Ecuaciones diferenciales estocásticas con condición final y soluciones de viscosidad de EDPS semilineales de segundo orden

El objetivo de este documento es recopilar algunos resultados clasicos sobre existencia y unicidad ´ de soluciones de ecuaciones diferenciales estocasticas (EDEs) con condici ´ on final (en ingl ´ es´ Backward stochastic differential equations) con particular enfasis en el caso de coeficientes mon ´ otonos, y su cone- ´ xion con soluciones de viscosidad de sistemas de ecuaciones diferenciales parciales (EDPs) parab ´ olicas ´ y el´ıpticas semilineales de segundo orden.

Boundary value problems for the N-wave interaction equations

We consider boundary value problems for the N-wave interaction equations in one and two space dimensions, posed for x [greater-or-equal, slanted] 0 and x,y [greater-or-equal, slanted] 0, respectively. Following the recent work of Fokas, we develop an inverse scattering formalism to solve these problems by considering the simultaneous spectral analysis of the two ordinary differential equations in the associated Lax pair. The solution of the boundary value problems is obtained through the solution of a local Riemann–Hilbert problem in the one-dimensional case, and a nonlocal Riemann–Hilbert problem in the two-dimensional case.

Similarity solutions of the shallow water equations

A one-dimensional shock (bore) reflection problem is discussed for the two-dimensional shallow water equations with cylindrical symmetry. The differential equations for a similarity solution are derived and solved numerically in conjunction with the Rankine-Hugoniot shock relations.

Optimal control of nonlinear differential algebraic equation systems

A novel iterative procedure is described for solving nonlinear optimal control problems subject to differential algebraic equations. The procedure iterates on an integrated modified linear quadratic model based problem with parameter updating in such a manner that the correct solution of the original non-linear problem is achieved. The resulting algorithm has a particular advantage in that the solution is achieved without the need to solve the differential algebraic equations . Convergence aspects are discussed and a simulation example is described which illustrates the performance of the technique. 1. Introduction When modelling industrial processes often the resulting equations consist of coupled differential and algebraic equations (DAEs). In many situations these equations are nonlinear and cannot readily be directly reduced to ordinary differential equations.

On the Convergence of Two-Stage Iterative Processes for Solving Linear Equations

This paper considers two-stage iterative processes for solving the linear system $Af = b$. The outer iteration is defined by $Mf^{k + 1} = Nf^k + b$, where $M$ is a nonsingular matrix such that $M - N = A$. At each stage $f^{k + 1} $ is computed approximately using an inner iteration process to solve $Mv = Nf^k + b$ for $v$. At the $k$th outer iteration, $p_k $ inner iterations are performed. It is shown that this procedure converges if $p_k \geqq P$ for some $P$ provided that the inner iteration is convergent and that the outer process would converge if $f^{k + 1} $ were determined exactly at every step. Convergence is also proved under more specialized conditions, and for the procedure where $p_k = p$ for all $k$, an estimate for $p$ is obtained which optimizes the convergence rate. Examples are given for systems arising from the numerical solution of elliptic partial differential equations and numerical results are presented.

Axonal velocity distributions in neural feld equations

By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.

Finite-dimensional global attractors in Banach spaces

We provide bounds on the upper box-counting dimension of negatively invariant subsets of Banach spaces, a problem that is easily reduced to covering the image of the unit ball under a linear map by a collection of balls of smaller radius. As an application of the abstract theory we show that the global attractors of a very broad class of parabolic partial differential equations (semilinear equations in Banach spaces) are finite-dimensional. (C) 2010 Elsevier Inc. All rights reserved.

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF NONLINEAR SCHRODINGER EQUATIONS

This paper proves the multiplicity of positive solutions for the following class of quasilinear problems: {-epsilon(p)Delta(p)u+(lambda A(x) + 1)vertical bar u vertical bar(p-2)u = f(u), R(N) u(x)>0 in R(N), where Delta(p) is the p-Laplacian operator, N > p >= 2, lambda and epsilon are positive parameters, A is a nonnegative continuous function and f is a continuous function with subcritical growth. Here, we use variational methods to get multiplicity of positive solutions involving the Lusternick-Schnirelman category of intA(-1)(0) for all sufficiently large lambda and small epsilon.

Soliton solutions for quasilinear Schrodinger equations with critical growth

In this paper we establish the existence of standing wave solutions for quasilinear Schrodinger equations involving critical growth. By using a change of variables, the quasilinear equations are reduced to semilinear one. whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v. In the proof that v is nontrivial, the main tool is the concentration-compactness principle due to P.L. Lions together with some classical arguments used by H. Brezis and L. Nirenberg (1983) in [9]. (C) 2009 Elsevier Inc. All rights reserved.

Lower semicontinuity of attractors for non-autonomous dynamical systems

This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

Dynamics in dumbbell domains II. The limiting problem

In this work we continue the analysis of the asymptotic dynamics of reaction-diffusion problems in a dumbbell domain started in [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2) (2006) 551-597]. Here we study the limiting problem, that is, an evolution problem in a ""domain"" which consists of an open, bounded and smooth set Omega subset of R(N) with a curve R(0) attached to it. The evolution in both parts of the domain is governed by a parabolic equation. In Omega the evolution is independent of the evolution in R(0) whereas in R(0) the evolution depends on the evolution in Omega through the continuity condition of the solution at the junction points. We analyze in detail the linear elliptic and parabolic problem, the generation of linear and nonlinear semigroups, the existence and structure of attractors. (C) 2009 Elsevier Inc. All rights reserved.

Dynamics in dumbbell domains III. Continuity of attractors

In this paper we conclude the analysis started in [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597] and continued in [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)] concerning the behavior of the asymptotic dynamics of a dissipative reaction-diffusion equation in a dumbbell domain as the channel shrinks to a line segment. In [J.M. Arrieta, AN Carvalho. G. Lozada-Cruz, Dynamics in dumbbell domains I. Continuity of the set of equilibria, J. Differential Equations 231 (2006) 551-597], we have established an appropriate functional analytic framework to address this problem and we have shown the continuity of the set of equilibria. In [J.M. Arrieta, AN Carvalho, G. Lozada-Cruz. Dynamics in dumbbell domains II. The limiting problem, J. Differential Equations 247 (1) (2009) 174-202 (this issue)], we have analyzed the behavior of the limiting problem. In this paper we show that the attractors are Upper semicontinuous and, moreover, if all equilibria of the limiting problem are hyperbolic, then they are lower semicontinuous and therefore, continuous. The continuity is obtained in L(p) and H(1) norms. (C) 2008 Elsevier Inc. All rights reserved.

Strongly damped wave problems: Bootstrapping and regularity of solutions

The aim of the article is to present a unified approach to the existence, uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in Banach spaces. Our results are applied next to a number of examples appearing in literature, which fall into the class of strongly damped semilinear wave equations. The present work essentially extends the results on the existence and regularity of solutions to such problems. Previously, these problems have been considered mostly within the Hilbert space setting and with the main part operators being selfadjoint. In this article we present a more general approach, involving sectorial operators in reflexive Banach spaces. (C) 2008 Elsevier Inc. All rights reserved.

Numerical prediction of three-dimensional time-dependent viscoelastic extrudate swell using differential and algebraic models

This study investigates the numerical simulation of three-dimensional time-dependent viscoelastic free surface flows using the Upper-Convected Maxwell (UCM) constitutive equation and an algebraic explicit model. This investigation was carried out to develop a simplified approach that can be applied to the extrudate swell problem. The relevant physics of this flow phenomenon is discussed in the paper and an algebraic model to predict the extrudate swell problem is presented. It is based on an explicit algebraic representation of the non-Newtonian extra-stress through a kinematic tensor formed with the scaled dyadic product of the velocity field. The elasticity of the fluid is governed by a single transport equation for a scalar quantity which has dimension of strain rate. Mass and momentum conservations, and the constitutive equation (UCM and algebraic model) were solved by a three-dimensional time-dependent finite difference method. The free surface of the fluid was modeled using a marker-and-cell approach. The algebraic model was validated by comparing the numerical predictions with analytic solutions for pipe flow. In comparison with the classical UCM model, one advantage of this approach is that computational workload is substantially reduced: the UCM model employs six differential equations while the algebraic model uses only one. The results showed stable flows with very large extrudate growths beyond those usually obtained with standard differential viscoelastic models. (C) 2010 Elsevier Ltd. All rights reserved.