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Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data. (English) Zbl 1421.35203

Summary: This paper is concerned with weighted energy estimates for solutions to wave equation \(\partial_t^2u-\Delta u+a(x)\partial_tu=0\) with space-dependent damping term \(a(x)=|x|^{-\alpha}\) \((\alpha\in [0,1])\) in an exterior domain \(\Omega\) having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polynomials are given and these decay rates are almost sharp, even when the initial data do not have compact support in \(\Omega\). The crucial idea is to use special solution of \(\partial_tu=|x|^\alpha\Delta u\) including Kummer’s confluent hypergeometric functions.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
47B25 Linear symmetric and selfadjoint operators (unbounded)
35B45 A priori estimates in context of PDEs
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[1] Arendt, W., Goldstein, R. G. and Goldstein, J. A., Outgrowths of Hardy’s inequality, in Recent Advances in Differential Equations and Mathematical Physics, , Vol. 412 (American Mathematics Society, 2006), pp. 51-68. · Zbl 1113.26017
[2] Beals, R. and Wong, R., Special Functions: A Graduate Text., Vol. 126 (Cambridge University Press, 2010). · Zbl 1222.33001
[3] Cattaneo, C., Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée, C. R. Acad. Sci.247 (1958) 431-433. · Zbl 1339.35135
[4] Chill, R. and Haraux, A., An optimal estimate for the difference of solutions of two abstract evolution equations, J. Differential Equations193 (2003) 385-395. · Zbl 1042.34090
[5] Hosono, T. and Ogawa, T., Large time behavior and \(L^p-L^q\) estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations203 (2004) 82-118. · Zbl 1049.35134
[6] Ikawa, M., Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan20 (1968) 580-608. · Zbl 0172.14304
[7] Ikawa, M., Hyperbolic Partial Differential Equations and Wave Phenomena (American Mathematical Society, 2000). · Zbl 0948.35004
[8] Ikehata, R., Diffusion phenomenon for linear dissipative wave equations in an exterior domain, J. Differential Equations186 (2002) 633-651. · Zbl 1017.35059
[9] Ikehata, R., Some remarks on the wave equation with potential type damping coefficients, Int. J. Pure Appl. Math.21 (2005) 19-24. · Zbl 1163.35427
[10] Ikehata, R. and Tanizawa, K., Global existence of solutions for semilinear damped wave equations in \(\mathbb{R}^N\) with noncompactly supported initial data, Nonlinear Anal.61 (2005) 1189-1208. · Zbl 1073.35171
[11] Karch, G., Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math.143 (2000) 175-197. · Zbl 0964.35022
[12] Li, T.-T., Nonlinear heat conduction with finite speed of propagation, in China-Japan Symposium on Reaction-Diffusion Equations and their Applications and Computational Aspects (World Scientific Publishing Co. Inc., 1997), pp. 81-91. · Zbl 0959.35088
[13] Lin, J., Nishihara, K. and Zhai, J., \(L^2\)-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations248 (2010) 403-422. · Zbl 1184.35213
[14] Marcati, P. and Nishihara, K., The \(L^p-L^q\) estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations191 (2003) 445-469. · Zbl 1031.35031
[15] Matsumura, A., On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci.12 (1976) 169-189. · Zbl 0356.35008
[16] Matsumura, A., Energy decay of solutions of dissipative wave equations, Proc. Japan Acad. Ser. A53 (1977) 232-236. · Zbl 0387.35041
[17] E. Mitidieri, A simple approach to Hardy inequalities, Mat. Zametki67 (2000) 563-572 (in Russian); Math. Notes67 (2000) 479-486. · Zbl 0964.26010
[18] Mochizuki, K., Scattering theory for wave equations with dissipative terms, Publ. Res. Inst. Math. Sci.12 (1976) 383-390. · Zbl 0357.35067
[19] Narazaki, T., \(L^p-L^q\) estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan56 (2004) 585-626. · Zbl 1059.35073
[20] Nishihara, K., \(L^p-L^q\) estimates of solutions to the damped wave equation in \(3\)-dimensional space and their application, Math. Z.244 (2003) 631-649. · Zbl 1023.35078
[21] Nishihara, K., Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, Comm. Partial Differential Equations35 (2010) 1402-1418. · Zbl 1204.35122
[22] Radu, P., Todorova, G. and Yordanov, B., Higher order energy decay rates for damped wave equations with variable coefficients, Discrete Contin. Dyn. Syst. Ser. S2 (2009) 609-629. · Zbl 1181.35024
[23] Radu, P., Todorova, G. and Yordanov, B., Decay estimates for wave equations with variable coefficients, Trans. Amer. Math. Soc.362 (2010) 2279-2299. · Zbl 1194.35065
[24] Rauch, J. and Taylor, M., Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J.24 (1974) 79-86. · Zbl 0281.35012
[25] Saeki, A. and Ikehata, R., Remarks on the decay rate for the energy of the dissipative linear wave equations in exterior domains, SUT J. Math.36 (2000) 267-277. · Zbl 1158.35327
[26] Sobajima, M. and Wakasugi, Y., Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain, J. Differential Equations261 (2016) 5690-5718. · Zbl 1356.35125
[27] Sobajima, M. and Wakasugi, Y., Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain, AIMS Math.2 (2017) 1-15. · Zbl 1428.35203
[28] Todorova, G. and Yordanov, B., Critical exponent for a nonlinear wave equation with damping, J. Differential Equations174 (2001) 464-489. · Zbl 0994.35028
[29] Todorova, G. and Yordanov, B., Weighted \(L^2\)-estimates for dissipative wave equations with variable coefficients, J. Differential Equations246 (2009) 4497-4518. · Zbl 1173.35032
[30] Vernotte, P., Les paradoxes de la théorie continue de l’équation de la chaleur, C. R. Acad. Sci. Paris246 (1958) 3154-3155. · Zbl 1341.35086
[31] Wakasugi, Y., Small data global existence for the semilinear wave equation with space-time dependent damping, J. Math. Anal. Appl.393 (2012) 66-79. · Zbl 1246.35133
[32] Wakasugi, Y., On diffusion phenomena for the linear wave equation with space-dependent damping, J. Hyperbolic Differential Equations11 (2014) 795-819. · Zbl 1312.35026
[33] Yang, H. and Milani, A., On the diffusion phenomenon of quasilinear hyperbolic waves, Bull. Sci. Math.124 (2000) 415-433. · Zbl 0959.35126
[34] Zuazua, E., Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations15 (1990) 205-235. · Zbl 0716.35010
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