On a non-homogeneous and non-linear heat equation

We consider the Cauchy-problem for a parabolic equation of the following type:\[\frac{\partial u}{\partial t} = \Delta u + f (u, \lvert x \rvert) \text{,}\]where $x \in \mathbb{R}^n , n \gt 2, f = f(u, \lvert x \rvert)$ is supercritical. We supplement this equation by the initial condition $u(x, 0) = \phi$, and we allow $\phi$ to be either bounded or unbounded in the origin but smaller than stationary singular solutions. We discuss local existence and long time behaviour for the solutions $u(t, x; \phi)$ for a wide class of non-homogeneous non-linearities $f$. We show that in the supercritical case, ground states with slow decay lie on the threshold between initial data corresponding to blow-up solutions, and the basin of attraction of the null solution. Our results extend previous ones in that we allow $f$ to be a Matukuma-type potential and in that we allow it to depend on u in a more general way.

We explore such a threshold in the subcritical case too, and we obtain a result which is new even for the model case $f(u) = u {\lvert u \rvert}^{q - 2}$. We find a family of initial data $\psi (x)$ which have fast decay (i.e. $\sim {\lvert x \rvert}^{2-n}$, are arbitrarily small in $L^\infty$- norm, but which correspond to blow-up solutions.

Keywords

Cauchy-problem, semi-linear heat equation, singular solutions, stability

2010 Mathematics Subject Classification

35B40, 35B60, 35Kxx

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Published 10 December 2015