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
In this paper we study the source-type solution for the heat equation with convection： ut = △u ＋ b·▽un for （x,t） ∈ ST→ RN × （0,T] and u（x,0） = δ（x） for x ∈ RN, where δ（x） denotes Dirac measure in = RN,N 2,n 0 and b = （b1,...,bN） ∈ RN is a vector. It is shown that there exists a critical number pc = N＋2 such that the source-type solution to the above problem exists and is unique if 0 N n 〈 pc and there exists a unique similarity source-type solution in the case n = N＋1 , while such a solution does not exist N if n 〉 pc. Moreover, the asymptotic behavior of the solution near the origin is studied. It is shown that when 0 〈 n 〈 N＋1 the convection is too weak and the short time behavior of the source-type solution near the origin N is the same as that for the heat equation without convection.
