44 research outputs found
Selfsimilar solutions in a sector for a quasilinear parabolic equation
We study a two-point free boundary problem in a sector for a quasilinear
parabolic equation. The boundary conditions are assumed to be spatially and
temporally "self-similar" in a special way. We prove the existence, uniqueness
and asymptotic stability of an expanding solution which is self-similar at
discrete times. We also study the existence and uniqueness of a shrinking
solution which is self-similar at discrete times.Comment: 23 page
Dynamics of Patterns
This workshop focused on the dynamics of nonlinear waves and spatio-temporal patterns, which arise in functional and partial differential equations. Among the outstanding problems in this area are the dynamical selection of patterns, gaining a theoretical understanding of transient dynamics, the nonlinear stability of patterns in unbounded domains, and the development of efficient numerical techniques to capture specific dynamical effects
Uniqueness from pointwise observations in a multi-parameter inverse problem
In this paper, we prove a uniqueness result in the inverse problem of
determining several non-constant coefficients of one-dimensional
reaction-diffusion equations. Such reaction-diffusion equations include the
classical model of Kolmogorov, Petrovsky and Piskunov as well as more
sophisticated models from biology. When the reaction term contains an unknown
polynomial part of degree with non-constant coefficients our
result gives a sufficient condition for the uniqueness of the determination of
this polynomial part. This sufficient condition only involves pointwise
measurements of the solution of the reaction-diffusion equation and of its
spatial derivative at a single point during a
time interval In addition to this uniqueness result, we give
several counter-examples to uniqueness, which emphasize the optimality of our
assumptions. Finally, in the particular cases N=2 and we show that such
pointwise measurements can allow an efficient numerical determination of the
unknown polynomial reaction term
Pulsating fronts for nonlocal dispersion and KPP nonlinearity
In this paper we are interested in propagation phenomena for nonlocal
reaction-diffusion equations of the type: , where J is a probability density and f is a KPP
nonlinearity periodic in the x variables. Under suitable assumptions we
establish the existence of pulsating fronts describing the invasion of the 0
state by a heterogeneous state. We also give a variational characterization of
the minimal speed of such pulsating fronts and exponential bounds on the
asymptotic behavior of the solution.Comment: Annales de l'Institut Henri Poincar\'e Analyse non lin\'eaire (2011