Very singular solution and short time asymptotic behaviors of nonnegative singular solutions for heat equation with nonlinear convection

In this paper we study the following Cauchy problem:ut=uxx+(un)x,(x,t)∈R×(0,∞),u(x,0)=0,x≠0, where parameter n≥0. Its nonnegative solution is called singular solution when u(x,t) satisfies the equation in the sense of distribution, initial conditions in the classical sense and also u(x,t) exhibits a singularity at the origin (0,0). As we know, the singular solution is called source-type solution if the initial is Mδ(x), where δ(x) is Dirac measure and constant M>0. The solution is called very singular solution if it possesses more singularity than that of source-type solution at the origin. Here we focus on what happens in the interactive effect between the diffusion and convection in a whole physical process. We find critical values n2