Self-similar solutions for the heat equation with a positive non-Lipschitz continuous, semilinear source term

We investigate the existence of self-similar solutions for the parabolic equation ut=Δu+umHu, with 0≤m<1 and H the Heaviside graph, coupled with the initial datum ux,0=−cx211−m, with c>0. We analyze two cases: the problem in Rn , n>1, with m=0 and the problem in R when 0≤m<1. In the first case we extend the result of Gianni and Hulshof (1992) and show that there exist only two self-similar solutions changing sign, provided 00. These solutions are of great interest. Indeed, on one hand they prove that the problem does not admit uniqueness and on the other they prove that a single point where ux,0=0, for an initial datum which is otherwise negative, can generate a region where ux,t is positive.