A remark on removable singularity for nonlinear convection–diffusion equation. (February 2016) - Lu, GuoFu; Jiang, Liang
Abstract: In this paper we study the following Cauchy problem: u t = u x x + ( u n ) x, ( x, t ) ∈ R × ( 0, ∞ ), u ( x, 0 ) = δ ( x ), x ∈ R, where δ ( x ) is a Dirac measure and n ≥ 0 . Its solution is called source-type solution. Such singular solution plays an important role in the development of theory of nonlinear parabolic equations. However, there seems not to be perfect answer to the research. Here we focus on whether there is a critical exponent n 0 such that when n < n 0 there exists unique source-type solution, while n ≥ n 0 there is no source-type solution, and on what the singular expansions of source-type solutions at origin are. From a physical point of view, there are phenomena of the interactive effect between the diffusion and convection in a heat process, which is re-confirmed and described through mathematical analysis and numerical simulation. In addition, thanks to the entropy inequality, we get new proof of uniqueness and are able to extend our approaches to nonlinear parabolic-hyperbolic equations with Radon measure as initial datum.