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September  2015, 14(5): 1759-1780. doi: 10.3934/cpaa.2015.14.1759

Asymptotic profiles for a strongly damped plate equation with lower order perturbation

Ryo Ikehata 1, and  Marina Soga 1,

1. 

Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan, Japan

Received  August 2014 Revised  April 2015 Published  June 2015

We consider the Cauchy problem in $ R^n$ for a strongly damped plate equation with a lower oder perturbation. We derive asymptotic profiles of solutions with weighted $L^{1,\gamma}(R^n)$ initial velocity by using a new method introduced in [7].
Keywords: asymptotic profiles, strong damping, weighted $L^1$-initial velocity., lower order term, Plate equation, Fourier analysis, low frequency.
Mathematics Subject Classification: Primary: 35B40, 35L30; Secondary: 35B05, 35B2.
Citation: Ryo Ikehata, Marina Soga. Asymptotic profiles for a strongly damped plate equation with lower order perturbation. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1759-1780. doi: 10.3934/cpaa.2015.14.1759
References:
[1]

R. C. Charão, C. R. daLuz and R. Ikehata, New decay rates for a problem of plate dynamics with fractional damping, J. Hyperbolic Diff. Eqns, 10 (2013), 1-13. doi: 10.1142/S0219891613500203.  Google Scholar

[2]

M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the $L^p$-$L^q$ framework, J. Diff. Eqns, 256 (2014), 2307-2336. doi: 10.1016/j.jde.2014.01.002.  Google Scholar

[3]

M. D'Abbicco and M. Reissig, Semi-linear structural damped waves, Math. Meth. Appl. Sci., 32 (2014), 1570-1592. Google Scholar

[4]

L. C. Evans, Partial Differential Equations, Berkeley Mathematics Lecture Noes Vol. 3a, 1994. Google Scholar

[5]

D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003.  Google Scholar

[6]

R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Meth. Appl. Sci., 27 (2004), 865-889. doi: 10.1002/mma.476.  Google Scholar

[7]

R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Diff. Eqns, 257 (2014), 2159-2177. doi: 10.1016/j.jde.2014.05.031.  Google Scholar

[8]

R. Ikehata and M. Natsume, Energy decay estimates for wave equations with a fractional damping, Diff. Int. Eqns, 25 (2012), 939-956.  Google Scholar

[9]

R. Ikehata, G. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Diff. Eqns, 254 (2013), 3352-3368. doi: 10.1016/j.jde.2013.01.023.  Google Scholar

[10]

G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197.  Google Scholar

[11]

X. Lu and M. Reissig, Rates of decay for structural damped models with decreasing in time coefficients, Int. J. Dyn. Syst. Differ. Equ., 2 (2009), 21-55. doi: 10.1504/IJDSDE.2009.028034.  Google Scholar

[12]

C. R. daLuz, R. Ikehata and R. C. Charão, Asymptotic behavior for abstract evolution differential equations of second order,, \emph{J. Diff. Eqns}, ().   Google Scholar

[13]

S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, Cambridge, 1973.  Google Scholar

[14]

G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[15]

Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Meth. Appl. Sci., 23 (2000), 203-226. doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M.  Google Scholar

[16]

M. A. J. Silva and T. F. Ma, On a viscoelastic plate equation with history setting and perturbation of $p$-Laplacian type, IMA J. Appl. Math., 78 (2013), 1130-1146. doi: 10.1093/imamat/hxs011.  Google Scholar

[17]

H. Takeda and S. Yoshikawa, On the initial value problem of the semilinear beam equation with weak damping II, Asymptotic profiles, J. Diff. Eqns, 253 (2012), 3061-3080. doi: 10.1016/j.jde.2012.07.014.  Google Scholar

[18]

T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068.  Google Scholar

show all references

References:
[1]

R. C. Charão, C. R. daLuz and R. Ikehata, New decay rates for a problem of plate dynamics with fractional damping, J. Hyperbolic Diff. Eqns, 10 (2013), 1-13. doi: 10.1142/S0219891613500203.  Google Scholar

[2]

M. D'Abbicco and M. R. Ebert, Diffusion phenomena for the wave equation with structural damping in the $L^p$-$L^q$ framework, J. Diff. Eqns, 256 (2014), 2307-2336. doi: 10.1016/j.jde.2014.01.002.  Google Scholar

[3]

M. D'Abbicco and M. Reissig, Semi-linear structural damped waves, Math. Meth. Appl. Sci., 32 (2014), 1570-1592. Google Scholar

[4]

L. C. Evans, Partial Differential Equations, Berkeley Mathematics Lecture Noes Vol. 3a, 1994. Google Scholar

[5]

D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003.  Google Scholar

[6]

R. Ikehata, New decay estimates for linear damped wave equations and its application to nonlinear problem, Math. Meth. Appl. Sci., 27 (2004), 865-889. doi: 10.1002/mma.476.  Google Scholar

[7]

R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Diff. Eqns, 257 (2014), 2159-2177. doi: 10.1016/j.jde.2014.05.031.  Google Scholar

[8]

R. Ikehata and M. Natsume, Energy decay estimates for wave equations with a fractional damping, Diff. Int. Eqns, 25 (2012), 939-956.  Google Scholar

[9]

R. Ikehata, G. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Diff. Eqns, 254 (2013), 3352-3368. doi: 10.1016/j.jde.2013.01.023.  Google Scholar

[10]

G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197.  Google Scholar

[11]

X. Lu and M. Reissig, Rates of decay for structural damped models with decreasing in time coefficients, Int. J. Dyn. Syst. Differ. Equ., 2 (2009), 21-55. doi: 10.1504/IJDSDE.2009.028034.  Google Scholar

[12]

C. R. daLuz, R. Ikehata and R. C. Charão, Asymptotic behavior for abstract evolution differential equations of second order,, \emph{J. Diff. Eqns}, ().   Google Scholar

[13]

S. Mizohata, The Theory of Partial Differential Equations, Cambridge University Press, Cambridge, 1973.  Google Scholar

[14]

G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[15]

Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Meth. Appl. Sci., 23 (2000), 203-226. doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M.  Google Scholar

[16]

M. A. J. Silva and T. F. Ma, On a viscoelastic plate equation with history setting and perturbation of $p$-Laplacian type, IMA J. Appl. Math., 78 (2013), 1130-1146. doi: 10.1093/imamat/hxs011.  Google Scholar

[17]

H. Takeda and S. Yoshikawa, On the initial value problem of the semilinear beam equation with weak damping II, Asymptotic profiles, J. Diff. Eqns, 253 (2012), 3061-3080. doi: 10.1016/j.jde.2012.07.014.  Google Scholar

[18]

T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electro-magneto-fluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068.  Google Scholar

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