Abstract This paper is devoted to the study of pulsating travelling fronts for reaction‐diffusion‐advection equations in a general class periodic domains with underlying diffusion and velocity fields. Such move some arbitrarily given direction an unknown effective speed. The notion generalizes that planar or shear flows. Various existence, uniqueness monotonicity results are proved two classes reaction terms. Firstly, combustion‐type nonlinearity, it front exists its speed unique. Moreover,...

Abstract In this paper, we generalize the usual notions of waves, fronts, and propagation speeds in a very general setting. These new notions, which cover all situations, involve uniform limits, with respect to geodesic distance, family hypersurfaces that are parametrized by time. We prove existence such waves for some time‐dependent reaction‐diffusion equations, as well intrinsic properties, monotonicity uniqueness results almost‐planar fronts. The classification results, obtained under...

This paper is devoted to nonlinear propagation phenomena in general unbounded domains of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript N"><mml:semantics><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:annotation encoding="application/x-tex">\mathbb...

Most mathematical studies on expanding populations have focused the rate of range expansion a population. However, genetic consequences population remain an understudied body theory. Describing as traveling wave solution derived from classical reaction-diffusion model, we analyze spatio-temporal evolution its structure. We show that presence Allee effect (i.e., lower per capita growth at low densities) drastically modifies diversity, both in colonization front and behind it. With pushed...

Abstract We consider traveling waves for a nonlinear diffusion equation with bistable or multistable nonlinearity. The goal is to study how planar front interacts compact obstacle that placed in the middle of space ℝ N . As first step, we prove existence and uniqueness an entire solution behaves like wave approaching from infinity eventually reaching obstacle. This causes disturbance on shape front, but show will gradually recover its profile continue propagate same direction, leaving...

In this paper, we explain in simple PDE terms a famous result of Bramson about the logarithmic delay position solutions $u(t,x)$ Fisher-KPP reaction-diffusion equations $\mathbb{R}$, with respect to travelling front minimal speed. Our proof is based on comparison $u$ linearized Dirichlet boundary conditions at front, and without delay. analysis also yields large-time convergence along their level sets profile front.

This paper is about one-dimensional symmetry properties for some entire and bounded solutions of ∆u + f(u) = 0 in IR. We consider u such that −1 < 1 u(x1, · , xn) → ±1 as xn ±∞, uniformly with respect to x1, xn−1. Under conditions on f we prove the only depend variable xn. also discuss more general elliptic operators. The qualitative then strongly coefficients operator. These results extend higher dimensions operators a result Ghoussoub Gui [21] proved n ≤ 3. AMS Classification : 35B05,...

We consider the Fisher–KPP (for Kolmogorov–Petrovsky–Piskunov) equation with a nonlocal interaction term. establish condition on that allows for existence of non-constant periodic solutions, and prove uniform upper bounds solutions Cauchy problem, as well lower spreading rate compactly supported initial data.

This paper deals with the solutions defined for all time of KPP equation ut = uxx + f(u), 0 < u(x,t) 1, (x,t) ∈ ℝ2, where ƒ is a KPP-type nonlinearity in [0,1]: ƒ(0) ƒ(1) 0, ƒ′(0) > ƒ′(1) (0,1), and ƒ′(s) ≤ [0,1]. admits infinitely many traveling-wave-type solutions, increasing or decreasing x. It also that depend only on t. In this paper, we build four other manifolds solutions: One 5-dimensional, one 4-dimensional, two are 3-dimensional. Some these new obtained by considering traveling...

We prove the uniqueness, up to shifts, of pulsating traveling fronts for reaction-diffusion equations in periodic media with Kolmogorov–Petrovskiĭ–Piskunov type nonlinearities. These results provide particular a complete classification all KPP fronts. Furthermore, more general case monostable nonlinearities, we also derive several global stability properties and convergence solutions Cauchy problem front-like initial data. In particular, minimal speed, which is new result even when medium...

This paper is concerned with a lattice dynamical system modeling the evolution of susceptible and infective individuals at discrete niches. We prove existence traveling waves connecting disease-free state to non-trivial leftover concentrations. also characterize minimal speed we non-existence smaller speeds.

In this paper we study solutions to reaction-diffusion equations in the bistable case, defined on whole space dimension $N$. The existence of with cylindric symmetry is already known. Here prove uniqueness these whose level sets are curved Lipschitz graphs. Using a centre manifold-like argument, also give precise asymptotics at infinity. 2, classify all under weak conditions Finally, provide an alternative proof based continuation argument.

come from teaching and research institutions in France or abroad, public private centers.L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques niveau recherche, publiés ou non, émanant des établissements d'enseignement recherche français étrangers, laboratoires publics privés. Bistable pulsating fronts for reaction-diffusion equations a periodic habitat

Let u be a solution of the Fisher–KPP equation \partial_{t} u=\Delta u+f(u) , t&gt;0 x\in\mathbb{R}^{N} with an initial datum u_{0} . We address following question: does become locally planar as t\to+\infty ? Namely, u(t_{n},x_{n}+\cdot) converge uniformly, up to subsequences, towards one-dimensional function, for any sequence ((t_{n},x_{n}))_{n\in\mathbb{N}} in (0,+\infty)\times\mathbb{R}^{N} such that t_{n}\to+\infty n\to+\infty This question is spirit celebrated De Giorgi’s conjecture...

This work deals with the existence of solutions a reaction-diffusion equation in plane $\mathbb{R}^2$. The problem, whose unknowns are real c and function u, is following: $$\left\{ \begin{array}{rl} \Delta u-c\displaystyle{{\partial u}\over{\partial y}}+f(u)=0\ & \hbox{ }\rr^2,\\ \begin{array}{ll}\forall \vec k \in {\cal C} (-\vec{e}_2,\alpha), u(\lambda k) \displaystyle{\mathop{\longrightarrow}_{\lambda \to +\infty}} 0,\\ \forall (\vec{e}_2,\pi-\alpha), 1, \end{array}...

Let Ω be a bounded C 2 domain in R n , where is any positive integer, and let * the Euclidean ball centered at 0 having same Lebesgue measure as Ω.Consider operator L = -div(A∇) + v • ∇ V on with Dirichlet boundary condition, symmetric matrix field A W 1,∞ (Ω), vector ∞ (Ω, ) continuous function Ω.We prove that minimizing principal eigenvalue of when fixed A, vary under some constraints operators smooth radially coefficients.The which are satisfied by original coefficients new ones expressed...