[CvGmt News] doppio seminario Punzo e Souplet 12/4

Daniele Castorina castorin at math.unipd.it
Wed Apr 6 16:59:14 CEST 2016

Carissimi,

vi segnalo che la prossima settimana inizieremo in anticipo (alle 14)
perchè avremo un doppio seminario di Equazioni Differenziali a Roma Tor
Vergata ovvero:
*
**Martedi 12 Aprile 2016, ore 14:00, Aula dal Passo**

**Fabio Punzo (Universita' della Calabria)**
**
Nonexistence of positive solutions for elliptic and parabolic equations
with a potential on Riemannian manifolds.**
**
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).**

**Martedi 12 Aprile 2016, ore 15:00, Aula dal Passo**
**
Philippe Souplet (Université Paris 13)**
**
Morrey spaces and classification of global solutions for a supercritical
semilinear heat equation. **
**
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 $t\to\infty$.

*

--
Daniele Castorina
Stanza 539 – Torre Archimede
Dipartimento di Matematica
Università di Padova
Via Trieste, 63 - 35121 Padova
Tel.: (+39) 0498271429
Email: castorin at math.unipd.it
Homepage: http://www.math.unipd.it/~castorin/

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