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Asymptotic stability of equilibrium state to the mixed initial-boundary value problem for quasilinear hyperbolic systems

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Abstract

Under the internal dissipative condition, the Cauchy problem for inhomogeneous quasilinear hyperbolic systems with small initial data admits a unique global C 1 solution, which exponentially decays to zero as t → +∞, while if the coefficient matrix Θ of boundary conditions satisfies the boundary dissipative condition, the mixed initial-boundary value problem with small initial data for quasilinear hyperbolic systems with nonlinear terms of at least second order admits a unique global C 1 solution, which also exponentially decays to zero as t → +∞. In this paper, under more general conditions, the authors investigate the combined effect of the internal dissipative condition and the boundary dissipative condition, and prove the global existence and exponential decay of the C 1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with small initial data. This stability result is applied to a kind of models, and an example is given to show the possible exponential instability if the corresponding conditions are not satisfied.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Yanzhao Li & Cunming Liu

  2. School of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, China

    Cunming Liu

Authors
  1. Yanzhao Li
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  2. Cunming Liu
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Correspondence to Yanzhao Li.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11326159, 11401421), the China Postdoctoral Science Foundation (No. 2014M560287) and the Shanxi Scholarship Council of China (No. 2013-045).

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Li, Y., Liu, C. Asymptotic stability of equilibrium state to the mixed initial-boundary value problem for quasilinear hyperbolic systems. Chin. Ann. Math. Ser. B 36, 323–344 (2015). https://doi.org/10.1007/s11401-015-0942-4

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  • DOI: https://doi.org/10.1007/s11401-015-0942-4

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