November 2018, 38(11): 5921-5941. doi: 10.3934/dcds.2018257

The diffusion phenomenon for damped wave equations with space-time dependent coefficients

Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA

Received  April 2018 Published  August 2018

We introduce a method to study the long-time behavior of solutions to damped wave equations, where the coefficients of the equations are space-time dependent. We show that solutions exhibit the diffusion phenomenon, connecting their asymptotic behaviors with the asymptotic behaviors of solutions to corresponding parabolic equations. Sharp decay estimates for solutions to damped wave equations are given, and decay estimates for derivatives of solutions are also discussed.

Citation: Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257
References:
[1]

R. Chill and A. Haraux, An optimal estimate for the difference of solutions of two abstract evolution equations, J. Differential Equations, 193 (2003), 385-395. doi: 10.1016/S0022-0396(03)00057-3.

[2]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J. 1964.

[3]

M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan, 20 (1968), 580-608. doi: 10.2969/jmsj/02040580.

[4]

M. Ikawa, Hyperbolic Partial Differential Equations and Wave Phenomena, American Mathematical Society, Providence, R. I., 2000.

[5]

R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain, J. Differential Equations, 186 (2002), 633-651. doi: 10.1016/S0022-0396(02)00008-6.

[6]

R. Ikehata, Energy decay of solutions for the semilinear dissipative wave equations in an exterior domain, Funkcial. Ekvac., 44 (2001), 487-499.

[7]

R. Ikehata, Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain, J. Differential Equations, 188 (2003), 390-405. doi: 10.1016/S0022-0396(02)00101-8.

[8]

R. Ikehata, Some remarks on the wave equation with potential type damping coefficients, Int. J. Pure Appl. Math., 21 (2005), 19-24. Available from: https://ijpam.eu/contents/2005-21-1/3/3.pdf.

[9]

R. Ikehata and K. Nishihara, Diffusion phenomenon for second order linear evolution equations, Studia Math., 158 (2003), 153-161. doi: 10.4064/sm158-2-4.

[10]

R. IkehataG. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Differential Equations, 254 (2013), 3352-3368. doi: 10.1016/j.jde.2013.01.023.

[11]

M. Khader, Nonlinear dissipative wave equations with space-time dependent potential, Nonlinear Anal., 74 (2011), 3945-3963. doi: 10.1016/j.na.2011.02.044.

[12]

J. LinK. Nishihara and J. Zhai, L2-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations, 248 (2010), 403-422. doi: 10.1016/j.jde.2009.09.022.

[13]

J. LinK. Nishihara and J. Zhai, Decay property of solutions for damped wave equations with space-time dependent damping term, J. Math. Anal. Appl., 374 (2011), 602-614. doi: 10.1016/j.jmaa.2010.09.032.

[14]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976/77), 169-189. doi: 10.2977/prims/1195190962.

[15]

K. Mochizuki and M. Nakao, Total energy decay for the wave equation in exterior domains with a dissipation near infinity, J. Math. Anal. Appl., 326 (2007), 582-588. doi: 10.1016/j.jmaa.2006.01.086.

[16]

K. Mochizuki and H. Nakazawa, Energy decay of solutions to the wave equations with linear dissipation localized near infinity, Publ. Res. Inst. Math. Sci., 37 (2001), 441-458. doi: 10.2977/prims/1145477231.

[17]

M. Nakao, Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations, Math. Z., 238 (2001), 781-797. doi: 10.1007/s002090100275.

[18]

K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, Comm. Partial Differential Equations, 35 (2010), 1402-1418. doi: 10.1080/03605302.2010.490285.

[19]

K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations, J. Math. Anal. Appl., 360 (2009), 412-421. doi: 10.1016/j.jmaa.2009.06.065.

[20]

H. Nishiyama, Remarks on the asymptotic behavior of the solution to damped wave equations, J. Differential Equations, 261 (2016), 3893-3940. doi: 10.1016/j.jde.2016.06.014.

[21]

K. Ono, Decay estimates for dissipative wave equations in exterior domains, J. Math. Anal. Appl., 286 (2003), 540-562. doi: 10.1016/S0022-247X(03)00489-X.

[22]

P. RaduG. Todorova and B. Yordanov, Diffusion phenomenon in Hilbert spaces and applications, J. Differential Equations, 250 (2011), 4200-4218. doi: 10.1016/j.jde.2011.01.024.

[23]

P. RaduG. Todorova and B. Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 609-629. doi: 10.3934/dcdss.2009.2.609.

[24]

P. RaduG. Todorova and B. Yordanov, The generalized diffusion phenomenon and applications, SIAM J. Math. Anal., 48 (2016), 174-203. doi: 10.1137/15M101525X.

[25]

M. Reissig and J. Wirth, LpLq decay estimates for wave equations with monotone time dependent dissipation, in Mathematical Models of Phenomena and Evolution Equations (ed. N. Yamada), RIMS Kokyuroku Nr. 1475, 91-106, Kyoto University, 2006. Available from: arXiv: math/0508549.

[26]

M. Sobajima and Y. Wakasugi, Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain, J. Differential Equations, 261 (2016), 5690-5718. doi: 10.1016/j.jde.2016.08.006.

[27]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933.

[28]

G. Todorova and B. Yordanov, Weighted L2-estimates for dissipative wave equations with variable coefficients, J. Differential Equations, 246 (2009), 4497-4518. doi: 10.1016/j.jde.2009.03.020.

[29]

J. Wirth, Wave equations with time-dependent dissipation. Ⅰ. Non-effective dissipation, J. Differential Equations, 222 (2006), 487-514. doi: 10.1016/j.jde.2005.07.019.

[30]

J. Wirth, Wave equations with time-dependent dissipation. Ⅱ. Effective dissipation, J. Differential Equations, 232 (2007), 74-103. doi: 10.1016/j.jde.2006.06.004.

[31]

T. Yamazaki, Diffusion phenomenon for abstract wave equations with decaying dissipation, Asymptotic analysis and singularities-hyperbolic and dispersive PDEs and fluid mechanics, Adv. Stud. Pure Math., 47-1, Math. Soc. Japan, Tokyo, 2007,363-381.

show all references

References:
[1]

R. Chill and A. Haraux, An optimal estimate for the difference of solutions of two abstract evolution equations, J. Differential Equations, 193 (2003), 385-395. doi: 10.1016/S0022-0396(03)00057-3.

[2]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N. J. 1964.

[3]

M. Ikawa, Mixed problems for hyperbolic equations of second order, J. Math. Soc. Japan, 20 (1968), 580-608. doi: 10.2969/jmsj/02040580.

[4]

M. Ikawa, Hyperbolic Partial Differential Equations and Wave Phenomena, American Mathematical Society, Providence, R. I., 2000.

[5]

R. Ikehata, Diffusion phenomenon for linear dissipative wave equations in an exterior domain, J. Differential Equations, 186 (2002), 633-651. doi: 10.1016/S0022-0396(02)00008-6.

[6]

R. Ikehata, Energy decay of solutions for the semilinear dissipative wave equations in an exterior domain, Funkcial. Ekvac., 44 (2001), 487-499.

[7]

R. Ikehata, Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain, J. Differential Equations, 188 (2003), 390-405. doi: 10.1016/S0022-0396(02)00101-8.

[8]

R. Ikehata, Some remarks on the wave equation with potential type damping coefficients, Int. J. Pure Appl. Math., 21 (2005), 19-24. Available from: https://ijpam.eu/contents/2005-21-1/3/3.pdf.

[9]

R. Ikehata and K. Nishihara, Diffusion phenomenon for second order linear evolution equations, Studia Math., 158 (2003), 153-161. doi: 10.4064/sm158-2-4.

[10]

R. IkehataG. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Differential Equations, 254 (2013), 3352-3368. doi: 10.1016/j.jde.2013.01.023.

[11]

M. Khader, Nonlinear dissipative wave equations with space-time dependent potential, Nonlinear Anal., 74 (2011), 3945-3963. doi: 10.1016/j.na.2011.02.044.

[12]

J. LinK. Nishihara and J. Zhai, L2-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations, 248 (2010), 403-422. doi: 10.1016/j.jde.2009.09.022.

[13]

J. LinK. Nishihara and J. Zhai, Decay property of solutions for damped wave equations with space-time dependent damping term, J. Math. Anal. Appl., 374 (2011), 602-614. doi: 10.1016/j.jmaa.2010.09.032.

[14]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976/77), 169-189. doi: 10.2977/prims/1195190962.

[15]

K. Mochizuki and M. Nakao, Total energy decay for the wave equation in exterior domains with a dissipation near infinity, J. Math. Anal. Appl., 326 (2007), 582-588. doi: 10.1016/j.jmaa.2006.01.086.

[16]

K. Mochizuki and H. Nakazawa, Energy decay of solutions to the wave equations with linear dissipation localized near infinity, Publ. Res. Inst. Math. Sci., 37 (2001), 441-458. doi: 10.2977/prims/1145477231.

[17]

M. Nakao, Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations, Math. Z., 238 (2001), 781-797. doi: 10.1007/s002090100275.

[18]

K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, Comm. Partial Differential Equations, 35 (2010), 1402-1418. doi: 10.1080/03605302.2010.490285.

[19]

K. Nishihara and J. Zhai, Asymptotic behaviors of solutions for time dependent damped wave equations, J. Math. Anal. Appl., 360 (2009), 412-421. doi: 10.1016/j.jmaa.2009.06.065.

[20]

H. Nishiyama, Remarks on the asymptotic behavior of the solution to damped wave equations, J. Differential Equations, 261 (2016), 3893-3940. doi: 10.1016/j.jde.2016.06.014.

[21]

K. Ono, Decay estimates for dissipative wave equations in exterior domains, J. Math. Anal. Appl., 286 (2003), 540-562. doi: 10.1016/S0022-247X(03)00489-X.

[22]

P. RaduG. Todorova and B. Yordanov, Diffusion phenomenon in Hilbert spaces and applications, J. Differential Equations, 250 (2011), 4200-4218. doi: 10.1016/j.jde.2011.01.024.

[23]

P. RaduG. Todorova and B. Yordanov, Higher order energy decay rates for damped wave equations with variable coefficients, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 609-629. doi: 10.3934/dcdss.2009.2.609.

[24]

P. RaduG. Todorova and B. Yordanov, The generalized diffusion phenomenon and applications, SIAM J. Math. Anal., 48 (2016), 174-203. doi: 10.1137/15M101525X.

[25]

M. Reissig and J. Wirth, LpLq decay estimates for wave equations with monotone time dependent dissipation, in Mathematical Models of Phenomena and Evolution Equations (ed. N. Yamada), RIMS Kokyuroku Nr. 1475, 91-106, Kyoto University, 2006. Available from: arXiv: math/0508549.

[26]

M. Sobajima and Y. Wakasugi, Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain, J. Differential Equations, 261 (2016), 5690-5718. doi: 10.1016/j.jde.2016.08.006.

[27]

G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933.

[28]

G. Todorova and B. Yordanov, Weighted L2-estimates for dissipative wave equations with variable coefficients, J. Differential Equations, 246 (2009), 4497-4518. doi: 10.1016/j.jde.2009.03.020.

[29]

J. Wirth, Wave equations with time-dependent dissipation. Ⅰ. Non-effective dissipation, J. Differential Equations, 222 (2006), 487-514. doi: 10.1016/j.jde.2005.07.019.

[30]

J. Wirth, Wave equations with time-dependent dissipation. Ⅱ. Effective dissipation, J. Differential Equations, 232 (2007), 74-103. doi: 10.1016/j.jde.2006.06.004.

[31]

T. Yamazaki, Diffusion phenomenon for abstract wave equations with decaying dissipation, Asymptotic analysis and singularities-hyperbolic and dispersive PDEs and fluid mechanics, Adv. Stud. Pure Math., 47-1, Math. Soc. Japan, Tokyo, 2007,363-381.

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