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Stability of planar diffusion wave for nonlinear evolution equation

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Abstract

It is known that the one-dimensional nonlinear heat equation \(u_t = f(u)_{x_1 x_1 }\), f′(u) > 0, u(±∞, t) = u −, u + ≠ u_ has a unique self-similar solution \(\bar u\left( {\tfrac{{x_1 }} {{\sqrt {1 + t} }}} \right)\). In multi-dimensional space, \(\bar u\left( {\tfrac{{x_1 }} {{\sqrt {1 + t} }}} \right)\) is called a planar diffusion wave. In the first part of the present paper, it is shown that under some smallness conditions, such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation: u t − Δf(u) = 0, x ∈ ℝn. The optimal time decay rate is obtained. In the second part of this paper, it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping: u tt + u t − Δf(u) = 0, x ∈ ℝn. The time decay rate is also obtained. The proofs are given by an elementary energy method.

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Authors and Affiliations

  1. National Natural Science Foundation of China, Beijing, 100085, China

    Cheng He

  2. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    FeiMin Huang

  3. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Yan Yong

Authors
  1. Cheng He
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  2. FeiMin Huang
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  3. Yan Yong
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Corresponding author

Correspondence to Yan Yong.

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Dedicated to the NSFC-CNRS Chinese-French summer institute on fluid mechanics in 2010

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He, C., Huang, F. & Yong, Y. Stability of planar diffusion wave for nonlinear evolution equation. Sci. China Math. 55, 337–352 (2012). https://doi.org/10.1007/s11425-011-4345-z

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  • DOI: https://doi.org/10.1007/s11425-011-4345-z

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