On a non-homogeneous and non-linear heat equation

TL;DR

This paper investigates the behavior of solutions to a supercritical, non-homogeneous, nonlinear heat equation, identifying thresholds between blow-up and decay, and extending previous results to more general nonlinearities and initial conditions.

Contribution

It extends the analysis of nonlinear heat equations to include supercritical, non-homogeneous cases with broader initial conditions and identifies new threshold solutions and decay behaviors.

Findings

Ground states with slow decay mark the threshold between blow-up and decay.

Existence of initial data above and below the threshold with arbitrarily small differences.

Both threshold-related solutions exhibit fast decay, contrary to expected slow decay.

Abstract

We consider the Cauchy-problem for a parabolic equation of the following type: \begin{equation*} \frac{\partial u}{\partial t}= \Delta u+ f(u,|x|), \end{equation*} where $f=f(u,|x|)$ is supercritical. We supply 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 blowing up initial data and the basin of attraction of the null solution. Our results extend previous ones allowing Matukuma-type potential and more generic dependence on $u$. Then, we further explore such a threshold in the subcritical case too. We find two families of initial data $\zeta(x)$ and $\psi(x)$ which are respectively above and below the threshold, and have arbitrarily small distance in $L^{\infty}$ norm, whose existence is new even for $f(u,r)=u^{q-1}$. Quite surprisingly both $\zeta(x)$ and $\psi(x)$ have fast decay (i.e. $\sim |x|^{2-n}$), while the expected critical asymptotic behavior is slow decay (i.e. $\sim |x|^{2/q-2}$).