59 research outputs found
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
Hardy spaces for the conjugated Beltrami equation in a doubly connected domain
International audienceWe consider Hardy spaces associated to the conjugated Beltrami equation on doubly connected planar domains. There are two main differences with previous studies. First, while the simple connectivity plays an important role in the simply connected case, the multiple connectivity of the domain leads to unexpected difficulties. In particular, we make strong use of a suitable parametrization of an analytic function in a ring by its real part on one part of the boundary and by its imaginary part on the other. Then, we allow the coefficient in the conjugated Beltrami equation to belong to for some , while it was supposed to be Lipschitz in the simply connected case. We define Hardy spaces associated with the conjugated Beltrami equation and solve the corresponding Dirichlet problem. The same problems for generalized analytic function are also solved
Global attractor and stabilization for a coupled PDE-ODE system
We study the asymptotic behavior of solutions of one coupled PDE-ODE system
arising in mathematical biology as a model for the development of a forest
ecosystem.
In the case where the ODE-component of the system is monotone, we establish
the existence of a smooth global attractor of finite Hausdorff and fractal
dimension.
The case of the non-monotone ODE-component is much more complicated. In
particular, the set of equilibria becomes non-compact here and contains a huge
number of essentially discontinuous solutions. Nevertheless, we prove the
stabilization of any trajectory to a single equilibrium if the coupling
constant is small enough
On the solvability of some systems of integro-differential equations with and without a drift
We prove the existence of solutions for some integro-differential systems
containing equations with and without the drift terms in the H^2 spaces by
virtue of the fixed point technique when the elliptic equations contain second
order differential operators with and without the Fredholm property, on the
whole real line or on a finite interval with periodic boundary conditions. Let
us emphasize that the study of the systems case is more complicated than of the
scalar situation and requires to overcome more cumbersome technicalities
SOLVABILITY OF SOME INTEGRO-DIFFERENTIAL EQUATIONS WITH DRIFT
We prove the existence in the sense of sequences of solutions for some integro-differential type equations involving the drift term in the appropriate H² spaces using the fixed point technique when the elliptic problems contain second order differential operators with and without Fredholm property. It is shown that, under the reasonable technical conditions, the convergence in L¹ of the integral kernels yields the existence and convergence in H² of solutions
The attractor for a nonlinear reaction-diffusion system in an unbounded domain
In this paper the quasilinear second order parabolic systems of reaction-diffusion type in an unbounded domain are considered. Our aim in this article is to study the long-time behaviour of parabolic systems for which the nonlinearity depends explicitely on the gradient of the unknown functions. To this end we give a systematic study of given parabolic systems and their attractors in weighted Sobolev spaces. Dependence of the Hausdorff dimension of attractors from weight of the Sobolev spaces are considered
EXPONENTIAL ATTRACTORS FOR NON-AUTONOMOUS DISSIPATIVE SYSTEM
In this paper we will introduce a version of exponential attractor for non-autonomous equations as a time dependent set with uniformly bounded finite fractal dimension which is positively invariant and attracts every bounded set at an exponential rate. This is a natural generalization of the existent notion for autonomous equations. A generation theorem will be proved under the assumption that the evolution operator is a compact perturbation of a contraction. In the second half of the paper, these results will be applied to some non-autonomous chemotaxis system
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