Franca:
In this talk we consider the long time behavior of positive solutions of the following Cauchy problem:
\[ \left\{ \begin{array}{l}
u_t=\Delta u + u^{q}\\
u(0,x)= \phi(x)\,,
\end{array} \right. \]
where $u :\mathbb{R} \times \mathbb{R}^{n} \to \mathbb{R}$, $q> \frac{n}{n-2}$ and of its generalization to spatial dependent potentials.
This equation can be regarded as a model for an exothermic reaction which may produce an explosion ($u$ is the temperature).
Hence we have two main expected behaviors: if "$\phi$ is large" $u$ blows up in finite time, while if "$\phi$ is small"
$u$ converges to $0$ for $t$ large. Our aim is to explore the threshold between these two behaviors.
In this context a key role is played by radial stationary solutions, i.e. Ground States, both regular and singular, and in particular by their separation properties.
In fact, roughly speaking, they determine the threshold between blowing up and fading solutions, and if suitable ordering properties are satisfied they gain some stability.If there will be time we will discuss some recent results concerning separation properties of the stationary problem where the Laplace operator is replaced by its $p$-Laplace generalization.
http://cvgmt.sns.it/seminar/814/
