# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

February 2017 , Volume 37 , Issue 2

Special issue on emergence and dynamics of patterns in nonlinear partial differential equations dedicated to Paul Fife

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2017, 37(2): i-iii doi: 10.3934/dcds.201702i +[Abstract](1102) +[HTML](562) +[PDF](84.4KB)
Abstract:
Professor Paul Chase Fife was born in Cedar City, Utah, on February 14, 1930. After undergraduate studies at the University of Chicago, he obtained a Master's degree in physics from the University of California Berkeley where he also received a Phi Beta Kappa Award. He then completed a PhD in Applied Mathematics at New York University, Courant Institute, in June 1959. While at NYU he met Jayne Winters, and they married on December 22, 1959. They then moved to Palo Alto, California, where Paul joined the Department of Mathematics at Stanford University.

2017, 37(2): 663-683 doi: 10.3934/dcds.2017028 +[Abstract](1543) +[HTML](1) +[PDF](554.3KB)
Abstract:

We study the stability of partitions involving two or more phases in convex domains under the assumption of at most two-phase contact, thus excluding in particular triple junctions. We present a detailed derivation of the second variation formula with particular attention to the boundary terms, and then study the sign of the principal eigenvalue of the Jacobi operator. We thus derive certain stability criteria, and in particular we recapture the Sternberg-Zumbrun result on the instability of the disconnected phases in the more general setting of several phases.

2017, 37(2): 685-723 doi: 10.3934/dcds.2017029 +[Abstract](2313) +[HTML](16) +[PDF](1665.2KB)
Abstract:

Non-local reaction-diffusion equations arise naturally to account for diffusions involving jumps rather than local diffusions related to Brownian motion. In ecology, long distance dispersal require such frameworks. In this work we study a one-dimensional non-local reaction-diffusion equation with bistable type reaction. The heterogeneity here is due to a gap, some finite region where there is decay. Outside this gap region the equation is a classical homogeneous (space independent) non-local reaction-diffusion equation. This type of problem is motivated by applications in ecology, sociology, and physiology. We first establish the existence of a generalized traveling front that approaches a traveling wave solution as t-∞, propagating in a heterogeneous environment. We then study the problem of obstruction of solutions. In particular, we study the propagation properties of the generalized traveling front with significant use of the work of Bates, Fife, Ren and Wang in [5]. As in the local diffusion case, we prove that obstruction is possible if the gap is sufficiently large. An interesting difference between the local dispersal and the non-local dispersal is that in the latter the obstructing steady states are discontinuous. We characterize these jump discontinuities and discuss the scaling between the range of the dispersal and the critical length of the gap observed numerically. We further explore other differences between the local and the non-local dispersal cases. In this paper, we illustrate these properties by numerical simulations and we state a series of open problems.

2017, 37(2): 725-742 doi: 10.3934/dcds.2017030 +[Abstract](1360) +[HTML](3) +[PDF](443.4KB)
Abstract:

We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain \begin{document}$Ω\subset\mathbb{R}^n.$\end{document} Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric solutions. In dimension \begin{document}$n=2$\end{document} an additional asymptotic result is obtained. These results are based on a pointwise estimate obtained for local minimizers of the Allen-Cahn energy.

2017, 37(2): 743-756 doi: 10.3934/dcds.2017031 +[Abstract](1362) +[HTML](5) +[PDF](543.8KB)
Abstract:

We consider a general form of reaction-dispersion equations with non-local or nonlinear dispersal operators and local reaction terms. Under some general conditions, we prove the non-existence of transition fronts, as well as some stretching properties at large time for the solutions of the Cauchy problem. These conditions are satisfied in particular when the reaction is monostable and when the dispersal operator is either the fractional Laplacian, a convolution operator with a fat-tailed kernel or a nonlinear fast diffusion operator.

2017, 37(2): 757-800 doi: 10.3934/dcds.2017032 +[Abstract](1480) +[HTML](7) +[PDF](879.0KB)
Abstract:

Classical models of pattern formation are based on diffusion-driven instability (DDI) of constant stationary solutions of reaction-diffusion equations, which leads to emergence of stable, regular Turing patterns formed around that equilibrium. In this paper we show that coupling reaction-diffusion equations with ordinary differential equations (ODE) may lead to a novel pattern formation phenomenon in which DDI causes destabilization of both constant solutions and Turing patterns. Bistability and hysteresis effects in the null sets of model nonlinearities yield formation of far from the equilibrium patterns with jump discontinuity. We derive conditions for stability of stationary solutions with jump discontinuity in a suitable topology which allows us to include the discontinuity points and leads to the definition of \begin{document}$(\varepsilon_0, A)$\end{document}-stability. Additionally, we provide conditions on stability of patterns in a quasi-stationary model reduction. The analysis is illustrated on the example of three-component model of receptor-ligand binding. The proposed model provides an example of a mechanism of de novo formation of far from the equilibrium patterns in reaction-diffusion-ODE models involving co-existence of DDI and hysteresis.

2017, 37(2): 801-827 doi: 10.3934/dcds.2017033 +[Abstract](1552) +[HTML](4) +[PDF](534.8KB)
Abstract:

This paper is devoted to construction of new solutions to the Cahn-Hilliard equation in \begin{document}$\mathbb R^d$\end{document}. Staring from the Delaunay unduloid \begin{document} $D_τ$ \end{document} with parameter \begin{document} $τ∈ (0,τ^*)$ \end{document} we find for each sufficiently small \begin{document} $\varepsilon$ \end{document} a solution \begin{document} $u$ \end{document} of this equation which is periodic in the direction of the \begin{document} $x_d$ \end{document} axis and rotationally symmetric with respect to rotations about this axis. The zero level set of \begin{document} $u$ \end{document} approaches as \begin{document} $\varepsilon \to 0$ \end{document} the surface \begin{document} $D_τ$ \end{document}. We use a refined version of the Lyapunov-Schmidt reduction method which simplifies very technical aspects of previous constructions for similar problems.

2017, 37(2): 829-858 doi: 10.3934/dcds.2017034 +[Abstract](2103) +[HTML](10) +[PDF](5771.2KB)
Abstract:

Spatially localized blooms of toxic plankton species have negative impacts on other organisms via the production of toxins, mechanical damage, or by other means. Such blooms are nowadays a worldwide spread environmental issue. To understand the mechanism behind this phenomenon, a two-prey (toxic and nontoxic phytoplankton)-one-predator (zooplankton) Lotka-Volterra system with diffusion has been considered in a previous paper. Numerical results suggest the occurrence of stable non-constant equilibrium solutions, that is, spatially localized blooms of the toxic prey. Such blooms appear for intermediate values of the rate of toxicity \begin{document}$μ$\end{document} when the ratio \begin{document}$D$\end{document} of the diffusion rates of the predator and the two prey is rather large. In this paper, we consider a one-dimensional limiting system (we call it a shadow system) in \begin{document}$(0,L)$\end{document} as \begin{document}$D \to \infty$\end{document} and discuss the existence and stability of non-constant equilibrium solutions with large amplitude when \begin{document}$μ$\end{document} is globally varied. We also show that the structure of non-constant equilibrium solutions sensitively depends on \begin{document}$L$\end{document} as well as \begin{document}$μ$\end{document}.

2017, 37(2): 859-878 doi: 10.3934/dcds.2017035 +[Abstract](1367) +[HTML](8) +[PDF](10042.2KB)
Abstract:

Algae in the ocean absorb carbon dioxide from the atmosphere and thus play an important role in the carbon cycle. An algal bloom occurs when there is a rapid increase in an algae population. We consider a reaction-advection-diffusion model for algal bloom density and present new proofs of existence and uniqueness results for the steady state solutions using techniques from dynamical systems. On the question of stability of the bloom profiles, we show that the only possible bifurcation would be due to an oscillatory instability.

2017, 37(2): 879-903 doi: 10.3934/dcds.2017036 +[Abstract](1769) +[HTML](20) +[PDF](466.8KB)
Abstract:

We aim at saying as much as possible about the spectra of three classes of linear diffusion operators involving nonlocal terms. In all but one cases, we characterize the minimum \begin{document} $λ_p$ \end{document} of the real part of the spectrum in two max-min fashions, and prove that in most cases \begin{document} $λ_p$ \end{document} is an eigenvalue with a corresponding positive eigenfunction, and is algebraically simple and isolated; we also prove that the maximum principle holds if and only if \begin{document} $λ_p>0$ \end{document} (in most cases) or \begin{document} $≥ 0$ \end{document} (in one case). We prove these results by an elementary method based on the strong maximum principle, rather than resorting to Krein-Rutman theory as did in the previous papers. In one case when it is impossible to characterize \begin{document} $λ_p$ \end{document} in the max-min fashion, we supply a complete description of the whole spectrum.

2017, 37(2): 905-914 doi: 10.3934/dcds.2017037 +[Abstract](1204) +[HTML](11) +[PDF](301.7KB)
Abstract:

In this paper, a free boundary problem related to cell motility is discussed. This free boundary problem consists of an interface equation for the domain evolution and a parabolic equation governing actin concentration in the domain. In [10] the existence of traveling wave solutions with disk-shaped domains were shown in a special situation where a polymerization rate is specified. In this paper, by relaxing the condition for the polymerization rate, the previous result is extended to the existence of traveling wave solutions with convex domains.

2017, 37(2): 915-944 doi: 10.3934/dcds.2017038 +[Abstract](1455) +[HTML](0) +[PDF](604.3KB)
Abstract:

We study the long time behavior of positive solutions of the Cauchy problem for nonlinear reaction-diffusion equations in \begin{document}$\mathbb{R}^N$\end{document} with bistable, ignition or monostable nonlinearities that exhibit threshold behavior. For \begin{document}$L^2$\end{document} initial data that are radial and non-increasing as a function of the distance to the origin, we characterize the ignition behavior in terms of the long time behavior of the energy associated with the solution. We then use this characterization to establish existence of a sharp threshold for monotone families of initial data in the considered class under various assumptions on the nonlinearities and spatial dimension. We also prove that for more general initial data that are sufficiently localized the solutions that exhibit ignition behavior propagate in all directions with the asymptotic speed equal to that of the unique one-dimensional variational traveling wave.

2017, 37(2): 945-961 doi: 10.3934/dcds.2017039 +[Abstract](1330) +[HTML](3) +[PDF](1209.2KB)
Abstract:

Clopidogrel is an anti-platelet compound that is widely used with aspirin to reduce the risk of cardiovascular incidents.In itself it is inactive; only after a biotransformation into its active metabolite clop-AM, does it inhibit platelet aggregation.Recently a system-pharmacological model has been proposed for the network of processes leading to reduced platelet aggregation.In this paper we present a mathematical analysis of this model and demonstrate how the complex pharmacokinetic modelcan be reduced to two simple coupled models, one for clopidogrel and one for clop-AM, yielding insight into the dynamicsof clop-AM and the impact of inter-individual differences on the level of inhibition.

2017, 37(2): 963-982 doi: 10.3934/dcds.2017040 +[Abstract](1352) +[HTML](14) +[PDF](410.1KB)
Abstract:

Cellular precipitation is a dynamic phase transition in solid solutions (such as alloys) where a metastable phase decomposes into two stable phases : an approximately planar (but corrugated) boundary advances into the metastable phase, leaving behind it interleaved plates (lamellas) of the two stable phases.

The forces acting on each interface (thermodynamic, elastic and surface tension) are modelled here using a first-order ODE, and the diffusion of solute along the interface by a second-order ODE, with boundary conditions at the triple junctions where three interfaces meet. Careful attention is paid to the approximations and physical assumptions used in formulating the model.

These equations, previously studied by approximate (mostly numerical) methods, have the peculiarity that \begin{document}$v,$\end{document} the velocity of advance of the interface, is not uniquely determined by the given physical data such as \begin{document}$c_0$\end{document}, the solute concentration in the metastable phase. It is hoped that our analytical treament will help to improve the understanding of this.

We show how to solve the equations exactly in the limiting case where \begin{document}$v=0$\end{document}. For larger \begin{document}$v$\end{document}, a successive approximation scheme is formulated. One result of the analysis is that there is just one value for \begin{document}$c_0$\end{document} at which \begin{document}$v$\end{document} can be vanishingly small.

2017, 37(2): 983-1012 doi: 10.3934/dcds.2017041 +[Abstract](1377) +[HTML](12) +[PDF](1025.6KB)
Abstract:

A ternary inhibitory system motivated by the triblock copolymer theoryis studied as a nonlocal geometric variational problem. The free energyof the system is the sum of two terms: the total size of the interfacesseparating the three constituents, and a longer ranging interaction energythat inhibits micro-domains from unlimited growth. In a particular parameterrange there is an assembly of many core-shells that exists as a stationaryset of the free energy functional. The cores form regions occupied by thefirst constituent of the ternary system, the shells form regionsoccupied by the second constituent, and the background is taken by thethird constituent. The constructive proof of the existence theorem revealsmuch information about the core-shell stationary assembly: asymptoticallyone can determine the sizes and locations of all the core-shells in theassembly. The proof also implies a kind of stability for the stationaryassembly.

2017, 37(2): 1013-1037 doi: 10.3934/dcds.2017042 +[Abstract](1368) +[HTML](5) +[PDF](531.3KB)
Abstract:

The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equations with time heterogeneous nonlinearity of ignition type. It is proven that such an equation admits space monotone transition fronts with finite speed and space regularity in the sense of uniform Lipschitz continuity. Our approach is first constructing a sequence of approximating front-like solutions and then proving that the approximating solutions converge to a transition front. We take advantage of the idea of modified interface location, which allows us to characterize the finite speed of approximating solutions in the absence of space regularity, and leads directly to uniform exponential decaying estimates.

2017, 37(2): 1039-1059 doi: 10.3934/dcds.2017043 +[Abstract](1219) +[HTML](2) +[PDF](491.1KB)
Abstract:

Using a singular perturbation based approach, we make rigorous the formal boundary layer asymptotic analysis of Turcotte, Spence and Bau from the early eighties for the vertical flow of an internally heated Boussinesq fluid in a vertical channel with viscous dissipation and pressure work. A key point in our proof is to establish the non-degeneracy of a special solution of the Painlevé-Ⅰ transcendent. To this end, we relate this problem to recent studies for the ground states of the focusing nonlinear Schrödinger equation in an annulus. We also relate our result to a particular case of the well known Lazer-McKenna conjecture from nonlinear analysis.

2017, 37(2): 1061-1073 doi: 10.3934/dcds.2017044 +[Abstract](1397) +[HTML](8) +[PDF](546.1KB)
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We study the interface dynamics and contact angle hysteresis in a two dimensional, chemically patterned channel described by the Cahn-Hilliard equation with a relaxation boundary condition. A system for the dynamics of the contact angle and contact point is derived in the sharp interface limit. We then analyze the behavior of the solution using the phase plane analysis. We observe the stick-slip of the contact point and the contact angle hysteresis. As the size of the pattern decreases to zero, the stick-slip becomes weaker but the hysteresis becomes stronger in the sense that one observes either the advancing contact angle or the receding contact angle without any switching in between. Numerical examples are presented to verify our analysis.

2017, 37(2): 1075-1107 doi: 10.3934/dcds.2017045 +[Abstract](1817) +[HTML](7) +[PDF](1121.9KB)
Abstract:

The diblock copolymer model is a fourth-order parabolic partial differential equation which models phase separation with fine structure. The equation is a gradient flow with respect to an extension of the standard van der Waals free energy functional which involves nonlocal interactions. Thus, the long-term dynamical behavior of the diblock copolymer model is described by its finite-dimensional attractor. However, even on one-dimensional domains, the full structure of the underlying equilibrium set is not fully understood. In this paper, we develop a rigorous computational approach for establishing the existence of equilibrium solutions of the diblock copolymer model. We consider both individual solutions, as well as pieces of solution branches in a parameter-dependent situation. The results are presented for the case of one-dimensional domains, and can easily be implemented using standard interval arithmetic packages.

2017, 37(2): 1109-1127 doi: 10.3934/dcds.2017046 +[Abstract](1468) +[HTML](5) +[PDF](708.2KB)
Abstract:

The dynamics are studied for the Keller-Segel's minimal chemotaxis model

on a bounded interval with homogeneous Neumann boundary conditions, where \begin{document}$\tau\geqslant 0$\end{document} and \begin{document}$k\gg 1$\end{document} are parameters and the total mass of \begin{document}$u$\end{document} is scaled to be one. In general, the dynamics can be divided into three stages: the first stage is very short in which \begin{document}$u$\end{document} quickly becomes a delta like function with mass concentrated near the point of global maximum of \begin{document}$v$\end{document}; in the second stage, the point of the global maximum of \begin{document}$v$\end{document} drifts towards the boundary of the domain and reaches it at the end of the second stage; in the third stage, the profile of the solution evolves to a steady state profile. This paper considers a special case in which the relaxation parameter \begin{document}$\tau$\end{document} is set to be zero, so the first stage takes no time. A free boundary problem describing the second stage is presented. Rigorous asymptotic behavior is proven for the third stage evolution.

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