103 research outputs found
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 . The
associated ergodic problem has Lipschitz continuous solutions if the analogue
of the ergodic constant is larger than a minimal value . 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 -dimensional reaction-diffusion equations of bistable type
We study the large time behaviour of the reaction-diffsuion equation
in spatial dimension , when the nonlinear term
is bistable and the initial datum is compactly supported. We prove the
existence of a Lipschitz function of the unit sphere, such that
converges uniformly in , as goes to infinity, to
, where is the unique 1D
travelling profile. This extends earlier results that identified the locations
of the level sets of the solutions with precision, or
identified precisely the level sets locations for almost radial initial data
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