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

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.
Citation: Thinh Tien Nguyen. Asymptotic limit and decay estimates for a class of dissipative linear hyperbolic systems in several dimensions. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-Heidelberg, 1976. Google Scholar

[3]

S. BianchiniB. 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. Google Scholar

[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. Google Scholar

[5]

G. CraciunA. 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. Google Scholar

[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. Google Scholar

[7]

T. Hosono and T. Ogawa, Large time behavior and Lp-Lq 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. Google Scholar

[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. Google Scholar

[9]

T. Kato, Perturbation Theory For Linear Operators, Springer-Verlag, Berlin, 1995. Google Scholar

[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. Google Scholar

[11]

P. Marcati and K. Nishihara, The Lp-Lq 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. Google Scholar

[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. Google Scholar

[13]

C. Mascia and T. T. Nguyen, Lp-Lq decay estimates for dissipative linear hyperbolic systems in 1D, J. Differential Equations, 263 (2017), 6189-6230. doi: 10.1016/j.jde.2017.07.011. Google Scholar

[14]

T. Narazaki, Lp-Lq 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. Google Scholar

[15]

K. Nishihara, Lp-Lq 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. Google Scholar

[16]

D. Serre, Matrices. Theory and Applications, Springer-Verlag, New York, 2002. Google Scholar

[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. Google Scholar

[18]

Y. UedaR. 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. Google Scholar

[19]

T. UmedaS. 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. Google Scholar

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. Google Scholar

[2]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-Heidelberg, 1976. Google Scholar

[3]

S. BianchiniB. 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. Google Scholar

[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. Google Scholar

[5]

G. CraciunA. 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. Google Scholar

[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. Google Scholar

[7]

T. Hosono and T. Ogawa, Large time behavior and Lp-Lq 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. Google Scholar

[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. Google Scholar

[9]

T. Kato, Perturbation Theory For Linear Operators, Springer-Verlag, Berlin, 1995. Google Scholar

[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. Google Scholar

[11]

P. Marcati and K. Nishihara, The Lp-Lq 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. Google Scholar

[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. Google Scholar

[13]

C. Mascia and T. T. Nguyen, Lp-Lq decay estimates for dissipative linear hyperbolic systems in 1D, J. Differential Equations, 263 (2017), 6189-6230. doi: 10.1016/j.jde.2017.07.011. Google Scholar

[14]

T. Narazaki, Lp-Lq 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. Google Scholar

[15]

K. Nishihara, Lp-Lq 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. Google Scholar

[16]

D. Serre, Matrices. Theory and Applications, Springer-Verlag, New York, 2002. Google Scholar

[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. Google Scholar

[18]

Y. UedaR. 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. Google Scholar

[19]

T. UmedaS. 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. Google Scholar

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