Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable

We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.     

Blow-up analysis for a nonlinear diffusion system

This paper deals with a nonlinear diffusion system coupled via nonlinear reaction terms of power type. As results of interactions among the multi-nonlinearities in the system described by six exponents, global boundedness and blow-up criteria of positive solutions are determined.Supported by the National Natural Science Foundation of China.Received: December 12, 2001; revised: May 6, December 3, 2002     

Two compressible immiscible fluids in porous media

We consider a model of flow of two compressible and immiscible phases in a three-dimensional porous media. The equations are obtained by the conservation of the mass of each phase. This model is treated in its general form with the whole nonlinear terms. The only assumption concerns the dependence of densities on a global pressure. We obtain the existence of weak solutions under different kinds of degeneracies of the capillary terms.     

Universal bounds for global solutions of a diffusion equation with a nonlocal reaction term

We consider the nonlinear heat equation with nonlocal reaction term in space , in smoothly bounded domains. We prove the existence of a universal bound for all nonnegative global solutions of this equation. Moreover, in contrast with similar recent results for equations with local reaction terms, this is shown to hold for all p>1. As an interesting by-product of our proof, we derive for this equation a smoothing effect under weaker assumptions than for corresponding problem with local reaction.     

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. Partially supported by project BFM2002-04572 (Spain). Partially supported by UBA grant EX046, CONICET and Fundación Antorchas (Argentina). Received: February 17, 2004; revised: July 5, 2004     

偏微分方程的吸引子

Abstract. In this paper the reaction-diffusion systems and the damped wave systems with non-linear terms of gradient form are studied ,and the conditions for making them to be gradient sys-tems are given. So the structure of attractors is clear and simple in some sense by the known re-sult. Some examples are given. Also the reaction-diffusion systems with general nonlinear termsare discussed.     

Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms

In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|^q−1u|∇u|^p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞.     

Generic hyperbolicity of stationary solutions for a reaction-diffusion system

In this paper, we study the generic hyperbolicity of equilibria of a reaction-diffusion system with respect to nonlinear terms in the set of C²-functions equipped with the Whitney Topology. To accomplish this, we combine Baire’s Lemma and the usual Transversality Theorem.     

Global attractor for a non-autonomous integro-differential equation in materials with memory

The long-time behavior of an integro-differential parabolic equation of diffusion type with memory terms, expressed by convolution integrals involving infinite delays and by a forcing term with bounded delay, is investigated in this paper. The assumptions imposed on the coefficients are weak in the sense that uniqueness of solutions of the corresponding initial value problems cannot be guaranteed. Then, it is proved that the model generates a multivalued non-autonomous dynamical system which possesses a pullback attractor. First, the analysis is carried out with an abstract parabolic equation. Then, the theory is applied to the particular integro-differential equation which is the objective of this paper. The general results obtained in the paper are also valid for other types of parabolic equations with memory.     

On dual and triple integral equations involving modified bessel functions

A generalisation of some tripe integral equations occuring in the solution of certain mixed boundary value problems involving the wave equation is investigated. It is found that the solution of the equations can be expressed in terms of the solution of a Fredholm intgral equation of the second kind and that a related pair of dual integral equations can be solved exactly     

Constructing zeros of accretive operators II

In this paper we study a homogenization problem for a time periodic boundary value problem concerning the quasi-stationary Maxwell equations with a non linear monotone magne tic characteristic. The main features of the problem are related to the vanishing of the conductivity inside each period so that the type of the equations is rapidly oscillating. The unknowns are a vector potential and a scalar potential. We show that the first one converges to zero up to terms of second order, while the second one converges to the solution of a suitable homogenized stationary equation (with time as a parameter). We show moreover that when the magnetic characteristic is linear and symmetric the second order terms in the asymptotic expansion of the vector potential can be identified and related to the time derivative of the limit scalar potential.     

Uniform blow-up rate for a nonlocal degenerate parabolic equations

In this paper, we introduce a new method for investigating the rate of blow-up of solutions of diffusion equations with nonlocal nonlinear reaction terms. In some cases, we prove that the solutions have global blow-up and the rate of blow-up is uniform in all compact subsets of the domain. In each case, the blow-up rate of _|u(t)|∞${\left|u\left(t\right)\right|}_{\infty }$ is precisely determined.     

Pattern formation in a nonlinear model for animal coats

Several models have been proposed for describing the formation of animal coat patterns. We consider reaction-diffusion models due to Murray, which rely on a Turing instability for the pattern selection. In this paper, we describe the early stages of the pattern formation process for large domain sizes. This includes the selection mechanism and the geometry of the patterns generated by the nonlinear system on one-, two-, and three-dimensional base domains. These results are obtained by an adaptation of results explaining the occurrence of spinodal decomposition in materials science as modeled by the Cahn-Hilliard equation. We use techniques of dynamical systems, viewing solutions of the reaction-diffusion model in terms of nonlinear semiflows. Our results are applicable to any parabolic system exhibiting a Turing instability.     

On the existence of traveling waves for delayed reaction-diffusion equations

We study the existence of traveling wave solutions for reaction-diffusion equations with nonlocal delay, where reaction terms are not necessarily monotone. The existence of traveling wave solutions for reaction-diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the Schauder fixed point theorem. The smoothness of upper and lower solutions is not required in this paper.     

Summability of divergent solutions of the n-dimensional heat equation

The Cauchy problem for n-dimensional complex heat equation is considered. The Borel summability of formal solutions is characterized in terms of analytic continuation with an appropriate growth condition of the spherical mean of the Cauchy data.     

Degenerate quasilinear pseudoparabolic equations with memory terms and variational inequalities

The existence of solutions of degenerate quasilinear pseudoparabolic equations, where the term _∂tu${\partial }_{t}u$ is replace by _∂tb(u)${\partial }_{t}b\left(u\right)$, with memory terms and quasilinear variational inequalities is shown. The existence of solutions of equations is proved under the assumption that the nonlinear function b$b$ is monotone and a gradient of a convex, continuously differentiable function. The uniqueness is proved for Lipschitz-continuous elliptic parts. The existence of solutions of quasilinear variational inequalities is proved under stronger assumptions, namely, the nonlinear function defining the elliptic part is assumed to be a gradient and the function b$b$ to be Lipschitz continuous.     

On mathematical foundation of the Brownian motor theory

The paper contains mathematical justification of basic facts concerning the Brownian motor theory. The homogenization theorems are proved for the Brownian motion in periodic tubes with a constant drift. The study is based on an application of the Bloch decomposition. The effective drift and effective diffusivity are expressed in terms of the principal eigenvalue of the Bloch spectral problem on the cell of periodicity as well as in terms of the harmonic coordinate and the density of the invariant measure. We apply the formulas for the effective parameters to study the motion in periodic tubes with nearly separated dead zones.     

A nonlinear equation in Banach spaces and applications to the well-posedness of Cauchy problems

We analyze a nonlinear equation in Banach spaces, with the nonlinearity composed of multiple terms of different degrees. We prove a theorem regarding the existence of solutions for such equations. Moreover, we show how this result may be applied to obtain the well-posedness of various parabolic initial value problems.     

Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations

The purpose of the paper is to extend the principal eigenvalue and principal eigenfunction theory for time independent and periodic parabolic equations to random and general nonautonomous ones. In the random case, a notion of principal Lyapunov exponent serving as an analog of principal eigenvalue is introduced. It is shown that the principal Lyapunov exponent is deterministic and of simple multiplicity. It is also shown that there is a one-dimensional invariant random subbundle corresponding to the solutions that are globally defined and of the same sign, which serves as an analog of principal eigenfunction. In addition, monotonicity of the principal Lyapunov exponent with respect to the zero-order terms both in the equation and in the boundary condition is proved. When the second- and first-order terms are deterministic, it is proved that the principal Lyapunov exponent is greater than or equal to the principal eigenvalue of the associated time-averaged equation. In the general nonautonomous case, the concepts of principal spectrum, which serves as an analog of principal eigenvalue, and principal Lyapunov exponents are introduced. As is known, the principal spectrum is a compact interval. It is proved in the paper that the principal spectrum contains all the principal Lyapunov exponents. When the second and first-order terms are time independent, a lower estimate of the infimum of the principal spectrum is given in terms of an associated time-averaged equation.     

Spectrum of some integro-differential operators and stability of travelling waves

Spectral properties of some integro-differential operators on R¹ are studied. Characterisation of the principal eigenvalue is obtained in terms of the positive eigenfunction. These results are used to prove local and global stability of travelling waves and to find their speed.