Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions

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April 2019, 39(4): 1651-1684. doi: 10.3934/dcds.2019073

Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions

Thinh Tien Nguyen

Gran Sasso Science Institute, Department of Mathematics, Viale Francesco Crispi 7, 67100 - L'Aquila, Italy

Received August 2017 Published January 2019

In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in, for instance, the velocity-jump processes in several dimensions. Given integers

$ n,d\ge 1 $

, let

$ \mathbf A: = (A^1,\dots,A^d)\in (\mathbb R^{n\times n})^d $

be a matrix-vector and let

$ B\in \mathbb R^{n\times n} $

be not necessarily symmetric but have one single eigenvalue zero, we consider the Cauchy problem for

$ n\times n $

linear systems having the form

$ \begin{equation*} \partial_{t}u+\mathbf A\cdot \nabla_{\mathbf x} u+Bu = 0,\qquad (\mathbf x,t)\in \mathbb R^d\times \mathbb R_+. \end{equation*} $

Under appropriate assumptions, we show that the solution

$ u $

is decomposed into

$ u = u^{(1)}+u^{(2)} $

such that the asymptotic profile of

$ u^{(1)} $

denoted by

$ U $

is a solution to a parabolic equation,

$ u^{(1)}-U $

decays at the rate

$ t^{-\frac d2(\frac 1q-\frac 1p)-\frac 12} $

as

$ t\to +\infty $

in any

$ L^p $

-norm and

$ u^{(2)} $

decays exponentially in

$ L^2 $

-norm, provided

$ u(\cdot,0)\in L^q(\mathbb R^d)\cap L^2(\mathbb R^d) $

for

$ 1\le q\le p\le \infty $

. Moreover,

$ u^{(1)}-U $

decays at the optimal rate

$ t^{-\frac d2(\frac 1q-\frac 1p)-1} $

as

$ t\to +\infty $

if the original system satisfies a symmetry property. The main proofs are based on the asymptotic expansions of the fundamental solution in the frequency space and the Fourier analysis.

Keywords: Large-time behavior, dissipative linear hyperbolic systems, asymptotic expansions, multi-dimensional space, decay estimates.

Mathematics Subject Classification: Primary: 35L45; Secondary: 35C20.

Citation: Thinh Tien Nguyen. Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1651-1684. doi: 10.3934/dcds.2019073

References:

[1]	H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin-Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.
[2]	J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-Heidelberg, 1976.
[3]	S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622. doi: 10.1002/cpa.20195.
[4]	A. Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova, 110 (2003), 103-117.
[5]	G. Craciun, A. Brown and A. Friedman, A dynamical system model of neurofilament transport in axons, J. Theoret. Biol., 237 (2005), 316-322. doi: 10.1016/j.jtbi.2005.04.018.
[6]	S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129-156. doi: 10.1093/qjmam/4.2.129.
[7]	T. Hosono and T. Ogawa, Large time behavior and L^p-L^q estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118. doi: 10.1016/j.jde.2004.03.034.
[8]	M. Kac, A stochastic model related to the telegrapher's equation. Reprinting of an article published in 1956, Papers arising from a Conference on Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972), Rocky Mountain J. Math., 4 (1974), 497–509. doi: 10.1216/RMJ-1974-4-3-497.
[9]	T. Kato, Perturbation Theory For Linear Operators, Springer-Verlag, Berlin, 1995.
[10]	S. Kawashima, Global existence and stability of solutions for discrete velocity models of the Boltzmann equation, North-Holland Math. Stud., 98 (1984), 59-85. doi: 10.1016/S0304-0208(08)71492-0.
[11]	P. Marcati and K. Nishihara, 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 Equations, 191 (2003), 445-469. doi: 10.1016/S0022-0396(03)00026-3.
[12]	C. Mascia, Exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity-jump processes, J. Differential Equations, 260 (2016), 401-426. doi: 10.1016/j.jde.2015.08.043.
[13]	C. Mascia and T. T. Nguyen, L^p-L^q decay estimates for dissipative linear hyperbolic systems in 1D, J. Differential Equations, 263 (2017), 6189-6230. doi: 10.1016/j.jde.2017.07.011.
[14]	T. Narazaki, L^p-L^q estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626. doi: 10.2969/jmsj/1191418647.
[15]	K. Nishihara, 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. doi: 10.1007/s00209-003-0516-0.
[16]	D. Serre, Matrices. Theory and Applications, Springer-Verlag, New York, 2002.
[17]	Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.
[18]	Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal., 205 (2012), 239-266. doi: 10.1007/s00205-012-0508-5.
[19]	T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068.

show all references

References:

[1]	H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin-Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.
[2]	J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-Heidelberg, 1976.
[3]	S. Bianchini, B. Hanouzet and R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60 (2007), 1559-1622. doi: 10.1002/cpa.20195.
[4]	A. Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova, 110 (2003), 103-117.
[5]	G. Craciun, A. Brown and A. Friedman, A dynamical system model of neurofilament transport in axons, J. Theoret. Biol., 237 (2005), 316-322. doi: 10.1016/j.jtbi.2005.04.018.
[6]	S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129-156. doi: 10.1093/qjmam/4.2.129.
[7]	T. Hosono and T. Ogawa, Large time behavior and L^p-L^q estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118. doi: 10.1016/j.jde.2004.03.034.
[8]	M. Kac, A stochastic model related to the telegrapher's equation. Reprinting of an article published in 1956, Papers arising from a Conference on Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972), Rocky Mountain J. Math., 4 (1974), 497–509. doi: 10.1216/RMJ-1974-4-3-497.
[9]	T. Kato, Perturbation Theory For Linear Operators, Springer-Verlag, Berlin, 1995.
[10]	S. Kawashima, Global existence and stability of solutions for discrete velocity models of the Boltzmann equation, North-Holland Math. Stud., 98 (1984), 59-85. doi: 10.1016/S0304-0208(08)71492-0.
[11]	P. Marcati and K. Nishihara, 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 Equations, 191 (2003), 445-469. doi: 10.1016/S0022-0396(03)00026-3.
[12]	C. Mascia, Exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity-jump processes, J. Differential Equations, 260 (2016), 401-426. doi: 10.1016/j.jde.2015.08.043.
[13]	C. Mascia and T. T. Nguyen, L^p-L^q decay estimates for dissipative linear hyperbolic systems in 1D, J. Differential Equations, 263 (2017), 6189-6230. doi: 10.1016/j.jde.2017.07.011.
[14]	T. Narazaki, L^p-L^q estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626. doi: 10.2969/jmsj/1191418647.
[15]	K. Nishihara, 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. doi: 10.1007/s00209-003-0516-0.
[16]	D. Serre, Matrices. Theory and Applications, Springer-Verlag, New York, 2002.
[17]	Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. doi: 10.14492/hokmj/1381757663.
[18]	Y. Ueda, R. Duan and S. Kawashima, Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application, Arch. Ration. Mech. Anal., 205 (2012), 239-266. doi: 10.1007/s00205-012-0508-5.
[19]	T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofluid dynamics, Japan J. Appl. Math., 1 (1984), 435-457. doi: 10.1007/BF03167068.

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