2,929 research outputs found
On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds
We consider the Fisher-KPP equation with a non-local interaction term. We
establish a condition on the interaction that allows for existence of
non-constant periodic solutions, and prove uniform upper bounds for the
solutions of the Cauchy problem, as well as upper and lower bounds on the
spreading rate of the solutions with compactly supported initial data
A Liouville theorem for the Euler equations in the plane
This paper is concerned with qualitative properties of bounded steady flows
of an ideal incompressible fluid with no stagnation point in the
two-dimensional plane R^2. We show that any such flow is a shear flow, that is,
it is parallel to some constant vector. The proof of this Liouville-type result
is firstly based on the study of the geometric properties of the level curves
of the stream function and secondly on the derivation of some estimates on the
at most logarithmic growth of the argument of the flow. These estimates lead to
the conclusion that the streamlines of the flow are all parallel lines
Asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains: two approaches
In this paper, we study the asymptotic behavior as of
solutions of semilinear elliptic equations in quarter- or half-spaces, for
which the value at is given. We prove the uniqueness and characterize
the one-dimensional or constant profile of the solutions at infinity. To do so,
we use two different approaches. The first one is a pure PDE approach and it is
based on the maximum principle, the sliding method and some new Liouville type
results for elliptic equations in the half-space or in the whole
space~. The second one is based on the theory of dynamical
systems
A Faber-Krahn inequality with drift
Let be a bounded domain in (,
), be the open Euclidean ball centered at 0 having
the same Lebesgue measure as , and with . If
denotes the principal eigenvalue of the operator
in with Dirichlet boundary condition, we
establish that where . Moreover, equality holds only when, up to
translation, and . This result can be
viewed as an isoperimetric inequality for the first eigenvalue of the Dirichlet
Laplacian with drift. It generalizes the celebrated Rayleigh-Faber-Krahn
inequality for the first eigenvalue of the Dirichlet Laplacian
Optimization of some eigenvalue problems with large drift
This paper is concerned with eigenvalue problems for non-symmetric elliptic
operators with large drifts in bounded domains under Dirichlet boundary
conditions. We consider the minimal principal eigenvalue and the related
principal eigenfunction in the class of drifts having a given, but large,
pointwise upper bound. We show that, in the asymptotic limit of large drifts,
the maximal points of the optimal principal eigenfunctions converge to the set
of points maximizing the distance to the boundary of the domain. We also show
the uniform asymptotic profile of these principal eigenfunctions and the
direction of their gradients in neighborhoods of the boundary
On the mean speed of bistable transition fronts in unbounded domains
This paper is concerned with the existence and further properties of
propagation speeds of transition fronts for bistable reaction-diffusion
equations in exterior domains and in some domains with multiple cylindrical
branches. In exterior domains we show that all transition fronts with complete
propagation propagate with the same global mean speed, which turns out to be
equal to the uniquely defined planar speed. In domains with multiple
cylindrical branches, we show that the solutions emanating from some branches
and propagating completely are transition fronts propagating with the unique
planar speed. We also give some geometrical and scaling conditions on the
domain, either exterior or with multiple cylindrical branches, which guarantee
that any transition front has a global mean speed
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