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: u(t) = u(xx) + (u(n))(x), (x, t) is an element of R x (0, infinity), 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 M delta(x), where delta(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 n(2) < n(1) < n(0) such that there exists unique source-type solution in the exponent range of n(2) < n < n(0), while there exists no nonnegative singular solution if n >= n(0). Only in the case of n(2) < n < n(1) there exists a very singular solution, but in the case of n >= n(1) or n < n(2) 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. (C) 2017 Elsevier Inc. All rights reserved.