Front propagation directed by a line of fast diffusion: large diffusion and large time asymptotics

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 case, to explicit algebraic computations. The goal of this paper is to prove that the same phenomenon holds, with a different type of nonlinearity, which precludes explicit computations. We discover a new transition phenomenon, that we explain in detail

Ergodic type problems and large time behaviour of unbounded solutions of Hamilton-Jacobi Equations

We study the large time behavior of Lipschitz continuous, possibly unbounded, viscosity solutions of Hamilton-Jacobi Equations in the whole space $\R^N$. The associated ergodic problem has Lipschitz continuous solutions if the analogue of the ergodic constant is larger than a minimal value $\lambda_{min}$. We obtain various large-time convergence and Liouville type theorems, some of them being of completely new type. We also provide examples showing that, in this unbounded framework, the ergodic behavior may fail, and that the asymptotic behavior may also be unstable with respect to the initial data

Convergence to Time-Periodic Solutions in Time-Periodic Hamilton–Jacobi Equations on the Circle

International audienceThe goal of this paper is to give a simple proof of the convergence to time-periodic states of the solutions of time-periodic Hamilton–Jacobi equations on the circle with convex Hamiltonian. Note that the period of limiting solutions may be greater than the period of the Hamiltonian

Front propagation in Fisher-KPP equations with fractional diffusion

We study in this note the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an important example being the fractional Laplacian. Contrary to what happens in the standard Laplacian case, where the stable state invades the unstable one at constant speed, we prove here that invasion holds at an exponential in time velocity. These results provide a mathematically rigorous justification of numerous heuristics about this model

Sharp large time behaviour in NN-dimensional reaction-diffusion equations of bistable type

We study the large time behaviour of the reaction-diffsuion equation $\partial_t u=\Delta u +f(u)$ in spatial dimension $N$, when the nonlinear term is bistable and the initial datum is compactly supported. We prove the existence of a Lipschitz function $s^\infty$ of the unit sphere, such that $u(t,x)$ converges uniformly in $\mathbb{R}^N$, as $t$ goes to infinity, to $U_{c_*}\bigg(|x|-c_*t + \frac{N-1}{c_*} \mathrm{ln}t + s^\infty\Big(\frac{x}{|x|}\Big)\bigg)$, where $U_{c*}$ is the unique 1D travelling profile. This extends earlier results that identified the locations of the level sets of the solutions with $o_{t\to+\infty}(t)$ precision, or identified precisely the level sets locations for almost radial initial data