160 research outputs found
Asymptotic decay under nonlinear and noncoercive dissipative effects for electrical conduction in biological tissues
We consider a nonlinear model for electrical conduction in biological
tissues. The nonlinearity appears in the interface condition prescribed on the
cell membrane.
The purpose of this paper is proving asymptotic convergence for large times
to a periodic solution when time-periodic boundary data are assigned. The
novelty here is that we allow the nonlinearity to be noncoercive. We consider
both the homogenized and the non-homogenized version of the problem
Variable Support Control for the Wave Equation: A Multiplier Approach
We study the controllability of the multidimensional wave equation in a
bounded domain with Dirichlet boundary condition, in which the support of the
control is allowed to change over time. The exact controllability is reduced to
the proof of the observability inequality, which is proven by a multiplier
method. Besides our main results, we present some applications
Alternating and variable controls for the wave equation
The present article discusses the exact observability of the wave equation
when the observation subset of the boundary is variable in time. In the
one-dimensional case, we prove an equivalent condition for the exact
observability, which takes into account only the location in time of the
observation. To this end we use Fourier series. Then we investigate the two
specific cases of single exchange of the control position, and of exchange at a
constant rate. In the multi-dimensional case, we analyse sufficient conditions
for the exact observability relying on the multiplier method. In the last
section, the multi-dimensional results are applied to specific settings and
some connections between the one and multi-dimensional case are discussed;
furthermore some open problems are presented.Comment: The original publication is available at www.esaim-cocv.org. The
copyright of this article belongs to ESAIM-COC
Flux through a time-periodic gate: Monte Carlo test of a homogenization result
We investigate via Monte Carlo numerical simulations and theoretical
considerations the outflux of random walkers moving in an interval bounded by
an interface exhibiting channels (pores, doors) which undergo an open/close
cycle according to a periodic schedule. We examine the onset of a limiting
boundary behavior characterized by a constant ratio between the outflux and the
local density, in the thermodynamic limit. We compare such a limit with the
predictions of a theoretical model already obtained in the literature as the
homogenization limit of a suitable diffusion problem
Some remarks on the Sobolev inequality in Riemannian manifolds
We investigate Sobolev and Hardy inequalities, specifically weighted
Minerbe's type estimates, in noncompact complete connected Riemannian manifolds
whose geometry is described by an isoperimetric profile. In particular, we
assume that the manifold satisfies the -hyperbolicity property, stated in
terms of a necessary integral Dini condition on the isoperimetric profile. Our
method seems to us to combine sharply the knowledge of the isoperimetric
profile and the optimal Bliss type Hardy inequality depending on the geometry
of the manifold. We recover the well known best Sobolev constant in the
Euclidean case.Comment: 16 page
Optimal decay rate for degenerate parabolic equations on noncompact manifolds
We consider an initial value problem for a doubly degenerate
parabolic equation in a noncompact Riemannian manifold. The
geometrical features of the manifold are coded in either a
Faber-Krahn inequality or a relative Faber-Krahn inequality. We
prove optimal decay and space-time local estimates of solutions. We
employ a simplified version of the by now classical local approach
by DeGiorgi, Ladyzhenskaya-Uraltseva, DiBenedetto which is of
independent interest even in the euclidean case
Asymptotic behavior for the filtration equation in domains with noncompact boundary
We consider the initial value boundary problem with zero Neumann data for an equation modelled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and non-negative. We show that the asymptotic profile for large times of u is one-dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem) we obtain a genuine multi-dimensional profile given by the well known Barenblatt solution
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