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 $p\in(2,\infty)$ and $q_*:=p-N/(N+1)$. We show that theasymptotic profile of compactly supported solutions is given by asource-type self-similar solution of the $p$-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

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

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 ∂_t u − ∆_p u + |∇u|^{p−1} = 0 in (0, ∞) × 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| → ∞, 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 − t)^{1/(2−p)} f_* (|x|), (t, x) ∈ (0, T_e) × R^N , as t → 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

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-

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

Pushed traveling fronts in monostable equations with monotone delayed reaction

We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. We prove the existence of the minimal speed of propagation, the uniqueness of wavefronts (up to a translation) and describe their asymptotics at $-\infty$. We also present a new uniqueness result for a class of nonlocal lattice equations.Comment: 17 pages, submitte

Modeling Peripheral Olfactory Coding in Drosophila Larvae

The Drosophila larva possesses just 21 unique and identifiable pairs of olfactory sensory neurons (OSNs), enabling investigation of the contribution of individual OSN classes to the peripheral olfactory code. We combined electrophysiological and computational modeling to explore the nature of the peripheral olfactory code in situ. We recorded firing responses of 19/21 OSNs to a panel of 19 odors. This was achieved by creating larvae expressing just one functioning class of odorant receptor, and hence OSN. Odor response profiles of each OSN class were highly specific and unique. However many OSN-odor pairs yielded variable responses, some of which were statistically indistinguishable from background activity. We used these electrophysiological data, incorporating both responses and spontaneous firing activity, to develop a Bayesian decoding model of olfactory processing. The model was able to accurately predict odor identity from raw OSN responses; prediction accuracy ranged from 12%–77% (mean for all odors 45.2%) but was always significantly above chance (5.6%). However, there was no correlation between prediction accuracy for a given odor and the strength of responses of wild-type larvae to the same odor in a behavioral assay. We also used the model to predict the ability of the code to discriminate between pairs of odors. Some of these predictions were supported in a behavioral discrimination (masking) assay but others were not. We conclude that our model of the peripheral code represents basic features of odor detection and discrimination, yielding insights into the information available to higher processing structures in the brain

Enhancement of immune response of HBsAg loaded poly(L-lactic acid) microspheres against Hepatitis B through incorporation of alum and chitosan

Purpose: Poly (L-lactic acid) (PLA) microparticles encapsulating Hepatitis B surface antigen (HBsAg) with alum and chitosan were investigated for their potential as a vaccine delivery system. Methods: The microparticles, prepared using a water-in-oil-in-water (w/o/w) double emulsion solvent evaporation method with polyvinyl alcohol (PVA) or chitosan as the external phase stabilising agent showed a significant increase in the encapsulation efficiency of the antigen. Results: PLA-Alum and PLA-chitosan microparticles induced HBsAg serum specific IgG antibody responses significantly higher than PLA only microparticles and free antigen following subcutaneous administration. Chitosan not only imparted a positive charge to the surface of the microparticles but was also able to increase the serum specific IgG antibody responses significantly. Conclusions: The cytokine assays showed that the serum IgG antibody response induced is different according to the formulation, indicated by the differential levels of interleukin 4 (IL-4), interleukin 6 (IL-6) and interferon gamma (IFN-γ). The microparticles eliciting the highest IgG antibody response did not necessarily elicit the highest levels of the cytokines IL-4, IL-6 and IFN-γ