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

In this paper, we consider the following equation ut=(um)xx+(un)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 n0=m+2 such that a unique source-type solution to this equation exists if 0≤n<n0 , while such a solution does not exist if n≥n0 (see our working paper). Our results give the perfect answer to such topical researches.

In this paper, we consider the following equation ut=（um）xx＋（un）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 n0=m＋2 such that a unique source-type solution to this equation exists if 0≤n