# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

All Issues

## Communications on Pure & Applied Analysis

2003 , Volume 2 , Issue 4

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2003, 2(4): 425-445 doi: 10.3934/cpaa.2003.2.425 +[Abstract](207) +[PDF](267.4KB)
Abstract:
The paper presents recent advances in $p$-regularity theory, which has been developing successfully for the last twenty years. The main result of this theory gives a detailed description of the structure of the zero set of an irregular nonlinear mapping. We illustrate the theory with an application to degenerate problems in different fields of mathematics, which substantiates the general applicability of the class of $p$-regular problems. Moreover, the connection between singular problems and nonlinear mappings is shown. Amongst the applications, the structure of $p$-factor-operators is used to construct numerical methods for solving degenerate nonlinear equations and optimization problems.
2003, 2(4): 447-459 doi: 10.3934/cpaa.2003.2.447 +[Abstract](172) +[PDF](210.9KB)
Abstract:
We study Hamilton-Jacobi equations with upper semicontinuous initial data without convexity assumptions on the Hamiltonian. We analyse the behavior of generalized u.s.c solutions at the initial time $t=0$, and find necessary and sufficient conditions on the Hamiltonian such that the solution attains the initial data along a sequence (right accessibility).
2003, 2(4): 461-479 doi: 10.3934/cpaa.2003.2.461 +[Abstract](159) +[PDF](241.9KB)
Abstract:
Setting the Homogenization of Hamilton Jacobi equations in the geometry of the Heisenberg group, we study the convergence toward a solution of the limit equation i.e. the solution of the effective Hamiltonian, in particular we estimate the rate of convergence. The periodicity of the fast variable and the dilation are both taken compatibly with the group.
2003, 2(4): 481-494 doi: 10.3934/cpaa.2003.2.481 +[Abstract](179) +[PDF](233.8KB)
Abstract:
We consider the Cauchy problem for the generalized porous medium equation

$u_t = \Delta \Phi(u)$ in R$^n \times [0,T]$

$u(x)=u_0(x)$ on R$^n$

with the nonlinearity $\Phi(u)$. For the case of $\Phi(u)=\sum_{i=1}^m c_i u^{\alpha_i}$, we show the existence of a solution which smoothness depends on the exponents $\alpha_i$. Regardless of the regularity of the solution, we show the free-boundary is smooth. We also extend similar results for $\Phi(u)$ as an infinite sum.

2003, 2(4): 495-509 doi: 10.3934/cpaa.2003.2.495 +[Abstract](182) +[PDF](233.6KB)
Abstract:
This paper deals with a general approach to the rate-independent processes which may display hysteretic behaviour.This approach based on the two energy functionals,namely potential and dissipation functionals.Under some natural assumptions on these functionals we prove the compactness of the set of stable points which in turn leads to the existence of solutions of the problems under consideration.We present an application of our results to ferromagnetic models.
2003, 2(4): 511-520 doi: 10.3934/cpaa.2003.2.511 +[Abstract](158) +[PDF](200.7KB)
Abstract:
In this paper we prove the exponential decay as time goes to infinity of regular solutions of the problem for the wave equations with memory and weak damping

$u_{t t}-\Delta u+\int^t_0g(t-s)\Delta u(s)ds + \alpha u_{t}=0$ in $\hat Q$

where $\hat Q$ is a non cylindrical domains of $\mathbb R^{n+1}$ $(n\ge1)$ with the lateral boundary $\hat{\sum}$ and $\alpha$ is a positive constant.

2003, 2(4): 521-537 doi: 10.3934/cpaa.2003.2.521 +[Abstract](155) +[PDF](392.6KB)
Abstract:
In the current paper the dynamics of a mixed parabolic-gradient system is examined. The system, which is a coupled system of parabolic equations and gradient equations, acts as a first model for the outgrowth of axons in a developing nervous system. For modeling considerations it is relevant to know the influence of the parameters in the system and the source profiles in the parabolic equations on the dynamics. These subjects are discussed together with an approximation which uses the quasi-steady-state solutions of the parabolic equations instead of the parabolic equations themselves.
Some of the findings are demonstrated by numerical simulations.
2003, 2(4): 539-566 doi: 10.3934/cpaa.2003.2.539 +[Abstract](196) +[PDF](305.4KB)
Abstract:
The present work is devoted to analyze the Dirichlet problem for quasilinear elliptic equation related to some Caffarelli-Kohn-Nirenberg inequalities. Precisely the problem under study is,

-div $( |x|^{-p\gamma}|\nabla u|^{p-2}\nabla u)=f(x, u)\in L^1(\Omega),\quad x\in \Omega$

$u(x)=0$ on $\partial \Omega,$

where $-\infty<\gamma<\frac{N-p}{p}$, $\Omega$ is a bounded domain in $\mathbb R^N$ such that $0\in\Omega$ and $f(x,u)$ is a Caratheodory function under suitable conditions that will be stated in each section.

2003, 2(4): 567-577 doi: 10.3934/cpaa.2003.2.567 +[Abstract](204) +[PDF](484.4KB)
Abstract:
In this paper we examine the dynamics of two time-delay coupled relaxation oscillators of the van der Pol type. By integrating the governing differential-delay equations numerically, we find the various phase-locked motions including the in-phase and out-of-phase modes. Our computations reveal that depending on the strength of coupling ($\alpha$) and the amount of time-delay ($\tau$), the in-phase (out-of-phase) mode may be stable or unstable. There are also values of $\alpha$ and $\tau$ for which the in-phase and out-of-phase modes are both stable leading to birhythmicity. The results are illustrated in the $\alpha$-$\tau$ parameter plane. Near the boundaries between stability and instability of the in-phase (out-of-phase) mode, many other types of phase-locked motions can occur. Several examples of these phase-locked states are presented.
2003, 2(4): 579-589 doi: 10.3934/cpaa.2003.2.579 +[Abstract](205) +[PDF](183.8KB)
Abstract:
Several comparison theorems for oscillation and nonoscillation of neutral difference equations with continuous arguments are established. Some known results are included and improved. All results obtained in this paper are new.
2003, 2(4): 591-599 doi: 10.3934/cpaa.2003.2.591 +[Abstract](209) +[PDF](199.5KB)
Abstract:
In this note, we present a fast communication of a new bifurcation theory for nonlinear evolution equations, and its application to Rayleigh-Bénard Convection. The proofs of the main theorems presented will appear elsewhere. The bifurcation theory is based on a new notion of bifurcation, called attractor bifurcation. We show that as the parameter crosses certain critical value, the system bifurcates from a trivial steady state solution to an attractor with dimension between $m$ and $m+1$, where $m+1$ is the number of eigenvalues crosses the imaginary axis. Based on this new bifurcation theory, we obtain a nonlinear theory for bifurcation and stability of the solutions of the Boussinesq equations, and the onset of the Rayleigh-Bénard convection. In particular, we show that the problem bifurcates from the trivial solution an attractor $\mathcal A_R$ when the Rayleigh number $R$ crosses the first critical Rayleigh number $R_c$ for all physically sound boundary conditions.

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