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Discrete & Continuous Dynamical Systems - A

2015 , Volume 35 , Issue 4

Special issue on dissipative systems and applications with emphasis on nonlocal or nonlinear diffusion problems

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Preface: Special issue on dissipative systems and applications with emphasis on nonlocal or nonlinear diffusion problems
Georg Hetzer and Wenxian Shen
2015, 35(4): i-iii doi: 10.3934/dcds.2015.35.4i +[Abstract](223) +[PDF](148.3KB)
The strong interest in infinite dimensional dissipative systems originated from the observation that the dynamics of large classes of partial differential equations and systems resembles the behavior known from the modern theory of finite-dimensional dynamical systems. Reaction-diffusion problems are typical examples in this context. In biological applications linear diffusion represents random dispersal of a species, but in many cases other dispersal strategies occur, which has led to models with cross diffusion and nonlocal dispersal.

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Asymptotic behavior for a nonlocal diffusion equation on the half line
Carmen Cortázar, Manuel Elgueta, Fernando Quirós and Noemí Wolanski
2015, 35(4): 1391-1407 doi: 10.3934/dcds.2015.35.1391 +[Abstract](78) +[PDF](424.6KB)
We study the large time behavior of solutions to a nonlocal diffusion equation, $u_t=J*u-u$ with $J$ smooth, radially symmetric and compactly supported, posed in $\mathbb{R}_+$ with zero Dirichlet boundary conditions. In the far-field scale, $\xi_1\le xt^{-1/2}\le \xi_2$ with $\xi_1,\xi_2>0$, the asymptotic behavior is given by a multiple of the dipole solution for the local heat equation, hence $tu(x,t)$ is bounded above and below by positive constants in this region for large times. The proportionality constant is determined from a conservation law, related to the asymptotic first momentum. In compact sets, after scaling the solution by a factor $t^{3/2}$, it converges to a multiple of the unique stationary solution of the problem that behaves as $x$ at infinity. The precise proportionality factor is obtained through a matching procedure with the far-field limit. Finally, in the very far-field, $x\ge t^{1/2} g(t)$ with $g(t)\to\infty$, the solution is proved to be of order $o(t^{-1})$.
Finite mass solutions for a nonlocal inhomogeneous dispersal equation
Carmen Cortázar, Manuel Elgueta, Jorge García-Melián and Salomé Martínez
2015, 35(4): 1409-1419 doi: 10.3934/dcds.2015.35.1409 +[Abstract](79) +[PDF](383.1KB)
In this paper we study the asymptotic behavior of the following nonlocal inhomogeneous dispersal equation $$ u_t(x,t) = \int_{\mathbb{R}} J\left(\frac{x-y}{g(y)}\right) \frac{u(y,t)}{g(y)} dy -u(x,t) \qquad x\in \mathbb{R},\ t>0, $$ where $J$ is an even, smooth, probability density, and $g$, which accounts for a dispersal distance, is continuous and positive. We prove that if $g(|y|)\sim a |y|$ as $|y|\to + \infty$ for some $0 < a < 1$, there exists a unique (up to normalization) positive stationary solution, which is in $L^1(\mathbb{R})$. On the other hand, if $g(|y|)\sim |y|^p$, with $p > 2$ there are no positive stationary solutions. We also establish the asymptotic behavior of the solutions of the evolution problem in both cases.
Nonlocal refuge model with a partial control
Jérôme Coville
2015, 35(4): 1421-1446 doi: 10.3934/dcds.2015.35.1421 +[Abstract](74) +[PDF](537.2KB)
In this paper, we analyse the structure of the set of positive solutions of an heterogeneous nonlocal equation of the form: $$ \int_{\Omega} K(x, y)u(y)\,dy -\int_ {\Omega}K(y, x)u(x)\, dy + a_0u+\lambda a_1(x)u -\beta(x)u^p=0 \quad \text{in}\quad \Omega$$ where $\Omega\subset \mathbb{R}^n$ is a bounded domain, $K\in C(\mathbb{R}^n\times \mathbb{R}^n) $ is non-negative, $a_i,\beta \in C(\Omega)$ and $\lambda\in\mathbb{R}$. Such type of equation appears in some studies of population dynamics where the population evolves in a partially controlled heterogeneous landscape and disperses on long ranges. Under some fairly general assumptions on $K,a_i$ and $\beta$, we first establish a necessary and sufficient condition for the existence of a unique positive solution. Then, we analyse the structure of the set of positive solutions $(\lambda,u_\lambda)$ depending on the presence or absence of a refuge zone (i.e $\omega$ so that $\beta_{|\omega}\equiv 0$).
On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions
Gregorio Díaz and Jesús Ildefonso Díaz
2015, 35(4): 1447-1468 doi: 10.3934/dcds.2015.35.1447 +[Abstract](105) +[PDF](454.2KB)
This paper deals with several qualitative properties of solutions of some stationary equations associated to the Monge--Ampère operator on the set of convex functions which are not necessarily understood in a strict sense. Mainly, we focus our attention on the occurrence of a free boundary (separating the region where the solution $u$ is locally a hyperplane, thus, the Hessian $D^{2}u$ is vanishing, from the rest of the domain). In particular, our results apply to suitable formulations of the Gauss curvature flow and of the worn stones problems intensively studied in the literature.
Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions
Raúl Ferreira and Julio D. Rossi
2015, 35(4): 1469-1478 doi: 10.3934/dcds.2015.35.1469 +[Abstract](79) +[PDF](364.1KB)
In this paper we prove decay estimates for solutions to a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions, that is, $$u_t (x,t)=\int_{\Omega\cup\Omega_0} J(x-y) |u(y,t)-u(x,t)|^{p-2}(u(y,t)-u(x,t))\, dy$$ for $(x,t)\in \Omega\times \mathbb{R}^+$ and $u(x,t)=0$ in $ \Omega_0\times \mathbb{R}^+$. The proof of these estimates is based on bounds for the associated first eigenvalue.
Existence and multiplicity of segregated solutions to a cell-growth contact inhibition problem
Gonzalo Galiano, Sergey Shmarev and Julian Velasco
2015, 35(4): 1479-1501 doi: 10.3934/dcds.2015.35.1479 +[Abstract](61) +[PDF](506.3KB)
We study the Dirichlet problem for the cross-diffusion system \[ \partial_tu_i=div(a_iu_i\nabla (u_1+u_2))+f_i(u_1,u_2),\quad i=1,2,\quad a_i=const>0, \] in the cylinder $Q=\Omega\times (0,T]$. It is assumed that the functions $f_1(r,0)$, $f_2(0,s)$ are locally Lipschitz-continuous and $f_1(0,s)=0$, $f_2(r,0)=0$. It is proved that for suitable initial data $u_0$, $v_0$ the system admits segregated solutions $(u_1,u_2)$ such that $u_i\in L^{\infty}(Q)$, $u_1+u_2\in C^{0}(\overline{Q})$, $u_1+u_2>0$ and $u_1\cdot u_2=0$ everywhere in $Q$. We show that the segregated solution is not unique and derive the equation of motion of the surface $\Gamma$ which separates the parts of $Q$ where $u_1>0$, or $u_2>0$. The equation of motion of $\Gamma$ is a modification of the Darcy law in filtration theory.
On a climatological energy balance model with continents distribution
Arturo Hidalgo and Lourdes Tello
2015, 35(4): 1503-1519 doi: 10.3934/dcds.2015.35.1503 +[Abstract](324) +[PDF](493.3KB)
We present some results on the mathematical treatment of a global two-dimensional diffusive climate model with land - sea distribution. The model is based on a long time averaged energy balance and leads to a nonlinear parabolic equation for the averaged surface temperature. The spatial domain is a compact two-dimensional Riemannian manifold without boundary simulating the Earth surface with land - sea configuration. In the oceanic areas the model is coupled with a deep ocean model. The coupling is given by a dynamic and diffusive boundary condition. We study the existence of a bounded weak solution and its numerical approximation.
Strong positivity of continuous supersolutions to parabolic equations with rough boundary data
Dung Le
2015, 35(4): 1521-1530 doi: 10.3934/dcds.2015.35.1521 +[Abstract](60) +[PDF](388.5KB)
Parabolic equations given on domains with corners are considered. Under very weak assumption on the coefficients, it will be shown that continuous nonnegative supersolutions are strictly positive.
Invasion entire solutions in a competition system with nonlocal dispersal
Wan-Tong Li, Li Zhang and Guo-Bao Zhang
2015, 35(4): 1531-1560 doi: 10.3934/dcds.2015.35.1531 +[Abstract](88) +[PDF](521.1KB)
This paper is concerned with invasion entire solutions of a Lotka-Volterra competition system with nonlocal dispersal, which formulate a new invasion way of the stronger species to the weaker one. We first give the asymptotic behavior of traveling wave solutions at infinity. Then by the comparison principle and sub-super solutions method, we establish the existence of invasion entire solutions which behave as two monotone waves with different speeds and coming from both sides of $x$-axis.
Spiraling bifurcation diagrams in superlinear indefinite problems
Julián López-Gómez, Marcela Molina-Meyer and Andrea Tellini
2015, 35(4): 1561-1588 doi: 10.3934/dcds.2015.35.1561 +[Abstract](113) +[PDF](2076.1KB)
This paper computes and discusses a series of intricate global bifurcation diagrams for a class of one-dimensional superlinear indefinite boundary value problems arising in population dynamics, under non-homogeneous boundary conditions, measured by $M>0$; the main bifurcation parameter being the amplitude $b$ of the superlinear terms.
Pattern formation in a cross-diffusion system
Yuan Lou, Wei-Ming Ni and Shoji Yotsutani
2015, 35(4): 1589-1607 doi: 10.3934/dcds.2015.35.1589 +[Abstract](126) +[PDF](450.5KB)
In this paper we study the Shigesada-Kawasaki-Teramoto model [17] for two competing species with cross-diffusion. We prove the existence of spectrally stable non-constant positive steady states for high-dimensional domains when one of the cross-diffusion coefficients is sufficiently large while the other is equal to zero.
Spreading speeds and traveling waves of nonlocal monostable equations in time and space periodic habitats
Nar Rawal, Wenxian Shen and Aijun Zhang
2015, 35(4): 1609-1640 doi: 10.3934/dcds.2015.35.1609 +[Abstract](101) +[PDF](537.9KB)
This paper is devoted to the investigation of spatial spreading speeds and traveling wave solutions of monostable evolution equations with nonlocal dispersal in time and space periodic habitats. It has been shown in an earlier work by the first two authors of the current paper that such an equation has a unique time and space periodic positive stable solution $u^*(t,x)$. In this paper, we show that such an equation has a spatial spreading speed $c^*(\xi)$ in the direction of any given unit vector $\xi$. A variational characterization of $c^*(\xi)$ is given. Under the assumption that the nonlocal dispersal operator associated to the linearization of the monostable equation at the trivial solution $0$ has a principal eigenvalue, we also show that the monostable equation has a continuous periodic traveling wave solution connecting $u^*(\cdot,\cdot)$ and $0$ propagating in any given direction of $\xi$ with speed $c>c^*(\xi)$.
Avoidance behavior in intraguild predation communities: A cross-diffusion model
Daniel Ryan and Robert Stephen Cantrell
2015, 35(4): 1641-1663 doi: 10.3934/dcds.2015.35.1641 +[Abstract](82) +[PDF](446.1KB)
A cross-diffusion model of an intraguild predation community in a two-dimensional bounded domain where the intraguild prey employs a fitness based avoidance strategy is examined. The avoidance strategy employed is to increase motility in response to negative local fitness. Global existence of trajectories and the existence of a compact global attractor is proved. It is shown that if the intraguild prey has positive fitness at any point in the habitat when trying to invade, then it will be uniformly persistent in the system if its avoidance tendency is sufficiently strong. This type of movement strategy can lead to coexistence states where the intraguild prey is marginalized to areas with low resource productivity while the intraguild predator maintains high densities in regions with abundant resources, a pattern observed in many real world intraguild predation systems. Additionally, the effects of fitness based avoidance on eigenvalues in more general systems are discussed.
On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications
Wenxian Shen and Xiaoxia Xie
2015, 35(4): 1665-1696 doi: 10.3934/dcds.2015.35.1665 +[Abstract](71) +[PDF](496.4KB)
This paper is to investigate the dependence of the principal spectrum points of nonlocal dispersal operators on underlying parameters and to consider its applications. In particular, we study the effects of the spatial inhomogeneity, the dispersal rate, and the dispersal distance on the existence of the principal eigenvalues, the magnitude of the principal spectrum points, and the asymptotic behavior of the principal spectrum points of nonlocal dispersal operators with Dirichlet type, Neumann type, and periodic boundary conditions in a unified way. We also discuss the applications of the principal spectral theory of nonlocal dispersal operators to the asymptotic dynamics of two species competition systems with nonlocal dispersal operators.
Clines with directional selection and partial panmixia in an unbounded unidimensional habitat
Linlin Su and Thomas Nagylaki
2015, 35(4): 1697-1741 doi: 10.3934/dcds.2015.35.1697 +[Abstract](163) +[PDF](2662.4KB)
In geographically structured populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into diallelic single-locus clines maintained by migration and arbitrary directional selection in an unbounded unidimensional habitat is investigated. The population density is uniform. Migration and selection are both weak; the former is homogeneous and symmetric. Suppose that the spatial factor $g(x)$ in the scaled selection term satisfies $g'(x)\ge 0$ for every $x$ and the limiting values $p_{\pm}=p(\pm\infty)$ of the equilibrium gene frequency $p(x)$ exist and satisfy $0 < p_- < p_+ < 1$. Then (i) $p_- < p(x) < p_+$ for every $x\in\mathbb{R}$; (ii) $p'(x)>0$ for every $x\in\mathbb{R}$; (iii) for each given pair $p_-$ and $p_+$, there exists at most one equilibrium $p(x)$; (iv) the existence and multiplicity of $p(x)$ are determined under various conditions; (v) given two pairs $p_{1\pm}$ and $p_{2\pm}$ such that $p_{1\pm}>p_{2\pm}$, the ordering $p_1(x)>p_2(x)$ holds for every $x\in\mathbb{R}$; (vi) if the factor $f(p)$ $(\ge 0)$ in the scaled selection term is unimodal, as is the case when the selection coefficients do not depend on $p$, and in some other situations, then $p_{1\pm}>p_{2\pm}$; and (vii) in a step-environment that changes sign at $x=0$, under some assumptions on $f(p)$, the equilibria satisfy $p''(x)>0$ if $ x<0$ and $p''(x)<0$ if $x>0$.
Traveling waves of a mutualistic model of mistletoes and birds
Chuncheng Wang, Rongsong Liu, Junping Shi and Carlos Martinez del Rio
2015, 35(4): 1743-1765 doi: 10.3934/dcds.2015.35.1743 +[Abstract](74) +[PDF](786.1KB)
The existences of an asymptotic spreading speed and traveling wave solutions for a diffusive model which describes the interaction of mistletoe and bird populations with nonlocal diffusion and delay effect are proved by using monotone semiflow theory. The effects of different dispersal kernels on the asymptotic spreading speeds are investigated through concrete examples and simulations.

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