52 research outputs found
Regularity of the extremal solution for singular p-Laplace equations
We study the regularity of the extremal solution to the singular
reaction-diffusion problem in , on
, where , , is a smooth bounded domain and is any positive, superlinear,
increasing and (asymptotically) convex nonlinearity. We provide a simple
proof of known and \textit{a priori} estimates for , i.e.
if , if and if
A global existence result for a Keller-Segel type system with supercritical initial data
We consider a parabolic-elliptic Keller-Segel type system, which is related
to a simplified model of chemotaxis. Concerning the maximal range of existence
of solutions, there are essentially two kinds of results: either global
existence in time for general subcritical () initial data,
or blow--up in finite time for suitably chosen supercritical
() initial data with concentration around finitely many
points. As a matter of fact there are no results claiming the existence of
global solutions in the supercritical case. We solve this problem here and
prove that, for a particular set of initial data which share large
supercritical masses, the corresponding solution is global and uniformly
bounded
A Liouville theorem for superlinear heat equations on Riemannian manifolds
We study the triviality of the solutions of weighted superlinear heat
equations on Riemannian manifolds with nonnegative Ricci tensor. We prove a
Liouville--type theorem for solutions bounded from below with nonnegative
initial data, under an integral growth condition on the weight
On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary
We derive the long time asymptotic of solutions to an evolutive
Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with
ergodic problems recently studied in \cite{bcr}. Our main assumption is an
appropriate degeneracy condition on the operator at the boundary. This
condition is related to the characteristic boundary points for linear operators
as well as to the irrelevant points for the generalized Dirichlet problem, and
implies in particular that no boundary datum has to be imposed. We prove that
there exists a constant such that the solutions of the evolutive problem
converge uniformly, in the reference frame moving with constant velocity ,
to a unique steady state solving a suitable ergodic problem.Comment: 12p
Regularity of stable solutions of -Laplace equations through geometric Sobolev type inequalities
In this paper we prove a Sobolev and a Morrey type inequality involving the
mean curvature and the tangential gradient with respect to the level sets of
the function that appears in the inequalities. Then, as an application, we
establish \textit{a priori} estimates for semi-stable solutions of in a smooth bounded domain . In
particular, we obtain new and bounds for the extremal solution
when the domain is strictly convex. More precisely, we prove that
if and if .Comment: 26 page
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