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Volume 21, Issue 2
Decay Rates Toward Stationary Waves of Solutions for Damped Wave Equations

Lili Fan , Hui Yin & Huijiang Zhao

J. Part. Diff. Eq., 21 (2008), pp. 141-172.

Published online: 2008-05

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  • Abstract
This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R_+ u_{tt}-u_{xx}+u_t+f(u)_x=0, t > 0, x ∈ R_+, u(0,x)=u_0(x)→ u_+, as x→+∞, u_t(0,x)=u_1(x), u(t,0)=u_b. For the non-degenerate case f'(u_+) < 0, it is shown in [1] that the above initialboundary value problem admits a unique global solution u(t, x) which converges to the stationary wave φ(x) uniformly in x ∈ R+ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. Moreover, by using the space-time weighted energy method initiated by Kawashima and Matsumura [2], the convergence rates (including the algebraic convergence rate and the exponential convergence rate) of u(t, x) toward φ(x) are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f'(u_b) < 0. The main purpose of this paper is devoted to discussing the case of f'(u_b) = 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.
  • Keywords

Damped wave equation stationary wave asymptotic stability decay rates space-time weighted energy method

  • AMS Subject Headings

35L70 35B35 35B40.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-21-141, author = {}, title = {Decay Rates Toward Stationary Waves of Solutions for Damped Wave Equations}, journal = {Journal of Partial Differential Equations}, year = {2008}, volume = {21}, number = {2}, pages = {141--172}, abstract = { This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R_+ u_{tt}-u_{xx}+u_t+f(u)_x=0, t > 0, x ∈ R_+, u(0,x)=u_0(x)→ u_+, as x→+∞, u_t(0,x)=u_1(x), u(t,0)=u_b. For the non-degenerate case f'(u_+) < 0, it is shown in [1] that the above initialboundary value problem admits a unique global solution u(t, x) which converges to the stationary wave φ(x) uniformly in x ∈ R+ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. Moreover, by using the space-time weighted energy method initiated by Kawashima and Matsumura [2], the convergence rates (including the algebraic convergence rate and the exponential convergence rate) of u(t, x) toward φ(x) are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f'(u_b) < 0. The main purpose of this paper is devoted to discussing the case of f'(u_b) = 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5275.html} }
TY - JOUR T1 - Decay Rates Toward Stationary Waves of Solutions for Damped Wave Equations JO - Journal of Partial Differential Equations VL - 2 SP - 141 EP - 172 PY - 2008 DA - 2008/05 SN - 21 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5275.html KW - Damped wave equation KW - stationary wave KW - asymptotic stability KW - decay rates KW - space-time weighted energy method AB - This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R_+ u_{tt}-u_{xx}+u_t+f(u)_x=0, t > 0, x ∈ R_+, u(0,x)=u_0(x)→ u_+, as x→+∞, u_t(0,x)=u_1(x), u(t,0)=u_b. For the non-degenerate case f'(u_+) < 0, it is shown in [1] that the above initialboundary value problem admits a unique global solution u(t, x) which converges to the stationary wave φ(x) uniformly in x ∈ R+ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. Moreover, by using the space-time weighted energy method initiated by Kawashima and Matsumura [2], the convergence rates (including the algebraic convergence rate and the exponential convergence rate) of u(t, x) toward φ(x) are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f'(u_b) < 0. The main purpose of this paper is devoted to discussing the case of f'(u_b) = 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.
Lili Fan , Hui Yin & Huijiang Zhao . (2019). Decay Rates Toward Stationary Waves of Solutions for Damped Wave Equations. Journal of Partial Differential Equations. 21 (2). 141-172. doi:
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