We complete the description, initiated in [6], of a free boundary travelling at constant speed in a half plane, where the propagation is controlled by a line having a large diffusion on its own. The main result of this work is that the free boundary is asymptotic to a line at infinity, whose angle to the horizontal is dicatated by the velocity of the wave imposed by the line. This helps understanding some rather counter-intuitive numerical simulations of [8].
Optimal extinction rates near the extinction time are derived for non-negative solutions to a fast diffusion equation with strong absorption, the power of the absorption exceeding that of the diffusion.
In this paper, we present an approach to characterising fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form uxx in usual reaction-diffusion equation, which represent linear diffusion, by terms of form phi(u)xx, representing nonlinear diffusion. We prove the convergence as k tends to infinity to a limit that is determined by the unique solution of a certain scalar nonlinear diffusion limit problem.
We study extinction profiles of solutions to fast diffusion equations with some initial data in the Marcinkiewicz space. The extinction profiles will be the singular solutions of their stationary equations.
In this paper, we present a surprising two-dimensional contraction family for porous medium and fast diffusion equations. This approach provides new a priori estimates on the solutions, even for the standard heat equation.
The system under study is a reaction-diffusion equation in a horizontal strip, coupled to a diffusion equation on its upper boundary via an exchange condition of the Robin type. This class of models was introduced by H. Berestycki, L. Rossi and the second author in order to model biological invasions directed by lines of fast diffusion. They proved, in particular, that the speed of invasion was enhanced by a fast diffusion on the line, the spreading velocity being asymptotically proportional to the square root of the fast diffusion coefficient. These results could be reduced, in the logistic c…
We prove existence and uniqueness of travelling waves for a reaction-diffusion system coupling a classical reaction-diffusion equation in a strip with a diffusion equation on a line. To do this we use a continuation method which leads to further insight into the system. In particular, the transition occurs through a singular perturbation which seems new in this context, connecting the system with a Wentzell type boundary value problem.
We give a simple proof of the $λ= d-2$ cases of the sharp Hardy-Littlewood-Sobolev inequality for $d\geq 3$, and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for $d=2$ via a monotone flow governed by the fast diffusion equation.
We establish a new property of Fisher-KPP type propagation in a plane, in the presence of a line with fast diffusion. We prove that the line enhances the asymptotic speed of propagation in a cone of directions. Past the critical angle given by this cone, the asymptotic speed of propagation coincides with the classical Fisher-KPP invasion speed. Several qualitative properties are further derived, such as the limiting behaviour when the diffusion on the line goes to infinity.
We investigate fast diffusions on finite directed graphs. We prove results in a way dual to presented in Bobrowski, A. Ann. Henri Poincaré (2012) 13(6): 1501-1510 and Bobrowski, A., Morawska, K. DCDS-B (2012), 17(7): 2313-2327, and obtain asymptotic behaviour of a diffusion semigroup on a graph in $ L^1 $ and $ L^2 $ as the diffusions' speed increases and the probability of a particle passing through a vertex decreases.
We focus on the spreading properties of solutions of monostable equations with fast diffusion. The nonlinear reaction term involves a weak Allee effect, which tends to slow down the propagation. We complete the picture of [3] by studying the subtle case where acceleration does occur and is induced by a combination of fast diffusion and of reactive growth. This requires the construction of new elaborate sub and supersolutions thanks to some underlying self-similar solutions.
We prove global Hölder gradient estimates for bounded positive weak solutions of fast diffusion equations in smooth bounded domains with the homogeneous Dirichlet boundary condition, which then lead us to establish their optimal global regularity. This solves a problem raised by Berryman and Holland in 1980.
We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on the spatial decay rates of initial data.
We study a Sobolev critical fast diffusion equation in bounded domains with the Brezis-Nirenberg effect. We obtain extinction profiles of its positive solutions, and show that the convergence rates of the relative error in regular norms are at least polynomial. Exponential decay rates are proved for generic domains. Our proof makes use of its regularity estimates, a curvature type evolution equation, as well as blow up analysis. Results for Sobolev subcritical fast diffusion equations are also obtained.
In this paper we consider an extension of the results in shape differentiation of semilinear equations with smooth nonlinearity presented in J.I. Díaz and D. Gómez-Castro: An Application of Shape Differentiation to the Effectiveness of a Steady State Reaction-Diffusion Problem Arising in Chemical Engineering. Electron. J. Differ. Equations in 2015 to the case in which the nonlinearities might be less smooth. Namely we will show that Gateaux shape derivatives exists when the nonlinearity is only Lipschitz continuous, and we will give a definition of the derivative when the nonlinearity has a bl…
We prove local higher integrability of the spatial gradient for solutions to obstacle problems of porous medium type in the fast diffusion case $m<1$. The result holds for the natural range of exponents that is known from other regularity results for porous medium type equations. We also cover the case of signed solutions.
We consider the dynamics of the Total Variation Flow (TVF) $u_t=÷(Du/|Du|)$ and of the Sign Fast Diffusion Equation (SFDE) $u_t=Δ\sign(u)$ in one spatial dimension. We find the explicit dynamic and sharp asymptotic behaviour for the TVF, and we deduce the one for the SFDE by an explicit correspondence between the two equations.
We consider a class of reaction-diffusion equations with a stochastic perturbation on the boundary. We show that in the limit of fast diffusion, one can rigorously approximate solutions of the system of PDEs with stochastic Neumann boundary conditions by the solution of a suitable stochastic/deterministic differential equation for the average concentration that involves reactions only. An interesting effect occurs, if the noise on the boundary does not change the averaging concentration, but is sufficiently large. Then surprising additional effective reaction terms appear. We focus on systems …
We study positive solutions of the super-fast diffusion equation in the whole space with initial data which are unbounded as $|x|\to\infty$. We find an explicit dependence of the slow temporal growth rate of solutions on the initial spatial growth rate. A new class of self-similar solutions plays a significant role in our analysis.
We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a subcritical exponent. We show that separable solutions are stable in some suitable sense by finding a class of functions which belong to their domain of attraction. For solutions in this class we establish optimal rates of convergence to separable solutions.
We investigate qualitative properties of local solutions $u(t,x)\ge 0$ to the fast diffusion equation, $\partial_t u =Δ(u^m)/m$ with $m<1$, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form $[0,T]\times\RR^d$. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low $m$ in the so-called very fast diffusion range, precisely for all $m\le m_c=(d-2)/d.$ The boundedness statements are true even for $m\le 0…
In this paper we study the asymptotic behavior of a very fast diffusion PDE in 1D with periodic boundary conditions. This equation is motivated by the gradient flow approach to the problem of quantization of measures introduced in \cite{CGI1}. We prove exponential convergence to equilibrium under minimal assumptions on the data, and we also provide sufficient conditions for $W_2$-stability of solutions.
The goal of this note is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy-Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities and rates for nonlinear diffusion equations.
In this paper we consider a reaction-diffusion equation of Fisher-KPP type inside an infinite cylindrical domain in $\mathbb{R}^{N+1}$, coupled with a reaction-diffusion equation on the boundary of the domain, where potentially fast diffusion is allowed. We will study the existence of an asymptotic speed of propagation for solutions of the Cauchy problem associated with such system, as well as the dependence of this speed on the diffusivity at the boundary and the amplitude of the cylinder. When $N=1$ the domain reduces to a strip between two straight lines. This models the effect of two roads…
This paper discusses finite time extinction for a perturbed fast diffusion equation with dynamic boundary conditions. The fast diffusion equation has the characteristic property of decay, such as the solution decays to zero in a finite amount of time depending upon the initial data. In the target problem, some $p$-th or $q$-th order perturbation term may work to blow up within this period. The problem arises from the conflict between the diffusion and the blow up, in the bulk and on the boundary. Firstly, the local existence and uniqueness of the solution are obtained. Finally, a result of fin…
Let $n\ge 3$ and $0<m<\frac{n-2}{n}$. We will extend the results of J.L. Vazquez and M. Winkler and prove the uniqueness of finite points blow-up solutions of the fast diffusion equation $u_t=Δu^m$ in both bounded domains and $\mathbb{R}^n\times (0,\infty)$. We will also construct initial data such that the corresponding solution of the fast diffusion equation in bounded domain oscillate between infinity and some positive constant as $t\to\infty$.
The rigorous asymptotics from reaction-cross-diffusion systems for three species with known entropy to cross-diffusion systems for two variables is investigated. The equations are studied in a bounded domain with no-flux boundary conditions. The global existence of very weak (integrable) solutions and the rigorous fast-reaction limit are proved. The limiting system inherits the entropy structure with an entropy that is not the sum of the entropies of the components. Uniform estimates are derived from the entropy inequality and the duality method.
We analyze long-time behavior of solutions to a class of problems related to very fast and singular diffusion porous medium equations having nonhomogeneous in space and time source terms with zero mean. In dimensions two and three, we determine critical values of porous medium exponent for the asymptotic $H^1$-convergence of the solutions to a unique nonhomogeneous positive steady state generally to hold.
We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a critical exponent. After a suitable rescaling which yields a non--linear Fokker--Planck equation, we find a continuum of algebraic rates of convergence to a self--similar profile. These rates depend explicitly on the spatial decay rates of initial data. This improves a previous result on slow convergence for the critical fast diffusion equation ({\sc Bonforte et al}. in Arch Rat Mech Anal 196:631--680, 2010) and provides answers to some open problems.
In the euclidean space, Sobolev and Hardy-Littlewood-Sobolev inequalities can be related by duality. In this paper, we investigate how to relate these inequalities using the flow of a fast diffusion equation in dimension $d\ge3$. The main consequence is an improvement of Sobolev's inequality when $d\ge5$, which involves the various terms of the dual Hardy-Littlewood-Sobolev inequality. In dimension $d=2$, Onofri's inequality plays the role of Sobolev's inequality and can also be related to its dual inequality, the logarithmic Hardy-Littlewood-Sobolev inequality, by a super-fast diffusion equat…
We propose here a new model to describe biological invasions in the plane when a strong diffusion takes place on a line. We establish the main properties of the system, and also derive the asymptotic speed of spreading in the direction of the line. For low diffusion, the line has no effect, whereas, past a threshold, the line enhances global diffusion in the plane and the propagation is directed by diffusion on the line. It is shown here that the global asymptotic speed of spreading in the plane, in the direction of the line, grows as the square root of the diffusion on the line. The model is …
We construct solutions of the fast diffusion equation, which exist for all $t\in\mathbb{R}$ and are singular on the set $Γ(t):= \{ ξ(s) ; -\infty <s \leq ct \}$, $c>0$, where $ξ\in C^3(\mathbb{R};\mathbb{R}^n)$, $n\geq 2$. We also give a precise description of the behavior of the solutions near $Γ(t)$.
A new model to describe biological invasion influenced by a line with fast diffusion has been introduced by H. Berestycki, J.-M. Roquejoffre and L. Rossi in 2012.The purpose of this article is to present a related model where the line of fast diffusion has a nontrivial range of influence, i.e. the exchanges between the line and the surrounding space has a nontrivial support. We show the existence of a spreading velocity depending on the diffusion on the line. Two intermediate model are also discussed. Then, we try to understand the influence of different exchange terms on this spreading speed.…
Existence and uniqueness of radially symmetric self-similar very singular solutions are proved for the singular diffusion equation with gradient absorption {equation*} \partial_t u -Δ_{p}u+|\nabla u|^q=0, \ \hbox{in} \ (0,\infty)\times\real^N, {equation*} where $2N/(N+1)
Singular limit problems of reaction-diffusion systems have been studied in cases where the effects of the reaction terms are very large compared with those of the other terms. Such problems appear in literature in various fields such as chemistry, ecology, biology, geology and approximation theory. In this paper, we deal with the singular limit of a general reaction-diffusion system including many problems in the literature. We formulate the problem, derive the limit equation and establish a rigorous mathematical theory.
We treat a model of population dynamics in a periodic environment presenting a fast diffusion line. This phenomenon is modelled via a "road-field" system, which is a system of coupled reaction-diffusion equations set in domains of different dimensions. Here, we consider for the first time the case of a reaction term depending on a spatial variable in a periodic fashion, which is of great interest for both its mathematical difficulties and for its applications. We derive necessary and sufficient conditions for the survival of the species in terms of the sign of a suitable generalised principal …
We find a continuum of extinction rates for solutions $u(y,τ)\ge 0$ of the fast diffusion equation $u_τ=Δu^m$ in a subrange of exponents $m\in (0,1)$. The equation is posed in $\ren$ for times up to the extinction time $T>0$. The rates take the form $\|u(\cdot,τ)\|_\infty\sim (T-τ)^θ$ \ for a whole interval of $θ>0$. These extinction rates depend explicitly on the spatial decay rates of initial data.
Reaction-Diffusion System Approximation to the Fast Diffusion Equation
Ryukoku University（龙谷大学）；IRMA, UMR 7501 Université de Strasbourg et CNRS（斯特拉斯堡大学与法国国家科学研究中心联合研究实验室）；Inria, IRMA, Université de Strasbourg, CNRS UMR 7501（法国国家信息与自动化研究所、斯特拉斯堡大学与法国国家科学研究中心联合研究实验室）
We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity.
We study the time asymptotic propagation of solutions to the reaction-diffusion cooperative systems with fractional diffusion. We prove that the propagation speed is exponential in time, and we find the precise exponent of propagation. This exponent depends on the smallest index of the fractional laplacians and on the principal eigenvalue of the matrix DF(0) where F is the reaction term. We also note that this speed does not depend on the space direction.
We study the velocity of travelling waves of a reaction-diffusion system coupling a standard reaction-diffusion equation in a strip with a one-dimensional diffusion equation on a line. We show that it grows like the square root of the diffusivity on the line. This generalises a result of Berestycki, Roquejoffre and Rossi in the context of Fisher-KPP propagation where the question could be reduced to algebraic computations. Thus, our work shows that this phenomenon is a robust one. The ratio between the asymptotic velocity and the square root of the diffusivity on the line is characterised as t…
We show that partial mass concentration can happen for stationary solutions of aggregation-diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions $N\geq6$. We then provide numerical computations which suggest the occurrence of ma…
A non self-similar change of coordinates provides improved matching asymptotics of the solutions of the fast diffusion equation for large times, compared to already known results, in the range for which Barenblatt solutions have a finite second moment. The method is based on relative entropy estimates and a time-dependent change of variables which is determined by second moments, and not by the scaling corresponding to the self-similar Barenblatt solutions, as it is usually done.
We will extend a recent result of B.Choi, P.Daskalopoulos and J.King. For any $n\ge 3$, $0<m<\frac{n-2}{n+2}$ and $γ>0$, we will construct subsolutions and supersolutions of the fast diffusion equation $u_t=\frac{n-1}{m}Δu^m$ in $\mathbb{R}^n\times (t_0,T)$, $t_0<T$, which decay at the rate $(T-t)^{\frac{1+γ}{1-m}}$ as $t\nearrow T$. As a consequence we obtain the existence of unique solution of the Cauchy problem $u_t=\frac{n-1}{m}Δu^m$ in $\mathbb{R}^n\times (t_0,T)$, $u(x,t_0)=u_0(x)$ in $\mathbb{R}^n$, which decay at the rate $(T-t)^{\frac{1+γ}{1-m}}$ as $t\nearrow T$ when $u_0$ satisfies …
We present a unified approach to characterising fast-reaction limits of systems of either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains, motivated by models of fast chemical reactions where either one or both reactant(s) is/are mobile. For appropriate initial data, solutions of four classes of problems each converge in the fast-reaction limit $k \to \infty$ to a self-similar limit profile that has one of four forms, depending on how many components diffuse and whether the spatial domain is a half or whole line.…
We consider singular-degenerate, multivalued stochastic fast diffusion equations with multiplicative Lipschitz continuous noise. In particular, this includes the stochastic sign fast diffusion equation arising from the Bak-Tang-Wiesenfeld model for self-organized criticality. A well-posedness framework based on stochastic variational inequalities (SVI) is developed, characterizing solutions to the stochastic sign fast diffusion equation, previously obtained in a limiting sense only. Aside from generalizing the SVI approach to stochastic fast diffusion equations we develop a new proof of well-p…
In this paper, we establish several local and global gradient estimates for the positive solution of Porous Medium Equations (PMEs) and Fast Diffusion Equations (FDEs). Our proof is probabilistic and uses martingale techniques.
We build the iDLA cluster using drifted random walks, and study the limiting shapes they exhibit, with the help of sandpile models. For constant drift, the normalised cluster converges to a canonical shape S, which can be termed a true heat ball, in that it gives rise to a mean value property for caloric functions. The existence and boundedness of such a shape answers a natural yet open question in PDE theory.
This paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results.
In this work we derive local gradient and Laplacian estimates of the Aronson-Bénilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discover some new entropy formulae for these equations.
This paper studies a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that considered a static growth potential, i.e., the initial growth potential is only advected by diffeomorphisms. We focus on the mathematical study of the corresponding system of coupled PDEs describing the joint dynamics of the diffeomorphic transformation together with the growth potential on the moving domain. Specifically, we prove the uniqueness and long time…
