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Source-type solution to nonlinear Fokker-Planck equation in one dimension

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Abstract

In this paper, we consider the following equation

$u_t = (u^m )_{xx} + (u^n )_x ,$

with the initial condition as Dirac measure. Attention is focused on existence, nonexistence, uniqueness and the asymptotic behavior near (0, 0) of solution to the Cauchy’s problem. The special feature of this equation lies in nonlinear convection effect, i.e., the equation possesses nonlinear hyperbolic character as well as degenerate parabolic one. The situation leads to a more sophisticated mathematical analysis. To our knowledge, the solvability of singular solution to the equation has not been concluded yet. Here based on the previous works by the authors, we show that there exists a critical number n 0 = m+2 such that a unique source-type solution to this equation exists if 0 ⩽ n < n 0, while such a solution does not exist if n ⩾ n 0 (see our working paper). Our results give the perfect answer to such topical researches.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Putian University, Putian, 351100, China

    GuoFu Lu

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  1. GuoFu Lu
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Correspondence to GuoFu Lu.

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Lu, G. Source-type solution to nonlinear Fokker-Planck equation in one dimension. Sci. China Math. 56, 1845–1868 (2013). https://doi.org/10.1007/s11425-013-4612-2

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  • DOI: https://doi.org/10.1007/s11425-013-4612-2

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