Travelling waves for diffusive and strongly competitive systems: relative motility and invasion speed

Our interest here is to find the invader in a two species, diffusive and competitive Lotka-Volterra system in the particular case of travelling wave solutions. We investigate the role of diffusion in homogeneous domains. We might expect a priori two different cases: strong interspecific competition and weak interspecific competition. In this paper, we study the first one and obtain a clear conclusion: the invading species is, up to a fixed multiplicative constant, the more diffusive one

Generalized transition fronts for one-dimensional almost periodic Fisher-KPP equations

This paper investigates the existence of generalized transition fronts for Fisher-KPP equations in one-dimensional, almost periodic media. Assuming that the linearized elliptic operator near the unstable steady state admits an almost periodic eigenfunction, we show that such fronts exist if and only if their average speed is above an explicit threshold. This hypothesis is satisfied in particular when the reaction term does not depend on x or (in some cases) is small enough. Moreover, except for the threshold case, the fronts we construct and their speeds are almost periodic, in a sense. When our hypothesis is no longer satisfied, such generalized transition fronts still exist for an interval of average speeds, with explicit bounds. Our proof relies on the construction of sub and super solutions based on an accurate analysis of the properties of the generalized principal eigenvalues

On the instability of threshold solutions of reaction-diffusion equations, and applications to optimization problems

The first part of this paper is devoted to the derivation of a technical result, related to the stability of the solution of a reaction-diffusion equation $u_t-\Delta u = f(x,u)$ on $(0,\infty)\times \mathbb{R}^N$, where the initial datum $u(0,x)=u_0(x)$ is such that $\lim_{t\to +\infty} u(t,x)=W(x)$ for all $x$, with $W$ a steady state in $H^1(\mathbb{R}^N)$. We characterize the perturbations $h$ such that, if $u^h$ is the solution associated with the initial datum $u_0+h$, then, if $h$ is small enough in a sense, one has $u^h(t,x)>W(x)$ (resp. $u(t,x)<W(x)$) for $t$ large. This condition depends on the sign of $\int_{\mathbb{R}^N} h(x)p(0,x)dx$, where $p$ is an adjoint solution, which satisfies a backward parabolic equation on $(0,\infty)$ and is uniquely defined [7]. We then provide two applications of our result. We first address an open problem stated in [8] when $N=1$ and $f$ is a bistable nonlinearity independent of $x$. Namely, we compute the derivative of the critical length $L^*(r)$ associated with the initial datum $\mathbf{I}_{(-L-r,-r)\cup (r,L+r)}$, that is the length $L$ above (resp. below) which $u(t,x)$ converges to $1$ (resp. $0$) as $t\to +\infty$. Lastly, again when $N=1$ and $f$ is a bistable nonlinearity independent of $x$, we prove the existence and characterize with a bathtub principle the initial datum $\underline{u}_0$ minimizing some cost function $\int_\mathbb{R} j(u_0)$ and guaranteeing at the same time that $\liminf_{t\to +\infty} u(t,x)>0$ for all $x$

What is the optimal shape of a fin for one dimensional heat conduction?

This article is concerned with the shape of small devices used to control the heat flowing between a solid and a fluid phase, usually called \textsl{fin}. The temperature along a fin in stationary regime is modeled by a one-dimensional Sturm-Liouville equation whose coefficients strongly depend on its geometrical features. We are interested in the following issue: is there any optimal shape maximizing the heat flux at the inlet of the fin? Two relevant constraints are examined, by imposing either its volume or its surface, and analytical nonexistence results are proved for both problems. Furthermore, using specific perturbations, we explicitly compute the optimal values and construct maximizing sequences. We show in particular that the optimal heat flux at the inlet is infinite in the first case and finite in the second one. Finally, we provide several extensions of these results for more general models of heat conduction, as well as several numerical illustrations

Propagation in a kinetic reaction-transport equation: travelling waves and accelerating fronts

In this paper, we study the existence and stability of travelling wave solutions of a kinetic reaction-transport equation. The model describes particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The boundedness of the velocity set appears to be a necessary and sufficient condition for the existence of positive travelling waves. The minimal speed of propagation of waves is obtained from an explicit dispersion relation. We construct the waves using a technique of sub- and supersolutions and prove their \eb{weak} stability in a weighted $L^2$ space. In case of an unbounded velocity set, we prove a superlinear spreading. It appears that the rate of spreading depends on the decay at infinity of the velocity distribution. In the case of a Gaussian distribution, we prove that the front spreads as $t^{3/2}$

Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations

We establish a large deviation theory for a velocity jump process, where new random velocities are picked at a constant rate from a Gaussian distribution. The Kolmogorov forward equation associated with this process is a linear kinetic transport equation in which the BGK operator accounts for the changes in velocity. We analyse its asymptotic limit after a suitable rescaling compatible with the WKB expansion. This yields a new type of Hamilton Jacobi equation which is non local with respect to velocity variable. We introduce a dedicated notion of viscosity solution for the limit problem, and we prove well-posedness in the viscosity sense. The fundamental solution is explicitly computed, yielding quantitative estimates for the large deviations of the underlying velocity-jump process {\em \`a la Freidlin-Wentzell}. As an application of this theory, we conjecture exact rates of acceleration in some nonlinear kinetic reaction-transport equations

Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations

International audienceThis paper is concerned with the study of the large-time behaviour of the solutions u of a class of one-dimensional reaction-diffusion equations with monostable reaction terms f , including in particular the classical Fisher-KPP nonlinearities. The nonnegative initial data u 0 (x) are chiefly assumed to be exponentially bounded as x tends to +∞ and separated away from the unstable steady state 0 as x tends to −∞. On the one hand, we give some conditions on u 0 which guarantee that, for some λ > 0, the quantity c λ = λ+f (0)/λ is the asymptotic spreading speed, in the sense that lim t→+∞ u(t, ct) = 1 (the stable steady state) if c c λ. These conditions are fulfilled in particular when u 0 (x) e λx is asymptotically periodic as x → +∞. On the other hand, we also construct examples where the spreading speed is not uniquely determined. Namely, we show the existence of classes of initial conditions u 0 for which the ω−limit set of u(t, ct + x) as t tends to +∞ is equal to the whole interval [0, 1] for all x ∈ R and for all speeds c belonging to a given interval (γ 1 , γ 2) with large enough γ 1 < γ 2 ≤ +∞