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1078-0947
eISSN:
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Discrete & Continuous Dynamical Systems - A
January 2008 , Volume 21 , Issue 1
A special issue dedicated to Edward Norman Dancer
on the occasion of his 60th birthday
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Professor Edward Norman Dancer, known to his friends and colleagues as Norm or Norman, was born in Bundaberg in north Queensland, Australia in December 1946. He graduated from the Australian National University in 1968 with first class honours, and continued to obtain a PhD from the University of Cambridge in 1972. He was appointed a Lecturer in 1973 at the University of New England, Armidale, where he received a Personal Chair in 1987. He left Armidale in 1993 to become a Professor of Mathematics at the University of Sydney, a position he has held since. He was elected a Fellow of the Australian Academy of Science (FAA) in 1996. He has held distinguished visiting professorships at many institutions in Europe and North America. In 2002 he received the prestigious Alexander von Humboldt Research Award, the highest prize awarded in Germany to foreign scientists.
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To understand the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, a spatial SIS reaction-diffusion model is studied, with the focus on the existence, uniqueness and particularly the asymptotic profile of the steady-states. First, the basic reproduction number $\R_{0}$ is defined for this SIS PDE model. It is shown that if $\R_{0} < 1$, the unique disease-free equilibrium is globally asymptotic stable and there is no endemic equilibrium. If $\R_{0} > 1$, the disease-free equilibrium is unstable and there is a unique endemic equilibrium. A domain is called high (low) risk if the average of the transmission rates is greater (less) than the average of the recovery rates. It is shown that the disease-free equilibrium is always unstable $(\R_{0} > 1)$ for high-risk domains. For low-risk domains, the disease-free equilibrium is stable $(\R_{0} < 1)$ if and only if infected individuals have mobility above a threshold value. The endemic equilibrium tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. Surprisingly, the density of susceptibles for this limiting disease-free equilibrium, which is always positive on the subdomain where the transmission rate is less than the recovery rate, must also be positive at some (but not all) places where the transmission rates exceed the recovery rates.
For $\Omega$ a bounded open set in $\R^N$ we consider the space $H^1_0(\bar{\Omega})=${$u_{|_{\Omega}}: u \in H^1(\R^N):$ $u(x)=0$ a.e. outside $\bar{\Omega}$}. The set $\Omega$ is called stable if $H^1_0(\Omega)=H^1_0(\bar{\Omega})$. Stability of $\Omega$ can be characterised by the convergence of the solutions of the Poisson equation
$ -\Delta u_n = f$ in $D(\Omega_n)^$´, $ u_n \in H^1_0(\Omega_n)$
and also the Dirichlet Problem with respect to $\Omega_n$ if $\Omega_n$ converges to $\Omega$ in a sense to be made precise. We give diverse results in this direction, all with purely analytical tools not referring to abstract potential theory as in Hedberg's survey article [Expo. Math. 11 (1993), 193--259]. The most complete picture is obtained when $\Omega$ is supposed to be Dirichlet regular. However, stability does not imply Dirichlet regularity as Lebesgue's cusp shows.
This paper is concerned with time-dependent reaction-diffusion equations of the following type:
$\partial_t u=$Δ$u+f(x-cte,u),t>0,x\in\R^N.$
These kind of equations have been introduced in [1] in
the case $N=1$ for studying the impact of a climate shift on the
dynamics of a biological species.
In the present paper, we first extend the results of
[1] to arbitrary dimension $N$ and to a greater
generality in the assumptions on $f$. We establish a necessary
and sufficient condition for the existence of travelling wave
solutions, that is, solutions of the type $u(t,x)=U(x-cte)$. This
is expressed in terms of the sign of the generalized principal eigenvalue $\l$ of
an associated linear elliptic operator in $\R^N$. With this
criterion, we then completely describe the large time dynamics for
this equation. In particular, we characterize situations in which
there is either extinction or persistence.
Moreover, we consider the problem obtained by adding a term
$g(x,u)$ periodic in $x$ in the direction $e$:
$\partial_t u=$Δ$u+f(x-cte,u)+g(x,u),t>0,x\in\R^N.$
Here, $g$ can be viewed as representing geographical characteristics of the territory which are not subject to shift. We derive analogous results as before, with $\l$ replaced by the generalized principal eigenvalue of the parabolic operator obtained by linearization about $u\equiv0$ in the whole space. In this framework, travelling waves are replaced by pulsating travelling waves, which are solutions of the form $U(t,x-cte)$, with $U(t,x)$ periodic in $t$. These results still hold if the term $g$ is also subject to the shift, but on a different time scale, that is, if $g(x,u)$ is replaced by $g(x-c'te,u)$, with $c'\in\R$.
We review some recent existence results for the elliptic problem $\Delta u + u^p =0$, $u>0$ in an exterior domain, $\Omega = \R^N\setminus \D$ under zero Dirichlet and vanishing conditions, where $\D$ is smooth and bounded, and $p>\frac{N+2}{N-2}$. We prove that the associated Dirichlet problem has infinitely many positive solutions. We establish analogous results for the standing-wave supercritical nonlinear Schrödinger equation $\Delta u - V(x)u + u^p = 0 $ where $V\ge 0$ and $V(x) = o(|x|^{-2})$ at infinity. In addition we present existence results for the Dirichlet problem in bounded domains with a sufficiently small spherical hole if $p$ differs from certain sequence of resonant values which tends to infinity.
Asymptotics of solutions to Schrödinger equations with singular dipole-type potentials are investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inverse-square functions. Both the linear and the semilinear (critical and subcritical) cases are considered.
We consider a stationary nonlinear Schröodinger equation with a repulsive delta-function impurity in one space dimension. This equation admits a unique positive solution and this solution is even. We prove that it is a minimizer of the associated energy on the subspace of even functions of $H^1(\R, \C)$, but not on all $H^1(\R, \C)$, and study its orbital stability.
For $N\geq3$ and $p>1$, we consider the nonlinear Schrödinger equation
$i\partial_{t}w+\Delta_{x}w+V(x) |w| ^{p-1}w=0\text{ where }w=w(t,x):\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{C}$
with a potential $V$ that decays at infinity like $| x|^{-b}$ for some $b\in (0,2)$. A standing wave is a solution of the form
$w(t,x)=e^{i\lambda t}u(x)\text{ where }\lambda>0\text{ and }u:\mathbb{R}^{N}\rightarrow\mathbb{R}.$
For $ 1 < p < 1+(4-2b)/(N-2)$, we establish the existence of a $C^1$-branch of standing waves parametrized by frequencies $\lambda $ in a right neighbourhood of $0$. We also prove that these standing waves are orbitally stable if $ 1 < p < 1+(4-2b)/N$ and unstable if $1+(4-2b)/N < p < 1+(4-2b)/(N-2)$.
Selfdual variational theory -- developed in [8] and [9] -- allows for the superposition of appropriate "boundary" Lagrangians with "interior" Lagrangians, leading to a variational formulation and resolution of problems with various linear and nonlinear boundary constraints that are not amenable to standard Euler-Lagrange theory. The superposition of several selfdual Lagrangians is also possible in many natural settings, leading to a variational resolution of certain differential systems. These results are applied to nonlinear transport equations with prescribed exit values, Lagrangian intersections of convex-concave Hamiltonian systems, initial-value problems of dissipative systems, as well as evolution equations with periodic and anti-periodic solutions.
Let us consider the problem
$-\Delta u+a(|x|)u=\lambda e^u$in$\ B_1,$ (0.1)
$u=0$ on$ \partial B_1.$
where $B_1$ is the unit ball in $R^N$, $N\ge2$, $\lambda>0$ and $a(|x|)\ge0$ is a smooth radial function.
Under some suitable assumptions on the regular part of the Green function of the operator
$-u''- \frac{N-1}{r}u+a(r)u$, we prove the existence of a radial solution to (0.1)
for $\lambda$ small enough.
We consider a class of $2m$ components competition-diffusion systems which involve $m$ parabolic equations as well as $m$ ordinary differential equation, and prove the strong convergence in $L^p$ of a subsequence of each component as the reaction coefficient tends to infinity. In the special case of $4$ components the solution of this system converges to that of a Stefan problem.
The precise dynamics of a reaction-diffusion model of autocatalytic chemical reaction is described. It is shown that exactly either one, two, or three steady states exists, and the solution of dynamical problem always approaches to one of steady states in the long run. Moreover it is shown that a global codimension one manifold separates the basins of attraction of the two stable steady states. Analytic ingredients include exact multiplicity of semilinear elliptic equation, the theory of monotone dynamical systems and the theory of asymptotically autonomous dynamical systems.
The global asymptotic stability with phase shift of traveling wave fronts of minimal speed, in short minimal fronts, is established for a large class of monostable lattice equations via the method of upper and lower solutions and a squeezing technique.
We consider a class of variational equations with exponential nonlinearities on compact surfaces. From considerations involving the Moser-Trudinger inequality, we characterize some sublevels of the Euler-Lagrange functional in terms of the topology of the surface and of the data of the equation. This is used together with a min-max argument to obtain existence results.
We consider the problem
$-\Delta u= |u|^{\frac4{N-2}} u \mbox{ in } \Omega \setminus \{B(\xi_1,\varepsilon)\cup B(\xi_2,\varepsilon)\},$
$ u = 0 \mbox{ on } \partial( \Omega \setminus
\{B(\xi_1,\varepsilon)\cup B(\xi_2,\varepsilon)\}),$
where $\Omega$ is a smooth bounded domain in $R^N$, $N\ge 3,$ $\xi_1,$ $\xi_2$ are different points in $\Omega$ and ε is a small positive parameter. We show that, for ε small enough, the equation has at least one pair of sign changing solutions, whose positive and negative parts concentrate at $\xi_1$ and $\xi_2$ as ε goes to zero.
We are interested in the time decay estimates of global solutions of the semilinear parabolic equation $u_t= \Delta u+|u|^{p-1}u$ in $\R^N\times\R^+$, where $p>1$. We find several new sufficient and/or necessary conditions guaranteeing that the solution for $t$ large behaves like the solution of the linear heat equation or has the self-similar decay. We are particularly interested in the behaviour of threshold solutions lying on the borderline between global existence and blow-up.
Using minimization arguments and a limit process, we construct a family of solutions which undergo an infinite number of transitions for an Allen-Cahn model equation.
We study positive solutions of the equation $\varepsilon^2\Delta u - u + u^{\frac{n+2}{n-2}} = 0$, where $n=3,4,5$, and $\varepsilon > 0$ is small, with Neumann boundary condition in a smooth bounded domain $\Omega \subset R^n$. We prove that, along some sequence $\{\varepsilon_j \}$ with $ \varepsilon_j \to 0$, there exists a solution with an interior bubble at an innermost part of the domain and a boundary layer on the boundary $\partial\Omega$.
Let $M^{N\times n}$ be the space of real $N\times n$ matrices. We construct non-negative quasiconvex functions $F:M^{N\times n}\to R_+$ of quadratic growth whose zero sets are the graphs $\Gamma_f$ of certain Lipschitz mappings $f:K\subset E\to$ $E^$⊥, where $E\subset M^{N\times n}$ is a linear subspace without rank-one matrices, $K$ a compact subset of $E$ with $E^$⊥ its orthogonal complement. We show that the gradients $DF:M^{N\times n}\to M^{N\times n}$ are strictly quasimonotone mappings and satisfy certain growth and coercivity conditions so that the variational integrals $u\to \int_{\Omega}F(Du(x))dx$ satisfy the Palais-Smale compactness condition in $W^{1,2}$. If $K$ is a smooth compact manifold of $E$ without boundary and the Lipschtiz mapping $f$ is of class $C^2$, then the closed $\epsilon$-neighbourhoods $(\Gamma_f)_\epsilon$ for small $\epsilon>0$ are quasiconvex sets.
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