381 research outputs found
Stationary localized structures and the effect of the delayed feedback in the Brusselator model
The Brusselator reaction-diffusion model is a paradigm for the understanding
of dissipative structures in systems out of equilibrium. In the first part of
this paper, we investigate the formation of stationary localized structures in
the Brusselator model. By using numerical continuation methods in two spatial
dimensions, we establish a bifurcation diagram showing the emergence of
localized spots. We characterize the transition from a single spot to an
extended pattern in the form of squares. In the second part, we incorporate
delayed feedback control and show that delayed feedback can induce a
spontaneous motion of both localized and periodic dissipative structures. We
characterize this motion by estimating the threshold and the velocity of the
moving dissipative structures.Comment: 18 pages, 11 figure
Negative Specific Heat of a Magnetically Self-Confined Plasma Torus
It is shown that the thermodynamic maximum entropy principle predicts
negative specific heat for a stationary magnetically self-confined
current-carrying plasma torus. Implications for the magnetic self-confinement
of fusion plasma are considered.Comment: 10p., LaTeX, 2 eps figure file
Steady states and dynamics of an autocatalytic chemical reaction model with decay
The dynamics and steady state solutions of an autocatalytic chemical reaction model with decay in the catalyst are considered. Nonexistence and existence of nontrivial steady state solutions are shown by using energy estimates, upper-lower solution method, and bifurcation theory. The effects of decay order, decay rate and diffusion rates to the dynamical behavior are discussed. (c) 2012 Published by Elsevier Inc
Multi-Peak Solutions for a Wide Class of Singular Perturbation Problems
In this paper we are
concerned with a wide class of singular perturbation problems arising
from such diverse fields as phase transitions,
chemotaxis, pattern formation,
population dynamics and chemical reaction theory.
We study the corresponding elliptic
equations in a bounded domain without any symmetry
assumptions. We assume that the
mean curvature of the boundary
has \overline{M} isolated, non-degenerate critical points.
Then we show that for any positive integer m\leq \overline{M}
there exists a stationary
solution with M local peaks which are attained on the boundary and
which lie close to these critical points.
Our method is based on Liapunov-Schmidt reduction
Stationary Multiple Spots for Reaction-Diffusion Systems
In this paper, we review
analytical methods for a rigorous study of the
existence and stability of stationary, multiple
spots for reaction-diffusion systems. We will
consider two classes of reaction-diffusion
systems: activator-inhibitor systems (such as
the Gierer-Meinhardt system) and
activator-substrate systems (such as the
Gray-Scott system or the Schnakenberg model).
The main ideas are presented in the context of
the Schnakenberg model, and these results are
new to the literature.
We will consider the systems in a
two-dimensional, bounded and smooth domain for small diffusion
constant of the activator.
Existence of multi-spots is proved using tools
from nonlinear functional analysis such as
Liapunov-Schmidt reduction and fixed-point
theorems. The amplitudes and positions of spots
follow from this analysis.
Stability is shown in two parts, for
eigenvalues of order one and eigenvalues
converging to zero, respectively. Eigenvalues
of order one are studied by deriving their
leading-order asymptotic behavior and reducing
the eigenvalue problem to a nonlocal eigenvalue
problem (NLEP). A study of the NLEP reveals a
condition for the maximal number of stable
spots.
Eigenvalues converging to zero are investigated
using a projection similar to Liapunov-Schmidt
reduction and conditions on the positions for
stable spots are derived. The Green's function
of the Laplacian plays a central role in the
analysis.
The results are interpreted in the biological,
chemical and ecological contexts. They are
confirmed by numerical simulations
Analysis of classes of superlinear semipositone problems with nonlinear boundary conditions
We study positive radial solutions for classes of steady state reaction diffusion problems on the exterior of a ball with both Dirichlet and nonlinear boundary conditions. We consider p-Laplacian problems (p>1) with reaction terms which are superlinear at infinity and semipositone. In the case p=2, using variational methods, we establish the existence of a solution, and via detailed analysis of the Green's function, we prove the positivity of the solution. In the case p[unequal to]2, we again use variational methods to establish the existence of a solution, but the positivity of the solution is achieved via sophisticated a priori estimates. In the case p[unequal to]2, the Green's function analysis is no longer available. Our results significantly enhance the literature on superlinear semipositone problems. Finally, we provide algorithms for the numerical generation of exact bifurcation curves for one-dimensional problems. In the autonomous case, we extend and analyze a quadrature method, and using nonlinear solvers in Mathematica, generate bifurcation curves. In the nonautonomous case, we employ shooting methods in Mathematica to generate bifurcation curves
Positive solutions of nonlinear elliptic boundary value problems
This dissertation focuses on the study of positive steady states to classes of nonlinear reaction diffusion (elliptic) systems on bounded domains as well as on exterior domains with Dirichlet boundary conditions. In particular, we study such systems in the challenging case when the reaction terms are negative at the origin, referred in the literature as semipositone problems. For the last 30 years, study of elliptic partial differential equations with semipositone structure has flourished not only for the semilinear case but also for quasilinear case. Here we establish several results that directly contribute to and enhance the literature of semipositone problems. In particular, we discuss existence, non-existence and multiplicity results for classes of superlinear as well as sublinear systems. We establish our results via the method of sub-super solutions, degree theory arguments, a priori bounds and energy analysis
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