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Discrete & Continuous Dynamical Systems - A

2012 , Volume 32 , Issue 11

Special Issue
on Nonlinear Elliptic and Parabolic Problems

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Preface
Julián López-Gómez
2012, 32(11): i-iii doi: 10.3934/dcds.2012.32.11i +[Abstract](139) +[PDF](111.9KB)
Abstract:
Nonlinear parabolic problems have a huge interest in mathematical sciences as they model a wide variety of real world systems whose deep understanding is a challenge for the advance of human learning. As a matter of fact, they have shown to be a milestone for the generation of knowledge and innovation in the theory of ecosystems, in fluid dynamics, in energy transfer, in reaction-diffusion, in chemotaxis, and even in the evaluation of stock options, where dispersal and diffusion coefficients are interchanged by volatility rates. As nonlinear elliptic equations describe the steady-state solutions of parabolic problems, their study is imperative for ascertaining the dynamics of all these mathematical models.

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Boundary estimates for solutions of weighted semilinear elliptic equations
Claudia Anedda and Giovanni Porru
2012, 32(11): 3801-3817 doi: 10.3934/dcds.2012.32.3801 +[Abstract](194) +[PDF](408.0KB)
Abstract:
Let $b(x)$ be a positive function in a bounded smooth domain $\Omega\subset R^N$, and let $f(t)$ be a positive non decreasing function on $(0,\infty)$ such that $\lim_{t\to\infty}f(t)=\infty$. We investigate boundary blow-up solutions of the equation $\Delta u=b(x)f(u)$. Under appropriate conditions on $b(x)$ as $x$ approaches $\partial\Omega$ and on $f(t)$ as $t$ goes to infinity, we find a second order approximation of the solution $u(x)$ as $x$ goes to $\partial\Omega$.
    We also investigate positive solutions of the equation $\Delta u+(\delta(x))^{2\ell}u^{-q}=0$ in $\Omega$ with $u=0$ on $\partial\Omega$, where $\ell\ge 0$, $q>3+2\ell$ and $\delta(x)$ denotes the distance from $x$ to $\partial\Omega$. We find a second order approximation of the solution $u(x)$ as $x$ goes to $\partial\Omega$.
Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$
Santiago Cano-Casanova
2012, 32(11): 3819-3839 doi: 10.3934/dcds.2012.32.3819 +[Abstract](179) +[PDF](461.4KB)
Abstract:
In this paper is proved that the Strong Maximum Principle is satisfied for a wide class of linear elliptic boundary value problems of mixed type in an annulus of $\mathbb{R}^N$, $N\geq 1$, provided it is thin enough. The coercive character of these boundary value problems is obtained thanks to the characterization of the Strong Maximum Principle found in [3], proving that the principal eigenvalue associated to each boundary value problem may be as large as we wish, independently of the weight on the boundary, by taking the annulus thin enough.
Dynamics of a reaction-diffusion-advection model for two competing species
Xinfu Chen, King-Yeung Lam and Yuan Lou
2012, 32(11): 3841-3859 doi: 10.3934/dcds.2012.32.3841 +[Abstract](381) +[PDF](447.2KB)
Abstract:
We study the dynamics of a reaction-diffusion-advection model for two competing species in a spatially heterogeneous environment. The two species are assumed to have the same population dynamics but different dispersal strategies: both species disperse by random diffusion and advection along the environmental gradient, but with different random dispersal and/or advection rates. Given any advection rates, we show that three scenarios can occur: (i) If one random dispersal rate is small and the other is large, two competing species coexist; (ii) If both random dispersal rates are large, the species with much larger random dispersal rate is driven to extinction; (iii) If both random dispersal rates are small, the species with much smaller random dispersal rate goes to extinction. Our results suggest that if both advection rates are positive and equal, an intermediate random dispersal rate may evolve. This is in contrast to the case when both advection rates are zero, where the species with larger random dispersal rate is always driven to extinction.
On domain perturbation for super-linear Neumann problems and a question of Y. Lou, W-M Ni and L. Su
E. N. Dancer
2012, 32(11): 3861-3869 doi: 10.3934/dcds.2012.32.3861 +[Abstract](185) +[PDF](316.5KB)
Abstract:
We prove a new domain variation result for Neumann problems and apply it to give new examples of non-uniqueness of positive solutions.
Anti-angiogenic therapy based on the binding to receptors
Manuel Delgado, Cristian Morales-Rodrigo and Antonio Suárez
2012, 32(11): 3871-3894 doi: 10.3934/dcds.2012.32.3871 +[Abstract](185) +[PDF](450.1KB)
Abstract:
This paper deals with a nonlinear system of partial differential equations modeling the effect of an anti-angiogenic therapy based on an agent that binds to specific receptors of the endothelial cells. We study the time-dependent problem as well as the stationary problem associated to it.
On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials
Veronica Felli, Alberto Ferrero and Susanna Terracini
2012, 32(11): 3895-3956 doi: 10.3934/dcds.2012.32.3895 +[Abstract](178) +[PDF](806.1KB)
Abstract:
The asymptotic behavior of solutions to Schrödinger equations with singular homogeneous potentials is investigated. Through an Almgren type monotonicity formula and separation of variables, we describe the exact asymptotics near the singularity of solutions to at most critical semilinear elliptic equations with cylindrical and quantum multi-body singular potentials. Furthermore, by an iterative Brezis-Kato procedure, pointwise upper estimate are derived.
Longtime behavior of solutions to chemotaxis-proliferation model with three variables
Doan Duy Hai and Atsushi Yagi
2012, 32(11): 3957-3974 doi: 10.3934/dcds.2012.32.3957 +[Abstract](208) +[PDF](421.7KB)
Abstract:
In this paper, we construct a global solution to a mathematical model presented by Murray [18] and investigate longtime behavior of solution. For any initial profile, the solution is proven to tend to a homogeneous stationary solution as $t \rightarrow \infty$. This result is highly congruent with the prediction in [18] which is said that the solution would tend to zero as $t \rightarrow \infty$.
Characterization of turing diffusion-driven instability on evolving domains
Georg Hetzer, Anotida Madzvamuse and Wenxian Shen
2012, 32(11): 3975-4000 doi: 10.3934/dcds.2012.32.3975 +[Abstract](194) +[PDF](508.5KB)
Abstract:
In this paper we establish a general theoretical framework for Turing diffusion-driven instability for reaction-diffusion systems on time-dependent evolving domains. The main result is that Turing diffusion-driven instability for reaction-diffusion systems on evolving domains is characterised by Lyapunov exponents of the evolution family associated with the linearised system (obtained by linearising the original system along a spatially independent solution). This framework allows for the inclusion of the analysis of the long-time behavior of the solutions of reaction-diffusion systems. Applications to two special types of evolving domains are considered: (i) time-dependent domains which evolve to a final limiting fixed domain and (ii) time-dependent domains which are eventually time periodic. Reaction-diffusion systems have been widely proposed as plausible mechanisms for pattern formation in morphogenesis.
Blow-up phenomena in reaction-diffusion systems
Monica Marras and Stella Vernier Piro
2012, 32(11): 4001-4014 doi: 10.3934/dcds.2012.32.4001 +[Abstract](194) +[PDF](389.1KB)
Abstract:
In this paper we deal with the blow-up phenomena of solutions to two different classes of reaction-diffusion systems coupled through nonlinearities with nonlinear boundary conditions. By using a differential inequality technique, we derive upper and lower bounds for the blow-up time, if blow-up occurs. Moreover by introducing suitable auxiliary functions, we give sufficient conditions on data in order to obtain global existence.
Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities
Jean Mawhin
2012, 32(11): 4015-4026 doi: 10.3934/dcds.2012.32.4015 +[Abstract](153) +[PDF](382.0KB)
Abstract:
Using a Lusternik-Schnirelman type multiplicity result for some indefinite functionals due to Szulkin, the existence of at least $n+1$ geometrically distinct T-periodic solutions is proved for the relativistic-type Lagrangian system $$(\phi(q'))' + \nabla_qF(t,q) = h(t),$$ where $\phi$ is an homeomorphism of the open ball $B_a \subset \mathbb{R}n$ onto $\mathbb{R}n$ such that $\phi(0) = 0$ and $\phi = \nabla \Phi$, $F$ is $T_j$-periodic in each variable $q_j$ and $h \in L^s(0,T;\mathbb{R}n)$ $(s > 1)$ has mean value zero. Application is given to the coupled pendulum equations $$\left(\frac{q'_j}{\sqrt{1 - \|q\|^2}}\right)' + A_j \sin q_j = h_j(t) \quad (j = 1,\ldots,n).$$ Similar results are obtained for the radial solutions of the homogeneous Neumann problem on an annulus in $\mathbb{R}n$ centered at $0$ associated to systems of the form $$\nabla \cdot \left(\frac{\nabla w_i}{\sqrt{1 - \sum_{j=1}^n \|\nabla w_j\|^2}}\right) + \partial_{w_j} G(\|x\|,w) = h_i(\|x\|), \quad (i = 1,\ldots,n),$$ involving the extrinsic mean curvature operator in a Minkovski space.
Asymptotic behavior of singular solutions for a semilinear parabolic equation
Shota Sato and Eiji Yanagida
2012, 32(11): 4027-4043 doi: 10.3934/dcds.2012.32.4027 +[Abstract](190) +[PDF](367.8KB)
Abstract:
We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. It is known that in some range of parameters, this equation has a family of singular steady states with ordered structure. Our concern in this paper is the existence of time-dependent singular solutions and their asymptotic behavior. In particular, we prove the convergence of solutions to singular steady states. The method of proofs is based on the analysis of a related linear parabolic equation with a singular coefficient and the comparison principle.
Periodic solutions for a class of second order ODEs with a Nagumo cubic type nonlinearity
Chiara Zanini and Fabio Zanolin
2012, 32(11): 4045-4067 doi: 10.3934/dcds.2012.32.4045 +[Abstract](167) +[PDF](711.7KB)
Abstract:
We prove the existence of multiple periodic solutions as well as the presence of complex profiles (for a certain range of the parameters) for the steady-state solutions of a class of reaction-diffusion equations with a FitzHugh-Nagumo cubic type nonlinearity. An application is given to a second order ODE related to a myelinated nerve axon model.

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