We investigate the well-posedness, the exponential stability, or the lack thereof, of thermoelastic systems in materials where, in contrast to classical thermoelastic models for Kirchhoff type plates, two temperatures are involved, related by an elliptic equation. The arising initial boundary value problems for different boundary conditions deal with systems of partial differential equations involving Schrödinger like equations, hyperbolic and elliptic equations, which have a different character compared to the classical one with the usual single temperature. Depending on the model -- with Fourier or with Cattaneo type heat conduction -- we obtain exponential resp. non-exponential stability, thus providing another examples where the change from Fourier's to Cattaneo's law leads to a loss of exponential stability.
We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to 1. We show that, under general hypotheses, these invariant manifolds are Gevrey with type related to some explicit constants. We provide examples of the optimality of our results as well as some applications to celestial mechanics, namely, the Sitnikov problem and the restricted planar three body problem.
In this paper we study the Sotomayor-Teixeira regularization of a general visible fold singularity of a planar Filippov system. Extending Geometric Fenichel Theory beyond the fold with asymptotic methods, we determine the deviation of the orbits of the regularized system from the generalized solutions of the Filippov one. This result is applied to the regularization of global sliding bifurcations as the Grazing-Sliding of periodic orbits and the Sliding Homoclinic to a Saddle, as well as to some classical problems in dry friction.; Roughly speaking, we see that locally, and also globally, the regularization of the bifurcations preserve the topological features of the sliding ones.
We consider positive semistable solutions u of Lu + f(u) = 0 with zero Dirichlet boundary condition, where L is a uniformly elliptic operator and f is an element of C-2 is a positive, nondecreasing, and convex nonlinearity which is super-linear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension n <= 9, but only established for n <= 4. In this paper we prove the L-infinity bound up to dimension n = 5 under the following further assumption on f: for every epsilon > 0, there exist T = T(epsilon) and C = C(epsilon) such that f '(t) < C f(t)(1+epsilon) for all t > T. This bound will follow from a L-p-estimate for f ' (u) for every p < 3 (and for all n >= 2). Under a similar but more restrictive assumption on f, we also prove the L-infinity estimate when n = 6. We remark that our results do not assume any lower bound on f '.
Acosta-Humànez, P.; Lazaro, J. Tomás; Morales, J.; Pantazi, C. Discrete and continuous dynamical systems. Series A Vol. 35, num. 5, p. 1767-1800 DOI: 10.3934/dcds.2015.35.1767 Data de publicació: 2015-05-01 Article en revista
We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Lienard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincare problem for some families is also approached.
We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian (-Delta)(s) with s > 1. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case s is an element of (0,1).; As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions (-Delta)(s)phi = lambda phi in Omega, phi equivalent to 0 in R-n\Omega.
This article provides sufficient conditions for the existence of periodic solutions with nonconstant sign in a family of polynomial, non-auto-nomous, firrst-order diferential equations that arise as a generalization of the Abel equation of the second kind.
Fedorov, Y.; García, L.; Vankerschaver, J. Discrete and continuous dynamical systems. Series A Vol. 33, num. 9, p. 4017-4040 DOI: 10.3934/dcds.2013.33.4017 Data de publicació: 2013-03 Article en revista
We consider the motion of a planar rigid body in a potential two-dimensional flow with a circulation and subject to a certain nonholonomic constraint. This model can be related to the design of underwater vehicles.
The equations of motion admit a reduction to a 2-dimensional nonlinear system, which is integrated explicitly. We show that the reduced system comprises both asymptotic and periodic dynamics separated by a critical value of the energy, and give a complete classification of types of the motion. Then we describe the whole variety of the trajectories of the body on the plane.
We present numerical algorithms that use small requirements of storage and operations to compute the iteration of cocycles over a rotation. We also show that these algorithms can be used to compute efficiently the stable and unstable bundles and the Lyapunov exponents of the cocycle.
Martin, P.; Sauzin, D.; Martinez-seara, Tere Discrete and continuous dynamical systems. Series A Vol. 31, num. 2, p. 301-372 DOI: 10.3934/dcds.2011.31.301 Data de publicació: 2011-10 Article en revista
The McMillan map is a one-parameter family of integrable symplectic
maps of the plane, for which the origin is a hyperbolic xed point
with a homoclinic loop, with small Lyapunov exponent when the parameter is
small. We consider a perturbation of the McMillan map for which we show
that the loop breaks in two invariant curves which are exponentially close one
to the other and which intersect transversely along two primary homoclinic
orbits. We compute the asymptotic expansion of several quantities related to
the splitting, namely the Lazutkin invariant and the area of the lobe between
two consecutive primary homoclinic points. Complex matching techniques are
in the core of this work. The coe cients involved in the expansion have a
resurgent origin, as shown in
Martin, P.; Sauzin, D.; Martinez-seara, Tere Discrete and continuous dynamical systems. Series A Vol. 31, num. 1, p. 165-207 DOI: 10.3934/dcds.2011.31.165 Data de publicació: 2011-09 Article en revista
Abstract. A sequence of \inner equations" attached to certain perturbations
of the McMillan map was considered in , their solutions were used in that
article to measure an exponentially small separatrix splitting. We prove here
all the results relative to these equations which are necessary to complete the
proof of the main result of . The present work relies on ideas from resur-
gence theory: we describe the formal solutions, study the analyticity of their
Borel transforms and use Ecalle's alien derivations to measure the discrepancy
between di erent Borel-Laplace sums.
We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation 1/2 in R n. Our energy estimates hold for every nonlinearity and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable.
As a consequence, in dimension , we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation in R n.