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 $u_t=\Delta u+u^{m}H\left(u\right)$, with $0\le m<1$ and $H$ the Heaviside graph, coupled with the initial datum $u\left(x,0\right)=-c{\left({\left(x\right)}^2\right)}^{\frac{1}{1-m}}$, with $c>0$. We analyze two cases: the problem in $R^n$ , $n>1$, with $m=0$ and the problem in $R$ when $0\le m<1$. 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 $0<c<c_{cr}$, with $c_{cr}$ obtained solving a specific algebraic equation depending on $n$. In the second case we prove that there exist at least two self-similar solutions of problem $u_t=u_{xx}+u^{m}H\left(u\right)$, $u\left(x,0\right)=-c{\left(x^2\right)}^{\frac{1}{1-m}}$, changing sign and evolving region where $u>0$. 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 $u\left(x,0\right)=0$, for an initial datum which is otherwise negative, can generate a region where $u\left(x,t\right)$ is positive.