239 research outputs found
Blowup solutions of Grushin's operator
In this note, we consider the blowup phenomenon of Grushin's operator. By
using the knowledge of probability, we first get expression of heat kernel of
Grushin's operator. Then by using the properties of heat kernel and suitable
auxiliary function, we get that the solutions will blow up in finite time.Comment:
Ricci flow of conformally compact metrics
In this paper we prove that given a smoothly conformally compact metric there
is a short-time solution to the Ricci flow that remains smoothly conformally
compact. We adapt recent results of Schn\"urer, Schulze and Simon to prove a
stability result for conformally compact Einstein metrics sufficiently close to
the hyperbolic metric.Comment: 26 pages, 2 figures. Version 2 includes stronger stability result and
fixes several typo
On a two-sidedly degenerate chemotaxis model with volume-filling effect
We consider a fully parabolic model for chemotaxis with volume-filling
effect and a nonlinear diffusion that degenerates in a two-sided fashion. We address
the questions of existence of weak solutions and of their regularity by using,
respectively, a regularization method and the technique of intrinsic scaling.Outstanding Young Investigators Award from the Research
Council of Norway; (J. M. Urbano) CMUC/FCT, Project
POCI/MAT/57546/200
Blowup in diffusion equations: A survey
AbstractThis paper deals with quasilinear reaction-diffusion equations for which a solution local in time exists. If the solution ceases to exist for some finite time, we say that it blows up. In contrast to linear equations blowup can occur even if the data are smooth and well-defined for all times. Depending on the equation either the solution or some of its derivatives become singular. We shall concentrate on those cases where the solution becomes unbounded in finite time. This can occur in quasilinear equations if the heat source is strong enough. There exist many theoretical studies on the question on the occurrence of blowup. In this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup. The asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power. A better knowledge would be useful also for their numerical treatment. Thus, not surprisingly, the numerical analysis of this type of problems is still at a rather early stage. The goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems
A Subelliptic Analogue of Aronson-Serrin's Harnack Inequality
We show that the Harnack inequality for a class of degenerate parabolic
quasilinear PDE \p_t u=-X_i^* A_i(x,t,u,Xu)+ B(x,t,u,Xu), associated to a
system of Lipschitz continuous vector fields in in \Om\times
(0,T) with \Om \subset M an open subset of a manifold with control
metric corresponding to and a measure follows from the basic
hypothesis of doubling condition and a weak Poincar\'e inequality. We also show
that such hypothesis hold for a class of Riemannian metrics g_\e collapsing
to a sub-Riemannian metric \lim_{\e\to 0} g_\e=g_0 uniformly in the parameter
\e\ge 0
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