave equations with timedependent coefficients, J. Nonlinear Math. Phys. 11 (2004), no. 4, 534548. MR 2098544 (2006b:35232)
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 Reissig, M. and Wirth, J., estimates for wave equations with monotone timedependent dissipation, Proceedings of the RIMS Symposium on Mathematical Models of Phenomena and Evolution Equations (to appear).
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 Todorova, G; Yordanov, B., Weighted Estimates for Dissipative Wave Equations with Variable Coefficients (to appear).
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 Wirth, J., Solution representations for a wave equation with weak dissipation, Math. Methods Appl. Sci. 27 (2004), no. 1, 101124. MR 2023397 (2004k:35040)
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 Wirth, J., Wave equations with timedependent dissipation. I. Noneffective dissipation, J. Diff. Eq. 222 (2006), 487514. MR 2208294 (2007d:35171)
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 Wirth, J., Wave equations with timedependent dissipation. II. Effective dissipation, J. Differential Equations 232 (2007), no. 1, 74103. MR 2281190 (2007k:35293)
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Additional Information
Petronela Radu
Affiliation:
Department of Mathematics and Statistics, University of NebraskaLincoln, 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
DOI:
http://dx.doi.org/10.1090/S0002994709047424
PII:
S 00029947(09)047424
Keywords:
Wave equations with variable coefficients,
linear dissipation,
decay rates,
subsolution
Received by editor(s):
October 4, 2007
Published electronically:
December 14, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
