243 research outputs found
Large time behavior for a quasilinear diffusion equation with critical gradient absorption
International audienceWe study the large time behavior of non-negative solutions to thenonlinear diffusion equation with critical gradient absorption\partial_t u-\Delta_{p}u+|\nabla u|^{q_*}=0 \quad \hbox{in} \(0,\infty)\times\mathbb{R}^N\ ,for and . We show that theasymptotic profile of compactly supported solutions is given by asource-type self-similar solution of the -Laplacian equation with suitable logarithmic time and space scales. In the process, we also get optimal decay rates for compactly supported solutions and optimal expansion rates for their supports that strongly improve previous results
Instantaneous shrinking and single point extinction for viscous Hamilton-Jacobi equations with fast diffusion
International audienceFor a large class of non-negative initial data, the solutions to the quasilinear viscous Hamilton-Jacobi equation in are known to vanish identically after a finite time when , the positivity set of is a bounded subset of even if in . This decay condition on is also shown to be optimal by proving that the positivity set of any solution emanating from a positive initial condition decaying at a slower rate as is the whole for all times. The time evolution of the positivity set is also studied: on the one hand, it is included in a fixed ball for all times if it is initially bounded (\emph{localization}). On the other hand, it converges to a single point at the extinction time for a class of radially symmetric initial data, a phenomenon referred to as \emph{single point extinction}. This behavior is in sharp contrast with what happens when ranges in and for which we show \emph{complete extinction}. Instantaneous shrinking and single point extinction take place in particular for the semilinear viscous Hamilton-Jacobi equation when and and seem to have remained unnoticed
Last passage percolation and traveling fronts
We consider a system of N particles with a stochastic dynamics introduced by
Brunet and Derrida. The particles can be interpreted as last passage times in
directed percolation on {1,...,N} of mean-field type. The particles remain
grouped and move like a traveling wave, subject to discretization and driven by
a random noise. As N increases, we obtain estimates for the speed of the front
and its profile, for different laws of the driving noise. The Gumbel
distribution plays a central role for the particle jumps, and we show that the
scaling limit is a L\'evy process in this case. The case of bounded jumps
yields a completely different behavior
Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation
We study a quasilinear parabolic Cauchy problem with a cumulative
distribution function on the real line as an initial condition. We call
'probabilistic solution' a weak solution which remains a cumulative
distribution function at all times. We prove the uniqueness of such a solution
and we deduce the existence from a propagation of chaos result on a system of
scalar diffusion processes, the interactions of which only depend on their
ranking. We then investigate the long time behaviour of the solution. Using a
probabilistic argument and under weak assumptions, we show that the flow of the
Wasserstein distance between two solutions is contractive. Under more stringent
conditions ensuring the regularity of the probabilistic solutions, we finally
derive an explicit formula for the time derivative of the flow and we deduce
the convergence of solutions to equilibrium.Comment: Stochastic partial differential equations: analysis and computations
(2013) http://dx.doi.org/10.1007/s40072-013-0014-
Self-similar extinction for a diffusive Hamilton-Jacobi equation with critical absorption
International audienceThe behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption â_t u â â_p u + |âu|^{pâ1} = 0 in (0, â) Ă R^N , and fast diffusion 2N/(N + 1) < p < 2. Given a non-negative and radially symmetric initial condition with a non-increasing profile which decays sufficiently fast as |x| â â, it is shown that the corresponding solution u to the above equation approaches a uniquely determined separate variable solution of the form U (t, x) = (T_e â t)^{1/(2âp)} f_* (|x|), (t, x) â (0, T_e) Ă R^N , as t â T_e , where T_e denotes the finite extinction time of u. A cornerstone of the convergence proof is an underlying variational structure of the equation. Also, the selected profile f_* is the unique non-negative solution to a second order ordinary differential equation which decays exponentially at infinity. A complete classification of solutions to this equation is provided, thereby describing all separate variable solutions of the original equation. One important difficulty in the uniqueness proof is that no monotonicity argument seems to be available and it is overcome by the construction of an appropriate Pohozaev functional
Reaction-diffusion systems and nonlinear waves
The authors investigate the solution of a nonlinear reaction-diffusion
equation connected with nonlinear waves. The equation discussed is more general
than the one discussed recently by Manne, Hurd, and Kenkre (2000). The results
are presented in a compact and elegant form in terms of Mittag-Leffler
functions and generalized Mittag-Leffler functions, which are suitable for
numerical computation. The importance of the derived results lies in the fact
that numerous results on fractional reaction, fractional diffusion, anomalous
diffusion problems, and fractional telegraph equations scattered in the
literature can be derived, as special cases, of the results investigated in
this article.Comment: LaTeX, 16 pages, corrected typo
- âŠ