Upper and lower solutions in quasilinear parabolic boundary value problems

The authors study a class of initial boundary value problems associated with parabolic quasilinear equations: by introducing special auxiliary functions, upper and lower solutions are obtained, which turn out to be sharp in the sense that they coincide with the solution in particular situations. To Larry Payne on the occasion of his 80th birthday. Received: February 3, 2004; revised: April 26, 2004 Partially supported by University of Cagliari     

Properties of non-simultaneous blow-up solutions in nonlocal parabolic equations

This paper deals with blow-up solutions in parabolic equations coupled via nonlocal nonlinearities, subject to homogeneous Dirichlet conditions. Firstly, some criteria on non-simultaneous and simultaneous blow-up are given, including four kinds of phenomena: (i) the existence of non-simultaneous blow-up; (ii) the coexistence of non-simultaneous and simultaneous blow-up; (iii) any blow-up must be simultaneous; (iv) any blow-up must be non-simultaneous. Next, total versus single point blow-up are classified completely. Moreover, blow-up rates are obtained for both non-simultaneous and simultaneous blow-up solutions.     

Blow-up properties of solutions for a multi-coupled parabolic system

This paper deals with the blow-up behavior of radial solutions to a parabolic system multi-coupled via inner sources and boundary flux. We first obtain a necessary and sufficient condition for the existence of non-simultaneous blow-up, and then find five regions of exponent parameters where both non-simultaneous and simultaneous blow-up may happen. In particular, nine simultaneous blow-up rates are established for different regions of parameters. It is interesting to observe that different initial data may lead to different simultaneous blow-up rates even with the same exponent parameters.     

Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle

In this work, we study surfaces over convex regions in ² which are evolving by the mean curvature flow. Here, we specify the angle of contact of the surface to the boundary cylinder. We prove that solutions converge to ones moving only by translation.Partially supported by the NSF grant no: DMS-9100383Partially supported by the NSF grant no: DMS 9108269.A01     

Non-simultaneous quenching in a system of heat equations coupled at the boundary

We study the solutions of a parabolic system of heat equations coupled at the boundary through a nonlinear flux. We characterize in terms of the parameters involved when non-simultaneous quenching may appear. Moreover, if quenching is non-simultaneous we find the quenching rate, which surprisingly depends on the flux associated to the other component.     

An asymptotic formula for the voltage potential in the case of a near-surface conductivity inclusion

We rigorously derive an asymptotic expansion of the steady-state voltage potentials in the presence of a conductivity inclusion of small volume that is close to a planar surface. This new formula is motivated by the practically important inverse problem of imaging a conductivity inclusion near a planar interface. Partly supported by ACI Jeunes Chercheurs (0693) from the Ministry of Education and Scientific Research, France. Partially supported by grant R02-2003-000-10012-0 from the Korea Science and Engineering Foundation. Received: December 2, 2003; revised: January 18, 2004     

Periodic solutions of a periodically forced and undamped beam resting on weakly elastic bearings

We study the problem of existence of periodic solutions to a partial differential equation modelling the behavior of an undamped beam subject to an external periodic force. We assume that the ordinary differential equation associated to the first two modes of vibration of the beam has a symmetric homoclinic solution. By using methods borrowed by dynamical systems theory we prove that, if the period is non resonant with the (infinitely many) internal periods of the PDE, the equation has a weak periodic solution of the same period as the external force. In particular we obtain continua of periodic solutions for the undamped beam in absence of external forces. This result may be considered as an infinite dimensional analogue of a result obtained in [16] concerning accumulation of periodic solutions to homoclinic orbits in finite dimensional reversible systems. Matteo Franca: Partially supported by G.N.A.M.P.A. – INdAM (Italy).     

Convergence to equilibrium for classical and quantum spin systems

Summary The paper is devoted to stochastic equations describing the evolution of classical and quantum unbounded spin systems on discrete lattices and on Euclidean spaces. Existence and asymptotic properties of the corresponding transition semigroups are studied in a unified way using the theory of dissipative operators on weighted Hilbert and Banach spaces. This paper is an enlarged and rewritten version of the paper [7].Partially supported by the Italian National Project MURST Problemi nonlinearinell' Analisi... and by DRET under contract 901636/A000/DRET/DSISR.Partially sponsored by the KBN grant 2 2003 91 02 and by the KBN grant 2PO3A 082 08     

Quenching rates for heat equations with coupled singular nonlinear boundary flux

We study finite time quenching for heat equations coupled via singular nonlinear bound-ary flux. A criterion is proposed to identify the simultaneous and non-simultaneous quenchings. In particular, three kinds of simultaneous quenching rates are obtained for different nonlinear exponent re-gions and appropriate initial data. This extends an original work by Pablo, Quir′os and Rossi for a heat system with coupled inner absorption terms subject to homogeneous Neumann boundary conditions.     

The Complete Determination of the Minimum Distance of Two-Point Codes on a Hermitian Curve

This is a continuation of the previous papers [3, 4, 5]. We finish determining the minimum distance of two-point codes on a Hermitian curve. Masaaki Homma: Partially supported by Grant-in-Aid for Scientific Research (15500017), JSPS. Seon Jeong Kim: Partially supported by Korea Research Foundation Grant (KRF-2004-041-C00016)     

Improved spatial decay bounds in generalized heat conduction

This paper deals with heat conduction in a semi-infinite cylinder using the generalized Maxwell-Cattaneo equations. Spatial decay bounds for the temperature and heat flux under two different types of boundary conditions are derived. For fixed time it is shown that in each case the solutions decay in appropriate measure like the exponential of a quadratic function of the distance from the base of the cylinder, whereas in previous work they had been shown to decay only at least as fast as the exponential of a linear function.Received: January 13, 2004     

An adaptive numerical method to handle blow-up in a parabolic system

We study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. In certain cases the system develops a blow-up singularity in finite time. Fixed mesh methods are not well suited to approximate the problem near the singularity. As an alternative to reproduce the behaviour of the continuous solution, we present an adaptive in space procedure. The scheme recovers the conditions for blow-up and non-simultaneous blow-up. It also gives the correct non-simultaneous blow-up rate and set. Moreover, the numerical simultaneous blow-up rates coincide with the continuous ones in the cases when the latter are known. Finally, we present numerical experiments that illustrate the behaviour of the adaptive method.     

Ginzburg-Landau equation and motion by mean curvature, II: Development of the initial interface

In this paper, we study the short time behavior of the solutions of a sequence of Ginzburg-Landau equations indexed by ∈. We prove that under appropriate assumptions on the initial data, solutions converge to ±1 in short time and behave like the one-dimensional traveling wave across the interface. In particular, energy remains uniformly bounded in ∈. Partially supported by the NSF Grant DMS-9200801 and by the Army Research Office through the Center for Nonlinear Analysis.     

Optimal conditions of non-simultaneous blow-up and uniform blow-up profiles in localized parabolic equations

This paper deals with localized parabolic equations , with homogeneous Dirichlet boundary conditions, where x₀ is any fixed point in a bounded domain of R^N. The optimal classification of non-simultaneous and simultaneous blow-up phenomena is proposed for all of the nonnegative exponents. Moreover, uniform blow-up profiles are obtained for all kinds of simultaneous blow-up solutions.     

Regularity of invariant measures for a class of perturbed Ornstein-Uhlenbeck operators

In the framework of [5] we prove regularity of invariant measures for a class of Ornstein-Uhlenbeck operators perturbed by a drift which is not necessarily bounded or Lipschitz continuous. Regularity here means that is absolutely continuous with respect to the Gaussian invariant measure of the unperturbed operator with the square root of the Radon-Nikodym density in the corresponding Sobolev space of order 1.Partially supported by the International Science Foundation (Grant M 38000), the Russian Foundation of Fundamental Research (Grant 94-01-01556), and EC-Science Project SC1^*CT92-0784.Partially supported by the Italian National Project MURST Problemi nonlineari nell'AnalisiPartially supported by the DFG(SFB-256-Bonn, SFB-343-Bielefeld) and EC-Science Project SC1^*CT92-0784.     

Spatial decay estimates for the Maxwell-Cattaneo equations with mixed boundary conditions

In this paper the authors derive spatial decay bounds for the temperature and heat flux as defined by the Generalized Maxwell-Cattaneo equations for heat conduction in a semi-infinite cylinder when the temperature and the tangential components of the heat flux vector vanish on the lateral surface of the cylinder. The results here supplement those previously found by the authors [5] when the heat flux vector was assumed to be zero on the lateral surface but no condition was imposed on the temperature.Received: February 7, 2002; revised: June 3, 2002     

Multiple vortices for a self-dual CP(1) Maxwell-Chern-Simons model

We prove the existence of at least two doubly periodic vortex solutions for a self-dual CP(1) Maxwell-Chern-Simons model. To this end we analyze a system of two elliptic equations with exponential nonlinearities. Such a system is shown to be equivalent to a fourth-order elliptic equation admitting a variational structure. Tonia Ricciardi: Partially supported by the MIUR National Project Variational Methods and Nonlinear Differential Equations     

Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω. Research supported by the RIP program of Oberwolfach Institut and by the Tunisian Ministry for Scientific Research and Technology (MRST) under Grant 02/UR/15-01. Research supported by the RIP program of Oberwolfach Institut. (Received: September 17, 2003; revised: February 26, 2004)     

Incomplete quenching of heat equations with coupled singular logarithms boundary fluxes

This paper deals with quenching phenomena for a heat equations with coupled singular logarithms boundary fluxes. Under appropriate hypotheses, the non-simultaneous quenching of the solution for the system is proved, and the estimates of quenching rates are given. Then we give a natural continuation of the solution (u,v) after the quenching time when the equations occurs non-simultaneous quenching. Moreover, we identify the heat equations verified by the continuation beyond quenching time, i.e., the equations occurs incomplete quenching.     

Incomplete quenching in a system of heat equations coupled at the boundary

In this paper we find a possible continuation for quenching solutions to a system of heat equations coupled at the boundary condition. This system exhibits simultaneous and non-simultaneous quenching. For non-simultaneous quenching our continuation is a solution of a parabolic problem with Neumann boundary conditions. We also give some results for simultaneous quenching and present some numerical experiments that suggest that the approximations are not uniformly bounded in this case.