In this work we study global well-posedness and large time behaviour for a typical reaction--diffusion system, which include degenerate diffusion, and whose non-linearities arise from chemical reactions. We show that there is an {\it indirect diffusion effect}, i.e. an effective diffusion for the non-diffusive species which is incurred by a combination of diffusion from diffusive species and reversible reactions between the species. Deriving new estimates for such degenerate reaction-diffusion system, we show, by applying the entropy method, that the solution converges exponentially to equilibrium, and provide explicit convergence rates and associated constants.
Global existence and boundedness for a degenerate chemotaxis system with indirect signal production via minimizing movement schemes
Osaka Metropolitan University（大阪公立大学）；Institut de Mathématiques de Toulouse (IMT) UMR5219, Université Toulouse III - Paul Sabatier（图卢兹第三大学-保罗·萨巴蒂尔大学）；Université Savoie Mont Blanc, CNRS（萨瓦-蒙布朗大学）
We consider a system of two reaction-diffusion equations coming out of reversible chemistry. When the reaction happens on the totality of the domain, it is known that exponential convergence to equilibrium holds. We show in this paper that this exponential convergence also holds when the reaction holds only on a given open set of a ball, thanks to an observation estimate deduced by logarithmic convexity.
We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or twice; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviours at the points where the diffusivity …
This paper establishes the spectral stability in exponentially weighted spaces of smooth traveling monotone fronts for reaction diffusion equations of Fisher-KPP type with nonlinear degenerate diffusion coefficient. It is assumed that the former is degenerate, that is, it vanishes at zero, which is one of the equilibrium points of the reaction. A parabolic regularization technique is introduced in order to locate a subset of the compression spectrum of the linearized operator around the wave, whereas the point spectrum is proved to be stable with the use of energy estimates. Detailed asymptoti…
In this paper we are interested in a degenerate parabolic system of reaction-diffusion equations arising in biology when studying cell adhesion at the protein level. In this modeling the unknown is the couple of the distribution laws of the freely diffusing proteins and of the fixed ones. Under sufficient conditions on the aggregation and unbinding probabilities, we prove the existence of solution of the considered system, as well as their positivity, boundedness and uniqueness. Moreover, we discuss the stability of the equilibrium solution. Finally, we show that the simpler and particular cho…
We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and several properties of traveling-wave solutions to such an equation. In particular, we provide a sharp estimate for the minimal speed of the profiles and improve previous results about the regularity of wavefronts. Moreover, we show the existence of an infinite number of semi-wavefronts with the same speed.
This paper presents a full classification of the short-time behavior of the interfaces in the Cauchy problem for the nonlinear second order degenerate parabolic PDE \[ u_t-Δu^m +b u^β=0, \ x\in \mathbb{R}^N, 0<t<T \] with nonnegative initial function $u_0$ such that \[ supp~u_0 = \{|x|<R\}, \ u_0 \sim C(R-|x|)^α, \quad{as} \ |x|\to R-0, \] where $m>1, C,α, β>0, b \in \mathbb{R}$. Interface surface $t=η(x)$ may shrink, expand or remain stationary depending on the relative strength of the diffusion and reaction terms near the boundary of support, expressed in terms of the parameters $m,β, α, sig…
A degenerate reaction-diffusion SIR model in interconnected regions
Laboratoire MISI, FST Settat, Univ. Hassan I（米西实验室，塞塔特高等师范学院，哈桑一世大学）；MAMCS Group, FST Errachidia, Moulay Ismail University of Meknes（MAMCS小组，埃拉迪希亚高等师范学院，穆莱·伊斯梅尔梅克内斯大学）；Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University（数学系，达尔马赫拉斯理学院，西迪穆罕默德·本阿卜杜拉大学）
We derive a precise motion law for fronts of solutions to scalar one-dimensional reaction-diffusion equations with equal depth multiple-wells, in the case the second derivative of the potential vanishes at its minimizers. We show that, renormalizing time in an algebraic way, the motion of fronts is governed by a simple system of ordinary differential equations of nearest neighbor interaction type. These interactions may be either attractive or repulsive. Our results are not constrained by the possible occurrence of collisions nor splittings. They present substantial differences with the result…
We consider the singular perturbation problem $$ Δu_ε=β_ε(u_ε), $$ where $β_ε(s)=\frac{1}εβ(\frac{s}ε)$, $β$ is a Lipschitz continuous function such that $β>0$ in $(0, 1)$, $β\equiv 0$ outside $(0, 1)$ and $\int_0^1β(s) ds={1/2}$. We construct an example exhibiting a {\em degenerate singularity} as $ε_k\searrow 0$. More precisely, there is a sequence of solutions $u_{ε_k}\to u$ as $k\to \infty$, and there exists $x^0\in\partial\{u>0\}$ such that $$ \frac{u(x^0+r\cdot)}{r} \to 0 \textrm{as} r\to 0.$$ Known results suggest that this singularity must be {\em unstable}, which makes it hard to capt…
In this paper, we investigate the expanding patterns and spreading speed of solutions of farmer and hunter-gatherer model which is a three-component degenerate reaction-diffusion system. Ecologically speaking, since the lifestyle of agriculture and settlement allows for a larger population, after an initially localized population of farmers migrated into a region occupied by hunter-gatherers, the Neolithic transition from hunter-gatherers to farmers happened. This model was proposed by J. Elias, K. Humayun and M. Mimura in 2018. By numerical simulations, a travelling wave solution with a certa…
We consider a degenerate partial differential equation arising in population dynamics, namely the porous medium equation with a bistable reaction term. We study its asymptotic behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We prove the rapid formation of transition layers which then propagate. We prove the convergence to a sharp interface limit whose normal velocity, at each point, is that of the underlying degenerate travelling wave.
We show interior Hölder continuity for a class of quasi-linear degenerate reaction-diffusion equations. The diffusion coefficient in the equation has a porous medium type degeneracy and its primitive has a singularity. The reaction term is locally bounded except in zero. The class of equations we analyse is motivated by a model that describes the growth of biofilms. Our method is based on the original proof of interior Hölder continuity for the porous medium equation. We do not restrict ourselves to solutions that are limits in the weak topology of a sequence of approximate continuous solution…
This paper is devoted to the analysis of non-negative solutions for a generalisation of the parabolic equation with porous medium like nonlinear diffusion and nonlinear nonlocal reaction. We investigate under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and aggregation modelled by nonlinear reaction, occurs. Precisely, we exhibit that the qualitative behavior of solutions is decided by the nonlinear diffusion which is chosen in such a way that its scaling and the reaction term coincide, i.e. that there is a critical exponent $m+2/n$ fo…
Propagation-Window Bounds in Degenerate Plant-Consumer Reaction-Diffusion Systems
National Taiwan University（台湾大学）；National Center for Theoretical Science（国家理论科学中心）；International School for Advanced Studies (SISSA)（高级研究国际学院）；Soochow University（东吴大学）；University of Illinois Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校）
Let us consider a nonlinear degenerate reaction-diffusion equation with application to climate science. After proving that the solution remains nonnegative at any time, when the initial state is nonnegative, we prove the approximate controllability between nonnegative states at any time via multiplicative controls, that is, using as control the reaction coefficient.
We study a Dirichlet problem driven by the degenerate fractional p-Laplacian and involving a nonlinear reaction, which depends on a positive parameter. The reaction is assumed to be (p-1)-sublinear near the origin and (p-1)-superlinear at infinity (including the concave-convex case). Following a variational approach based on a combination of critical point theory and suitable truncation techniques, we prove a bifurcation-type result for the existence of positive solutions.
We consider a one-dimensional reaction-diffusion equation of Fisher-Kolmogoroff-Petrovsky-Piscounoff type. We investigate the effect of the interaction between the nonlinear diffusion coefficient and the reaction term on the existence and nonexistence of travelling waves. Our diffusion coefficient is allowed to be degenerate or singular at both equilibrium points, 0 and 1, while the reaction term need not be differentiable. These facts influence the existence and qualitative properties of travelling waves in a substantial way.
We solve and characterize the Lagrange multipliers of a reaction-diffusion system in the Gibbs simplex of R^{N+1} by considering strong solutions of a system of parabolic variational inequalities in R^N. Exploring properties of the two obstacles evolution problem, we obtain and approximate a N-system involving the characteristic functions of the saturated and/or degenerated phases in the nonlinear reaction terms. We also show continuous dependence results and we establish sufficient conditions of non-degeneracy for the stability of those phase subregions.
In this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric framework. We use a constructive method, alternating minimizing movements for the Wasserstein distance and for the Fisher-Rao distance, and prove existence of weak solutions for general scalar reaction-diffusion-advection equations. We extend the approach to systems of multiple interacting species, and also consider an application to a very degenerate diffusion problem involving a Gamma-limit. Moreover, some numerical simulations are included.
We focus on the (sharp) threshold phenomena arising in some reaction-diffusion equations supplemented with some compactly supported initial data. In the so-called ignition and bistable cases, we prove the first sharp quantitative estimate on the (sharp) threshold values. Furthermore, numerical explorations allow to conjecture some refined estimates. Last we provide related results in the case of a degenerate monostable nonlinearity "not enjoying the hair trigger effect". AMS Subject Classifications: 35K57 (Reaction-diffusion equations), 35K15 (Initial value problems for second-order parabolic …
A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation
EDF R&D（法国电力集团研发部）；Laboratoire de Mathématiques, Université de Paris-Sud 11（巴黎第十一大学数学实验室）；CNRS and Laboratoire de Mathématiques, Université de Paris-Sud 11（法国国家科学研究中心及巴黎第十一大学数学实验室）
In this paper we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy problems. In particular, we consider a one-dimensional semilinear degenerate reaction-diffusion equation in divergence form governed via the coefficient of the \-reaction term (bilinear or multiplicative control). The above one-dimensional equation is degenerate since the diffusion coefficient is positive on the interior of the spatial domain and vanishes at the boundary points. Furthermore, two different kinds of degenerate diffusion coefficient are distinguished and studied …
We study reaction-diffusion equations in one spatial dimension and with general (space- or time-) inhomogeneous mixed bistable-ignition reactions. For those satisfying a simple quantitative hypothesis, we prove existence and uniqueness of transition fronts, as well as convergence of "typical" solutions to the unique transition front (the existence part even extends to mixed bistable-ignition-monostable reactions). These results also hold for all pure ignition reactions without any other hypotheses, but not for all pure bistable reactions. In fact, we find examples of either spatially or tempor…
In this paper we obtain well-posedness for a class of semilinear weakly degenerate reaction-diffusion systems with Robin boundary conditions. This result is obtained through a Gagliardo-Nirenberg interpolation inequality and some embedding results for weighted Sobolev spaces.
We study the boundedness and convergence to equilibrium of weak solutions to reaction-diffusion systems with nonlinear diffusion. The nonlinear diffusion is of porous medium type and the nonlinear reaction terms are assumed to grow polynomially and to dissipate (or conserve) the total mass. By utilising duality estimates, the dissipation of the total mass and the smoothing effect of the porous medium equation, we prove that if the exponents of the nonlinear diffusion terms are high enough, then weak solutions are bounded, locally Hölder continuous and their $L^{\infty}(Ω)$-norm grows in time a…
We analyze families of primal high-order hybridizable discontinuous Galerkin (HDG) methods for solving degenerate (second-order) elliptic problems. One major trouble regarding this class of PDEs concerns its mathematical nature, which may be nonuniform over the domain. Due to the local degeneracy of the diffusion term, it can be purely hyperbolic in a subregion and elliptic in the rest. This problem is thus quite delicate to solve since the exact solution is discontinuous at interfaces separating both elliptic and hyperbolic parts. The proposed HDG method is developed in a unified and compact …
We study a non-local variant of a diffuse interface model proposed by Hawkins--Darrud et al. (2012) for tumour growth in the presence of a chemical species acting as nutrient. The system consists of a Cahn--Hilliard equation coupled to a reaction-diffusion equation. For non-degenerate mobilities and smooth potentials, we derive well-posedness results, which are the non-local analogue of those obtained in Frigeri et al. (European J. Appl. Math. 2015). Furthermore, we establish existence of weak solutions for the case of degenerate mobilities and singular potentials, which serves to confine the …
Quantitative estimates for the top Lyapunov exponents for systems of stochastic reaction-diffusion equations are proven. The treatment includes reaction potentials with degenerate minima. The proof relies on an asymptotic expansion of the invariant measure, with careful control on the resulting error terms. As a consequence of these estimates, synchronisation by noise is deduced for systems of stochastic reaction-diffusion equations for the first time.
In this work we investigate a coupled system of degenerate and nonlinear partial differential equations governing the transport of reactive solutes in groundwater. We show that the system admits a unique weak solution provided the nonlinear adsorption isotherm associated with the reaction process satisfies certain physically reasonable structural conditions. We conclude, moreover, that the solute concentrations stay non-negative if the source term is componentwise non-negative and investigate numerically the finite speed of propagation of compactly supported initial concentrations, in a two-co…
Convergence of a discrete-in-time Approximation to a Degenerate Parabolic-Hyperbolic System
Joanneum Research Forschungsgesellschaft mbH（约阿内姆研究有限公司）；Faculty of Advanced Science and Technology, Ryukoku University（龙谷大学先进科学技术学部）；Institute of Scientific Computing, Faculty of Mathematics, TUD Dresden University of Technology（德累斯顿工业大学数学学院科学计算研究所）
