65 research outputs found
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 on , where the
initial datum is such that
for all , with a steady state in . We characterize
the perturbations such that, if is the solution associated with the
initial datum , then, if is small enough in a sense, one has
(resp. ) for large. This condition depends on
the sign of , where is an adjoint
solution, which satisfies a backward parabolic equation on and is
uniquely defined [7]. We then provide two applications of our result. We first
address an open problem stated in [8] when and is a bistable
nonlinearity independent of . Namely, we compute the derivative of the
critical length associated with the initial datum
, that is the length above (resp.
below) which converges to (resp. ) as . Lastly,
again when and is a bistable nonlinearity independent of , we
prove the existence and characterize with a bathtub principle the initial datum
minimizing some cost function and
guaranteeing at the same time that for all
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 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
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 ReactionâDiffusion 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 +â and separated away from the unstable steady state 0 as x tends to ââ. 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 tâ+â 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 â +â. 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 Ïâlimit set of u(t, ct + x) as t tends to +â is equal to the whole interval [0, 1] for all x â R and for all speeds c belonging to a given interval (Îł 1 , Îł 2) with large enough Îł 1 < Îł 2 †+â
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