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

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 $\partial_t u-\Delta_p u+|\nabla u|^q=0$ in $(0,\infty)\times\mathbb{R}^N$ are known to vanish identically after a finite time when $2N/(N+1) 0$, the positivity set of $u(t)$ is a bounded subset of $\mathbb{R}^N$ even if $u_0 > 0$ in $\mathbb{R}^N$. This decay condition on $u_0$ 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 $|x|\to\infty$ is the whole $\mathbb{R}^N$ 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 $q$ ranges in $[p-1,p/2)$ and $p\in (2N/(N+1),2]$ 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 $p=2$ and $q\in (0,1)$ 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

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-

Future Imaginings: Organizing in Response to Climate Change

Climate change has rapidly emerged as a major threat to our future. Indeed the increasingly dire projections of increasing global average temperatures and escalating extreme weather events highlight the existential challenge that climate change presents for humanity. In this editorial article we outline how climate change not only presents real, physical threats but also challenges the way we conceive of the broader economic, political and social order. We asked ourselves (and the contributors to this special issue) how we can imagine alternatives to our current path of ever escalating greenhouse gas emissions and economic growth. Through reference to the contributions that make up this special issue, we suggest that critically engaging with the concept of social, economic and political imaginaries can assist in tackling the conceptual and organizational challenges climate change poses. Only by questioning current sanitised and market-oriented interpretations of the environment, and embracing the catharsis and loss that climate change will bring, can we open up space for new future imaginings

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

Neurotoxic Peptides from the Venom of the Giant Australian Stinging Tree

Stinging Trees from Australasia produce remarkably persistent and painful stings upon contact of their stiff epidermal hairs, called trichomes, with mammalian skin. Dendrocnide-induced acute pain typically lasts for several hours and intermittent painful flares can persist for days and weeks. Pharmacological activity has been attributed to small molecule neurotransmitters and inflammatory mediators, but these compounds alone cannot explain the observed sensory effects. We show here that the venoms of Australian Dendrocnide species contain heretofore unknown pain-inducing peptides that potently activate mouse sensory neurons and delay inactivation of voltage-gated sodium channels. These neurotoxins localize specifically to the stinging hairs and are miniproteins of 4 kDa whose 3D structure is stabilized in an inhibitory cystine knot motif, a characteristic shared with neurotoxins found in spider and cone snail venoms. Our results provide an intriguing example of inter-kingdom convergent evolution of animal and plant venoms with shared modes of delivery, molecular structure and pharmacology