Diffusion phenomenon in Hilbert spaces and applications

Abstract We prove an abstract version of the striking diffusion phenomenon that offers a strong connection between the asymptotic behavior of abstract parabolic and dissipative hyperbolic equations. An important aspect of our approach is that we use in a natural way spectral analysis without involving complicated resolvent estimates. Our proof of the diffusion phenomenon does not use the individual behavior of solutions; instead we show that only their difference matters. We estimate the Hilbert norm of the difference in terms of the Hilbert norm of solutions to the parabolic problems, which allows us to transfer the decay from the parabolic to the hyperbolic problem. The application of these estimates to operators with Markov property combined with a weighted Nash inequality yields explicit and sharp decay rates for hyperbolic problems with variable (x-dependent) coefficients in exterior domains. Our method provides new insight in this area of extensive research which was not well understood until now.

[1]  R. Racke,et al.  Decay rates for solutions of damped systems and generalized Fourier transforms. , 1990 .

[2]  N. Varopoulos,et al.  Hardy-Littlewood theory for semigroups , 1985 .

[3]  K. Ono.  Decay estimates for dissipative wave equations in exterior domains , 2003 .

[4]  S. Kawashima,et al.  Global existence and asymptotic behavior of solutions for quasi-linear dissipative plate equation , 2010, 1003.2885.

[5]  M. Nakao.  Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations , 2001 .

[6]  Petronela Radu,et al.  Higher order energy decay rates for damped wave equations with variable coefficients , 2008, 0811.2159.

[7]  Nakao Hayashi,et al.  On the critical nonlinear damped wave equation with large initial data , 2007 .

[8]  Alain Haraux,et al.  An optimal estimate for the difference of solutions of two abstract evolution equations , 2003 .

[9]  J. Wirth.  Asymptotic properties of solutions to wave equations with time-dependent dissipation , 2004 .

[10]  Ikehata Ryo.  Fast decay of solutions for linear wave equations with dissipation localized near infinity in an exterior domain , 2003 .

[11]  R. Ikehata.  Diffusion phenomenon for linear dissipative wave equations in an exterior domain , 2002 .

[12]  Jens Wirth.  Solution representations for a wave equation with weak dissipation , 2002 .

[13]  A. Matsumura,et al.  On the Asymptotic Behavior of Solutions of Semi-linear Wave Equations , 1976 .

[14]  B. Turesson,et al.  Nonlinear Potential Theory and Weighted Sobolev Spaces , 2000 .

[15]  E. DiBenedetto.  Degenerate Parabolic Equations , 1993 .

[16]  A. Milani,et al.  On the Diffusion Phenomenon of Quasilinear Hyperbolic Waves , 1980 .

[17]  R. Bañuelos,et al.  Heat kernel, eigenfunctions, and conditioned Brownian motion in planar domains , 1989 .

[18]  M. Nakao.  Decay of solutions of the wave equation with a local degenerate dissipation , 1996 .

[19]  G. Burton.  Sobolev Spaces , 2013 .

[20]  M. Nakao,et al.  Total energy decay for the wave equation in exterior domains with a dissipation near infinity , 2007 .

[21]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[22]  Thierry Coulhon.  Inégalités de Gagliardo-Nirenberg pour les semi-groupes d'opérateurs et applications , 1992 .

[23]  B. Yordanov,et al.  Weighted L2-estimates for dissipative wave equations with variable coefficients , 2009 .

[24]  Enrique Zuazua,et al.  ASYMPTOTIC EXPANSION FOR DAMPED WAVE EQUATIONS WITH PERIODIC COEFFICIENTS , 2001 .

[25]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[26]  Thierry Coulhon,et al.  Ultracontractivity and Nash Type Inequalities , 1996 .

[27]  W. Müller.  The heat kernel , 1987 .

[28]  J. Nash,et al.  PARABOLIC EQUATIONS. , 1957, Proceedings of the National Academy of Sciences of the United States of America.

[29]  B. Yordanov,et al.  Decay estimates for wave equations with variable coefficients , 2009 .

[30]  Taeko Yamazaki,et al.  Diffusion phenomenon for abstract wave equations with decaying dissipation , 2007 .

[31]  Jens Wirth,et al.  Wave equations with time-dependent dissipation II. Effective dissipation , 2006 .

[32]  G. Karch.  Asymptotics of solutions to a convection—diffusion equation on the half-line , 2000, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[33]  K. Mochizuki,et al.  Energy Decay of Solutions to the Wave Equations with Linear Dissipation Localized Near Infinity , 2001 .

[34]  Kenji Nishihara,et al.  Asymptotic Behavior of Solutions of Quasilinear Hyperbolic Equations with Linear Damping , 1997 .

[35]  Grzegorz Karch,et al.  Selfsimilar profiles in large time asymptotics of solutions to damped wave equations , 2000 .

[36]  Kenji Nishihara,et al.  Diffusion phenomenon for second order linear evolution equations , 2003 .

[37]  A. Milani,et al.  On the diffusion phenomenonof quasilinear hyperbolic waves , 2000 .

[38]  Mohammed Kbiri Alaoui,et al.  On Degenerate Parabolic Equations , 2011, Int. J. Math. Math. Sci..

[39]  Tai-Ping Liu,et al.  Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping , 1992 .

[40]  E. Hille.  Functional Analysis And Semi-Groups , 1948 .

[41]  Kenji Nishihara,et al.  Convergence Rates to Nonlinear Diffusion Waves for Solutions of System of Hyperbolic Conservation Laws with Damping , 1996 .