| Contents |
| In this talk we consider a non-linear heat equation of the form
$$u_t= \Delta u+ f(u,|x|)$$
and we assume that the non-linearity $f$ is supercritical with respect to
$n/(n-2)$ in some weak sense.
We prove local existence of regular and singular solutions and
we use Fowler transformation in order to establish existence and
separation properties of radial Ground States. Then we use these results
to detect long time behavior of both radial and non-radial initial data.
In particular we describe the threshold separating initial data in the
basin of attraction of the null solution, from initial data which blow up
in finite time.
We show that such a border is made up by Ground States when $f$ is supercritical with respect to $(n+2)/(n-2)$, thus extending results by Wang, and Deng, Li, Liu to potential including the Matukuma case
$f(u,r)= \frac{u^{q}}{1+r^a}$.
We also obtain a family of sub and super-solutions which are new even for the original problem $f(u)=u^q$ for any $q>n/(n-2)$. |
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