Decay estimates for wave equations with variable coefficients

Authors:
Petronela Radu, Grozdena Todorova and Borislav Yordanov

Journal:
Trans. Amer. Math. Soc. **362** (2010), 2279-2299

MSC (2000):
Primary 35L05, 35L15; Secondary 37L15

Published electronically:
December 14, 2009

MathSciNet review:
2584601

Full-text PDF Free Access

Abstract | References | Similar Articles | 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.

**1.**Viorel Barbu,*Nonlinear semigroups and differential equations in Banach spaces*, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR**0390843****2.**Lawrence C. Evans,*Partial differential equations*, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR**1625845****3.**Shao Ji Feng and De Xing Feng,*Nonlinear internal damping of wave equations with variable coefficients*, Acta Math. Sin. (Engl. Ser.)**20**(2004), no. 6, 1057–1072. MR**2130371**, 10.1007/s10114-004-0394-3**4.**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****5.**Ryo Ikehata,*Local energy decay for linear wave equations with variable coefficients*, J. Math. Anal. Appl.**306**(2005), no. 1, 330–348. MR**2132904**, 10.1016/j.jmaa.2004.12.056**6.**Han Yang and Albert Milani,*On the diffusion phenomenon of quasilinear hyperbolic waves*, Bull. Sci. Math.**124**(2000), no. 5, 415–433 (English, with English and French summaries). MR**1781556**, 10.1016/S0007-4497(00)00141-X**7.**Cathleen S. Morawetz,*The decay of solutions of the exterior initial-boundary value problem for the wave equation*, Comm. Pure Appl. Math.**14**(1961), 561–568. MR**0132908****8.**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. 2, 585–626. MR**2048476**, 10.2969/jmsj/1191418647**9.**Michael Reissig,*𝐿_{𝑝}-𝐿_{𝑞} decay estimates for wave equations with time-dependent coefficients*, J. Nonlinear Math. Phys.**11**(2004), no. 4, 534–548. MR**2098544**, 10.2991/jnmp.2004.11.4.9**10.**Reissig, M. and Wirth, J.,*estimates for wave equations with monotone time-dependent dissipation*, Proceedings of the RIMS Symposium on Mathematical Models of Phenomena and Evolution Equations (to appear).**11.**Todorova, G; Yordanov, B.,*Weighted Estimates for Dissipative Wave Equations with Variable Coefficients*(to appear).**12.**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. 317–337. MR**2311533**, 10.1090/conm/426/08196**13.**Jens Wirth,*Solution representations for a wave equation with weak dissipation*, Math. Methods Appl. Sci.**27**(2004), no. 1, 101–124. MR**2023397**, 10.1002/mma.446**14.**Jens Wirth,*Wave equations with time-dependent dissipation. I. Non-effective dissipation*, J. Differential Equations**222**(2006), no. 2, 487–514. MR**2208294**, 10.1016/j.jde.2005.07.019**15.**Jens Wirth,*Wave equations with time-dependent dissipation. II. Effective dissipation*, J. Differential Equations**232**(2007), no. 1, 74–103. MR**2281190**, 10.1016/j.jde.2006.06.004

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35L05,
35L15,
37L15

Retrieve articles in all journals with MSC (2000): 35L05, 35L15, 37L15

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

DOI:
https://doi.org/10.1090/S0002-9947-09-04742-4

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.