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Self-similar solutions of quasilinear parabolic equations with nonlinear gradient terms

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Abstract

This paper is devoted to study the classification of self-similar solutions to the quasilinear parabolic equations with nonlinear gradient termsu t = Δ(u m) - uq∣Δu∣p withm ≥ 1,p,q > 0 andp +q >m. Form = 1, it is shown that the very singular self-similar solution exists if and only ifnq + (n +1)p <n + 2, and in case of existence, such solution is unique. Form > 1, it is shown that very singular self-similar solutions exist if and only if 1 <m < 2 andnq + (n + 1)p < 2 +mn, and such solutions have compact support if they exist. Moreover, the interface relation is obtained.

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, 210018, Nanjing, China

    Peihu Shi & Mingxin Wang

  2. Department of Mathematics, Xuzhou Normal University, 221116, Xuzhou, China

    Mingxin Wang

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  1. Peihu Shi
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  2. Mingxin Wang
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Correspondence to Mingxin Wang.

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Shi, P., Wang, M. Self-similar solutions of quasilinear parabolic equations with nonlinear gradient terms. Sci. China Ser. A-Math. 47, 659 (2004). https://doi.org/10.1360/02ys0245

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  • DOI: https://doi.org/10.1360/02ys0245

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