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Discrete & Continuous Dynamical Systems - A

2002 , Volume 8 , Issue 2

(A Special Issue: Current Developments in PDE)

Guest Editors: Carlos Conca, Manuel del Pino, Patricio Felmer, and Raul Manasevich

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Nonvariational elliptic systems
Claudianor O. Alves and Djairo G. De Figueiredo
2002, 8(2): 289-302 doi: 10.3934/dcds.2002.8.289 +[Abstract](164) +[PDF](163.4KB)
Abstract:
In this paper, we present new concepts of sublinearity and superlinearity for elliptic systems of the form

(P)$ \qquad\qquad -\Delta u = f(u, v), -\nabla v = g(u, v)$ in $\Omega$

$u = v = 0$ on $\partial \Omega$

where $\Omega$ is a smooth bounded domain and $f, g$ are continuous functions. Then we prove existence of positive solutions for such systems via topological methods.

Solitons and Bohmian mechanics
Vieri Benci
2002, 8(2): 303-317 doi: 10.3934/dcds.2002.8.303 +[Abstract](228) +[PDF](210.3KB)
Abstract:
This article proposes a set of ideas concerning the introduction of nonlinear analysis, particularly nonlinear PDE, in the theory of particles. The Quantum Mechanics theory is essentially a linear theory, due mainly to the fact that there was a the lack of nonlinear mathematics at the time of the discoveries in particle physics. The main idea is to perturb the Schroedinger equation by a nonlinear term. This nonlinear term has two main parts, a second order quasilinear differential operator responsible for the smoothing of the solutions and a nonlinear 0-order term with a singularity providing topology to the space. By minimizing the energy functional, solutions to the equation are obtained in each topological class. Then the qualitative properties of the soliton is analyzed. By rescaling arguments, the asymptotic behavior of the static solutions is studied. Next the evolution is studied, deriving stability of the soliton and the guidance formula. In this way the equations of Bohmian Mechanics are obtained. Most proofs are omitted, but in all cases a proper reference is given.
Some Dirichlet problems with bad coercivity
Lucio Boccardo
2002, 8(2): 319-329 doi: 10.3934/dcds.2002.8.319 +[Abstract](240) +[PDF](217.5KB)
Abstract:
Here I summarize and I translate in English my lectures, devoted to some Dirichlet problems with a common feature: bad coercivity.
Very simple examples are:

$-\Delta u = f(x)\in L^1(\Omega)\quad$ in $\Omega$

$u = 0\quad$ on $\partial \Omega$

since the term $\int_\Omega f(x)v(x)$ does not make sense, if $f\in L^1(\Omega), v\in W^{1,2}_0(\Omega)$;

-div$(\frac{\nabla u}{(1+|u|)^\theta})=f(x)\in L^2(\Omega)\quad$ in $\Omega$

$u = 0\quad$ on $\partial \Omega$

Since the term $\int_\Omega \frac{|\nabla v|^2}{(1+|v|)^\theta}$ goes to zero, if $v$ is large.

Topics in regularity and qualitative properties of solutions of nonlinear elliptic equations
Xavier Cabré
2002, 8(2): 331-359 doi: 10.3934/dcds.2002.8.331 +[Abstract](267) +[PDF](318.4KB)
Abstract:
In these notes we describe the Alexandroff-Bakelman-Pucci estimate and the Krylov-Safonov Harnack inequality for solutions of $Lu = f(x)$, where $L$ is a second order uniformly elliptic operator in nondivergence form with bounded measurable coefficients. It is the purpose of these notes to present several applications of these inequalities to the study of nonlinear elliptic equations.
The first topic is the maximum principle for the operator $L$, and its applications to the moving planes method and to symmetry properties of positive solutions of semilinear problems. The second topic is a short introduction to the regularity theory for solutions of fully nonlinear elliptic equations. We prove a $C^{1,\alpha}$ estimate for classical solutions, we introduce the notion of viscosity solution, and we study Jensen’s approximate solutions.
An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation
Jean Dolbeault
2002, 8(2): 361-380 doi: 10.3934/dcds.2002.8.361 +[Abstract](289) +[PDF](294.6KB)
Abstract:
The purpose of kinetic equations is the description of dilute particle gases at an intermediate scale between the microscopic scale and the hydrodynamical scale. By dilute gases, one has to understand a system with a large number of particles, for which a description of the position and of the velocity of each particle is irrelevant, but for which the decription cannot be reduced to the computation of an average velocity at any time $t\in \mathbb R$ and any position $x\in \mathbb R^d:$ one wants to take into account more than one possible velocity at each point, and the description has therefore to be done at the level of the phase space – at a statistical level – by a distribution function $f(t, x, v)$.
This course is intended to make an introductory review of the literature on kinetic equations. Only the most important ideas of the proofs will be given. The two main examples we shall use are the Vlasov-Poisson system and the Boltzmann equation in the whole space.
An overview on linear and nonlinear Dirac equations
Maria J. Esteban and Eric Séré
2002, 8(2): 381-397 doi: 10.3934/dcds.2002.8.381 +[Abstract](238) +[PDF](267.8KB)
Abstract:
In this paper we review the variational treatment of linear and nonlinear eigenvalue problems involving the Dirac operator. These problems arise when searching for bound states and bound energies of electrons submitted to external or self-consistent interactions in a relativistic framework. The corresponding energy functionals are totally indefinite and that creates difficulties to define variational principles related to those equations. Here we describe recent works dealing with this kind of problems. In particular we describe the solutions found to linear Dirac equations with external potential, the Maxwell-Dirac equations, some nonlinear Dirac equations with local nonlinear terms and the Dirac-Fock equations arising in the description of atoms and molecules.
The problem Of blow-up in nonlinear parabolic equations
Victor A. Galaktionov and Juan-Luis Vázquez
2002, 8(2): 399-433 doi: 10.3934/dcds.2002.8.399 +[Abstract](753) +[PDF](369.4KB)
Abstract:
The course aims at presenting an introduction to the subject of singularity formation in nonlinear evolution problems usually known as blowup. In short, we are interested in the situation where, starting from a smooth initial configuration, and after a first period of classical evolution, the solution (or in some cases its derivatives) becomes infinite in finite time due to the cumulative effect of the nonlinearities. We concentrate on problems involving differential equations of parabolic type, or systems of such equations.
A first part of the course introduces the subject and discusses the classical questions addressed by the blow-up theory. We propose a list of main questions that extends and hopefully updates on the existing literature. We also introduce extinction problems as a parallel subject.
In the main bulk of the paper we describe in some detail the developments in which we have been involved in recent years, like rates of growth and pattern formation before blow-up, the characterization of complete blow-up, the occurrence of instantaneous blow-up (i.e., immediately after the initial moment) and the construction of transient blow-up patterns (peaking solutions), as well as similar questions for extinction.
In a final part we have tried to give an idea of interesting lines of current research. The survey concludes with an extensive list of references. Due to the varied and intense activity in the field both aspects are partial, and reflect necessarily the authors' tastes.
O.D.E. type behavior of blow-up solutions of nonlinear heat equations
Frank Merle and Hatem Zaag
2002, 8(2): 435-450 doi: 10.3934/dcds.2002.8.435 +[Abstract](204) +[PDF](268.2KB)
Abstract:
We show that all global (in time and in space) and bounded solutions of the vector-valued equation

$\frac{\partial w}{\partial s}=\Delta w-\frac{1}{2}y \cdot \nabla w -\frac{w}{p-1}+|w|^{p-1}w$

(where $w : \mathbb R^N\times \mathbb R \to \mathbb R^M, p> 1$ and $(N - 2)p < N + 2$) are independent of space and completely explicit. We then derive from this various uniform estimates and a uniform localization property for blow-up solutions of $\partial_t u=\Delta u + |u|^{p-1}u$.

Nonlinear parabolic differential equations and inequalities
Wolfgang Walter
2002, 8(2): 451-468 doi: 10.3934/dcds.2002.8.451 +[Abstract](225) +[PDF](243.4KB)
Abstract:
The paper starts with a short review on the history of PDEs and their related inequalities. We present a unified approach to the basic parabolic differential inequalities. It starts in Section 2 with the famous Lemma of Nagumo and leads to recent and new result on parabolic equations with singular nonlinear elliptic operators (7) that include the p-Laplacian and the operator of capillary surfaces. Sections 3 and 4 deal with the one-dimensional case and radial solutions. A Semi-Strong Minimum Principle is found in Section 6.
Controllability of partial differential equations and its semi-discrete approximations
Enrique Zuazua
2002, 8(2): 469-513 doi: 10.3934/dcds.2002.8.469 +[Abstract](244) +[PDF](400.0KB)
Abstract:
In these notes we analyze some problems related to the controllability and observability of partial differential equations and its space semidiscretizations. First we present the problems under consideration in the classical examples of the wave and heat equations and recall some well known results. Then we analyze the $1-d$ wave equation with rapidly oscillating coefficients, a classical problem in the theory of homogenization. Then we discuss in detail the null and approximate controllability of the constant coefficient heat equation using Carleman inequalities. We also show how a fixed point technique may be employed to obtain approximate controllability results for heat equations with globally Lipschitz nonlinearities. Finally we analyze the controllability of the space semi-discretizations of some classical PDE models: the Navier-Stokes equations and the $1-d$ wave and heat equations. We also present some open problems.

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