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

Abstract

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<n1<n0 such that there exists unique source-type solution in the exponent range of 0<n<n0, while there exists no nonnegative singular solution if n≥n0. Only in the case of n2<n<n1 there exists a very singular solution, but in the case of n≥n1 or n≤n2 there is no solution that exhibits more singular than source-type solution at the origin. Furthermore we describe the short time asymptotic behavior of the singular solutions when such Cauchy problem is solvability for source-type solution or very singular solution.