<html>
<head>
<meta content="text/html; charset=UTF-8" http-equiv="Content-Type">
</head>
<body text="#000000" bgcolor="#FFFFFF">
Carissimi,<br>
<br>
vi segnalo che la prossima settimana inizieremo in anticipo (alle
14) perchè avremo un doppio seminario di Equazioni Differenziali a
Roma Tor Vergata ovvero:<br>
<b><br>
</b><b>Martedi 12 Aprile 2016, ore 14:00, Aula dal Passo</b><b><br>
<br>
</b><b>Fabio Punzo (Universita' della Calabria)</b><b><br>
</b><b><br>
Nonexistence of positive solutions for elliptic and parabolic
equations with a potential on Riemannian manifolds.</b><b><br>
</b><b><br>
Abstract:In this talk I will present some results concerning
nonexistence of nonnegative, nontrivial weak solutions for a class
of quasilinear elliptic and parabolic differential inequalities
with a potential on complete, noncompact Riemannian manifolds. In
particular, we investigate the interplay between the geometry of
the underlying manifold, the (power) nonlinearity and the behavior
of the potential at infinity in obtaining nonexistence of
nonnegative solutions. Such results have been recently obtained in
collaboration with P. Mastrolia (Università di Milano) and D.
Monticelli (Politecnico di Milano).</b><b><br>
<br>
</b><b>Martedi 12 Aprile 2016, ore 15:00, Aula dal Passo</b><b><br>
</b><b><br>
Philippe Souplet (Université Paris 13)</b><b><br>
</b><b><br>
Morrey spaces and classification of global solutions for a
supercritical semilinear heat equation. </b><b><br>
</b><b><br>
Abstract:We prove the boundedness of global classical solutions
for the semilinear heat equation $u_t-\Delta u= |u|^{p-1}u$ in the
whole space ${\bf R}^n$, with $n\ge 3$ and supercritical power
$p>(n+2)/(n-2)$. This is proved without any radial symmetry or
sign assumptions, unlike in all the previously known results, and
under decay assumptions on the initial data that are essentially
optimal in view of the known counter-examples. Moreover, we show
that any global classical solution has to decay in time faster
than $t^{-1/(p-1)}$, which is also optimal and in contrast with
the subcritical case. The proof relies on nontrivial modifications
of techniques developed by Chou-Du-Zheng (Calc. Var. PDE 2007) and
by Blatt-Struwe (IMRN 2015) for the case of convex bounded
domains. It is based on weighted energy estimates of Giga-Kohn
type, combined with an analysis of the equation in a suitable
Morrey space. We in particular simplify the approach of
Blatt-Struwe by establishing and using a result on global
existence and decay for small initial data in critical elliptic
Morrey spaces, rather than $\varepsilon$-regularity in parabolic
Morrey spaces. Our results are actually valid for any convex,
bounded or unbounded, smooth domain. As a consequence we also
prove that set of initial data producing global solutions is open
in the corresponding Morrey topology, and we show that the
so-called ``borderline'' weak solutions blow up in finite time and
then become classical again and decay as $t\to\infty$. <br>
<br>
</b>
<pre class="moz-signature" cols="72">--
Daniele Castorina
Stanza 539 – Torre Archimede
Dipartimento di Matematica
Università di Padova
Via Trieste, 63 - 35121 Padova
Tel.: (+39) 0498271429
Email: <a class="moz-txt-link-abbreviated" href="mailto:castorin@math.unipd.it">castorin@math.unipd.it</a>
Homepage: <a class="moz-txt-link-freetext" href="http://www.math.unipd.it/~castorin/">http://www.math.unipd.it/~castorin/</a></pre>
</body>
</html>