Bifurcaciones causadas por variación del dominio en problemas elípticos

Tesis Univ. Compl. Dpto de Ecuaciones funcionales. Dir. por Alfonso Casal Piga, leída en Madrid, el 16 de octubre de 1981.Fac. de Ciencias MatemáticasTRUEProQuestpu

An extinction delay mechanism for some evolution equations

We study the ”finite extinction phenomenon”(there exists t0 ≥ 0 such that u(t, x) ≡ 0 ∀t ≥ t0, a.e. x ∈ Ω) for solutions of parabolic reaction-diffusion equations of the type ∂u ∂t − k∆u+λb(t)f(u(t − τ, x)) = 0 and ordinary delayed differential equations (k = 0) with a delay term τ > 0

Cálculo diferencial de varias variables

NO DISPONEMOS DE COPIA DIGITALIZADA DE ESTE DOCUMENTODepto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

Finite extinction and null controllability via delayed feedback non-local actions

We give sufficient conditions to have the finite extinction for all solutions of linear parabolic reaction-diffusion equations of the type partial derivative u/partial derivative t - Lambda u = -M(t)u(t - tau, x) (1) with a delay term tau > 0, on Omega, an open set of R(N), M(t) is a bounded linear map on L(p)(Omega), u(t, x) satisfies a homogeneous Neumann or Dirichlet boundary condition. We apply this result to obtain distributed null controllability of the linear heat equation u(t) - Delta u = upsilon(t, x) by means of the delayed feedback term upsilon(t, x) = -M(t)u(t - tau, x)

Blow-up in some ordinary and partial differential equations with time-delay

Blow-up phenomena are analyzed for both the delay-differential equation (DDE) u'(t) = B'(t)u(t - tau), and the associated parabolic PDE (PDDE) partial derivative(t)u=Delta u+B'(t)u(t-tau,x), where B : [0, tau] -> R is a positive L(1) function which behaves like 1/vertical bar t - t*vertical bar(alpha), for some alpha is an element of (0, 1) and t* is an element of (0,tau). Here B' represents its distributional derivative. For initial functions satisfying u(t* - tau) > 0, blow up takes place as t NE arrow t* and the behavior of the solution near t* is given by u(t) similar or equal to B(t)u(t - tau), and a similar result holds for the PDDE. The extension to some nonlinear equations is also studied: we use the Alekseev's formula (case of nonlinear (DDE)) and comparison arguments (case of nonlinear (PDDE)). The existence of solutions in some generalized sense, beyond t = t* is also addressed. This results is connected with a similar question raised by A. Friedman and J. B. McLeod in 1985 for the case of semilinear parabolic equations

Hopf bifurcation and bifurcation from constant oscillations to a torus path for delayed complex Ginzburg-Landau equations

We consider the complex Ginzburg-Landau equation with feedback control given by some delayed linear terms (possibly dependent of the past spatial average of the solution). We prove several bifurcation results by using the delay as parameter. We start proving a Hopf bifurcation result for the equation without diffusion (the so-called Stuart-Landau equation) when the amplitude of the delayed term is suitably chosen. The diffusion case is considered first in the case of the whole space and later on a bounded domain with periodicity conditions. In the first case a linear stability analysis is made with the help of computational arguments (showing evidence of the fulfillment of the delicate transversality condition). In the last section the bifurcation takes place starting from an uniform oscillation and originates a path over a torus. This is obtained by the application of an abstract result over suitable functional spaces