Localization and blow-up of thermal waves in nonlinear heat conduction with peaking

The authors consider the initial-boundary value problem for the porous medium equation ut =(um)xx in (0,∞)×(0,T), where m>1, 00}as t↑T under the hypothesis that ψ(t)↑∞ as t↑T is investigated. The effect of localization of the blowing-up boundary function when lim sup t↑T ζ(t)<∞ is investigated. It is established that localization occurs if and only if lim sup t↑T (∫ t 0 ψ m (s)ds)/ψ(t)<∞, and some estimates concerning the asymptotic behaviour of the solution near the singular point t=T and in the blow-up set Ω={x≥0: lim sup t↑T u(x,t)=∞} are given. Various estimates from above and below on the length ω=supΩ of the blow-up set are obtained. These theorems make more precise some previous results concerning the localization of the boundary blowing-up function which were given in the book by A. A. Samarskiĭ, the reviewer et al. [Peaking modes in problems for quasilinear parabolic equations(Russian), "Nauka'', Moscow, 1987]. Proofs of the theorems are based on comparison with some explicit solutions and on construction of different kinds of weak sub- and supersolutions. The authors use some special integral identities and estimates of the solution and its derivatives by means of the maximum principle. A special comparison theorem above blow-up sets for different boundary functions is proved

Travelling waves in nonlinear diffusion-convection-reaction

The study of travelling waves or fronts has become an essential part of the mathematical analysis of nonlinear diffusion-convection-reaction processes. Whether or not a nonlinear second-order scalar reaction-convection-diffusion equation admits a travelling-wave solution can be determined by the study of a singular nonlinear integral equation. This article is devoted to demonstrating how this correspondence unifies and generalizes previous results on the occurrence of travelling-wave solutions of such partial differential equations. The detailed comparison with earlier results simultaneously provides a survey of the topic. It covers travelling-wave solutions of generalizations of the Fisher, Newell-Whitehead, Zeldovich, KPP and Nagumo equations, the Burgers and nonlinear Fokker-Planck equations, and extensions of the porous media equation. \u

BALANCING ECONOMIC CONSIDERATIONS IN SUSTAINABILITY OF AGRICULTURE

Agricultural and Food Policy,

On the localized wave patterns supported by convection-reaction-diffusion equation

A set of traveling wave solution to convection-reaction-diffusion equation is studied by means of methods of local nonlinear analysis and numerical simulation. It is shown the existence of compactly supported solutions as well as solitary waves within this family for wide range of parameter values

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

A singular nonlinear Volterra integral equation

This paper concerns the integral equationx(t) = f(t) + t0 g(s)/x(s) dsin which the functions and variables are real-valued and x is the unknown. The interest is in nonnegative continuous solutions of this equation for t ≥ 0 when f ∈ C([0,∞)), f(0) ≥ 0 and g ∈ L1(0, τ) for all τ ∈ (0,∞). Of particular interest is the singular case f(0) = 0. This equation arises in the study of travelling waves in nonlinear reaction-convection-diffusion processes. It is shown that the integral equation has none, one or an uncountable number of solutions. Subsequently, it is shown that, even if there is an infinite number of solutions, there is one which is maximal. Moreover, a method for constructing this particular solution is provided. This permits the establishment of necessary and sufficient conditions for the existence of a solution. Comparison principles for solutions of the equation with different sets of coefficients are then presented. Rather detailed analyses follow for the case that f(0) = 0 and g(s) ≤ 0 for almost all s in a right neighborhood of zero and for the case that f(0) = 0 and the inequality for g is reversed. These analyses demonstrate that the equation may indeed have none, one or an uncountable number of solutions, among other phenomena

Inducible chromatin priming is associated with the establishment of immunological memory in T cells

Immunological memory is a defining feature of vertebrate physiology, allowing rapid responses to repeat infections. However, the molecular mechanisms required for its establishment and maintenance remain poorly understood. Here, we demonstrated that the first steps in the acquisition of T-cell memory occurred during the initial activation phase of naïve T cells by an antigenic stimulus. This event initiated extensive chromatin remodeling that reprogrammed immune response genes toward a stably maintained primed state, prior to terminal differentiation. Activation induced the transcription factors NFAT and AP-1 which created thousands of new DNase I-hypersensitive sites (DHSs), enabling ETS-1 and RUNX1 recruitment to previously inaccessible sites. Significantly, these DHSs remained stable long after activation ceased, were preserved following replication, and were maintained in memory-phenotype cells. We show that primed DHSs maintain regions of active chromatin in the vicinity of inducible genes and enhancers that regulate immune responses. We suggest that this priming mechanism may contribute to immunological memory in T cells by facilitating the induction of nearby inducible regulatory elements in previously activated T cells