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Convergence to Sharp Traveling Waves of Solutions for Burgers-Fisher-KPP Equations with Degenerate Diffusion

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Abstract

This paper is concerned with the convergence to sharp traveling waves of solutions with semi-compactly supported initial data for Burgers-Fisher-KPP equations with degenerate diffusion. We characterize the motion of the free boundary in the long-time asymptotic of the solution to Cauchy problem and the convergence to sharp traveling wave with almost exponential decay rates. Here a key difficulty lies in the intrinsic presence of nonlinear advection effect. After providing the analysis of the nonlinear advection effect on the asymptotic propagation speed of the free boundary, we construct sub- and super-solutions with semi-compact supports to estimate the motion of the free boundary. The new method overcomes the difficulties of the non-integrability of the generalized derivatives of sharp traveling waves at the free boundary.

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Acknowledgements

The authors would like to express their sincere thanks for the referees’ valuable comments and suggestions, which led the paper a significant improvement. The research of T. Xu was supported by NSFC Grant No. 12301241, Guangzhou Basic and Applied Basic Research Foundation No. 202201010645. The research of S. Ji was supported by NSFC Grant Nos. 12271178 and 12171166, Guangzhou Basic and Applied Basic Research Foundation No. 2024A04J2022, and the Fundamental Research Funds for the Central Universities No. 2022ZYGXZR032. The research of M. Mei was supported in part by NSERC Grant RGPIN-2022-03374. The research of J. Yin was supported by NSFC Grant No. 12171166.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, 510006, Guangdong, People’s Republic of China

    Tianyuan Xu

  2. School of Mathematics, South China University of Technology, Guangzhou, 510641, Guangdong, People’s Republic of China

    Shanming Ji

  3. Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2, Canada

    Ming Mei

  4. Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6, Canada

    Ming Mei

  5. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Guangdong, People’s Republic of China

    Jingxue Yin

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Correspondence to Shanming Ji.

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Xu, T., Ji, S., Mei, M. et al. Convergence to Sharp Traveling Waves of Solutions for Burgers-Fisher-KPP Equations with Degenerate Diffusion. J Nonlinear Sci 34, 44 (2024). https://doi.org/10.1007/s00332-024-10021-x

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