# Weighted *L*^{2}-estimates of solutions for damped wave equations with variable coefficients

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## Abstract

The authors establish weighted *L*^{2}-estimates of solutions for the damped wave equations with variable coefficients *u*_{ tt }− divA(*x*)∇*u*+*au*_{ t } = 0 in *ℝ*^{ n } under the assumption a(*x*) ≥ a_{0}[1+*ρ*(*x*)]^{−l}, where a_{0} > 0, *l* < 1, *ρ*(*x*) is the distance function of the metric *g* = *A*^{−1}(*x*) on *ℝ*^{ n }. The authors show that these weighted *L*^{2}-estimates are closely related to the geometrical properties of the metric *g* = *A*^{−1}(*x*).

## Keywords

Distance function of a metric Riemannian metric wave equation with variable coefficients weighted*L*

^{2}-estimate

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## References

- [1]Ikehata R, Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain,
*J. Differential Equations*, 2003,**188**(2): 390–405.MathSciNetCrossRefzbMATHGoogle Scholar - [2]Matsumura A, On the asymptotic behavior of solutions of semi-linear wave equations,
*Publ. Res. Inst. Math. Sci.*, 1976,**12**(1): 169–189.MathSciNetCrossRefzbMATHGoogle Scholar - [3]Nakao M, Decay of solutions of the wave equation with a local nonlinear dissipation,
*Math. Ann.*, 1996,**305**(3): 403–417.MathSciNetCrossRefzbMATHGoogle Scholar - [4]Narazaki T, Lp-Lq estimates for damped wave equations and their applications to semi-linear problem,
*J. Math. Soc. Japan*, 2004,**56**: 585–626.MathSciNetCrossRefzbMATHGoogle Scholar - [5]Ono K, On global solutions and blow-up solutions of nonlinear Kirchhoff Strings with nonlinear dissipation,
*J. Math. Anal. Appl.*, 1997,**216**: 321–342.MathSciNetCrossRefzbMATHGoogle Scholar - [6]Radu P, Todorova G, and Yordanov B, Decay estimates for wave equations with variable coefficients,
*Trans. Amer. Math. Soc.*, 2010,**362**: 2279–2299.MathSciNetCrossRefzbMATHGoogle Scholar - [7]Todorova G and Yordanov B, Weighted
*L*_{2}-estimates for dissipative wave equations with variable coefficients,*J. Differential Equations*, 2009,**246**: 4497–4518.MathSciNetCrossRefzbMATHGoogle Scholar - [8]Yao P F, On the observatility inequality for exact controllability of wave equations with variable coefficients,
*SIAM J. Contr. and Optim.*, 1999,**37**(5): 1568–1599.CrossRefzbMATHGoogle Scholar - [9]Chai S, Guo Y, and Yao P F, Boundary feedback stabilization of shallow shells,
*SIAM J. Control Optim.*, 2003,**42**(1): 239–259.MathSciNetCrossRefzbMATHGoogle Scholar - [10]Chai S and Liu K, Observability inequalities for the transmission of shallow shells,
*Sys. Control Lett.*, 2006,**55**(9): 726–735.MathSciNetCrossRefzbMATHGoogle Scholar - [11]Chai S and Liu K, Boundary feedback stabilization of the transmission problem of Naghdi’s model,
*J. Math. Anal. Appl.*, 2006,**319**(1): 199–214.MathSciNetCrossRefzbMATHGoogle Scholar - [12]Chai S and Yao P E, Observability inequalities for thin shells,
*Sci. China Ser. A*, 2003,**46**(3): 300–311.MathSciNetzbMATHGoogle Scholar - [13]Feng S J and Feng D X, Nonlinear internal damping of wave equations with variable coefficients, Acta Math. Sin.,
*Engl. Ser.*, 2004,**20**(6): 1057–1072.MathSciNetzbMATHGoogle Scholar - [14]Lasiecka I, Triggiani R, and Yao P F, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients,
*J. Math. Anal. Appl.*, 1999,**235**(1): 13–57.MathSciNetCrossRefzbMATHGoogle Scholar - [15]Yao P F, Modeling and control in vibrational and structural dynamics: A differential geometric approach,
*Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series*, CRC Press, Boca Raton, FL, 2011.Google Scholar - [16]Gulliver R, Lasiecka I, LittmanW, et al., The case for differential geometry in the control of single and coupled PDEs: The structural acoustic chamber,
*Geometric Methods in Inverse Problems and PDE Control*, 73–181, IMA Vol. Math. Appl.,**137**, Springer-Verlag, New York, 2004.Google Scholar - [17]Yao P F, Global smooth solutions for the quasilinear wave equation with boundary dissipation,
*J. Differential Equations*, 2007,**241**: 62–93.MathSciNetCrossRefzbMATHGoogle Scholar - [18]Yao P F, Boundary controllability for the quasilinear wave equation,
*Appl. Math. Optim.*, 2010,**61**(2): 191–233.MathSciNetCrossRefzbMATHGoogle Scholar - [19]Zhang Z F and Yao P F, Global smooth solutions of the quasilinear wave equation with an internal velocity feedback,
*SIAM J. Control Optim.*, 2008,**47**(4): 2044–2077.MathSciNetCrossRefzbMATHGoogle Scholar - [20]Yao P F, Energy decay for the cauchy problem of the linear wave equation of variable coefficients with dissipation, Chinese Annals of Mathematics,
*Series B*, 2010,**31**(1): 59–70.MathSciNetzbMATHGoogle Scholar - [21]Zhang Z F, Fast decay of solutions for wave equations with localized dissipation on noncompact Riemannian manifolds,
*Nonlinear Analysis: Real World Applications*, 2016,**27**: 246–260.MathSciNetCrossRefzbMATHGoogle Scholar - [22]Schoen R and Yau S T, Lectures on differential geometry,
*Conference Proceedings and Lecture Notes in Geometry and Topology*, I. International Press, Cambridge, MA, 1994.Google Scholar

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