# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete and Continuous Dynamical Systems - B

October 2018 , Volume 23 , Issue 8

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2018, 23(8): 2951-2966 doi: 10.3934/dcdsb.2017128 +[Abstract](9127) +[HTML](844) +[PDF](481.89KB)
Abstract:

A food-chain model with Crowley-Martin functional response in the unstirred chemostat is considered. First, the global framework of coexistence solutions is discussed by the maximum principle and bifurcation theory. We obtain the sufficient and necessary conditions for coexistence of steady-state. Second, the stability and uniqueness of coexistence solutions are investigated by means of the combination of the perturbation theory and fixed point index theory. Our results indicate that if the magnitude of interference among predator is sufficiently large, the model has only one unique linearly stable coexistence solution when the maximal growth rate of predator belongs to certain range. Finally, some numerical simulations are carried out to verify and complement the theoretical results.

2018, 23(8): 2967-2988 doi: 10.3934/dcdsb.2017149 +[Abstract](9561) +[HTML](1296) +[PDF](566.06KB)
Abstract:

We consider fractional Navier-Stokes equations in a smooth bound-ed domain \begin{document} $Ω\subset\mathbb{R}^N$ \end{document}, \begin{document} $N≥2$ \end{document}. Following the geometric theory of abstract parabolic problems we give the detailed analysis concerning existence, uniqueness, regularization and continuation properties of the solution. For the original Navier-Stokes problem we construct next global solution of the Leray-Hopf type satisfying also Duhamel's integral formula. Focusing finally on the 3-D model with zero external force we estimate a time after which the latter solution regularizes to strong solution. We also estimate a time such that if a local strong solution exists until that time, then it exists for ever.

2018, 23(8): 2989-3021 doi: 10.3934/dcdsb.2017183 +[Abstract](6746) +[HTML](819) +[PDF](639.49KB)
Abstract:

Low angle grain boundaries can be modeled as arrays of line defects (dislocations) in crystalline materials. The classical continuum models for energetics and dynamics of curved grain boundaries are mainly based on those with equilibrium dislocation structures without the long-range elastic interaction, leading to a capillary force proportional to the local curvature of the grain boundary. The new continuum model recently derived by Zhu and Xiang (J. Mech. Phys. Solids, 69,175-194,2014) incorporates both the long-range dislocation interaction energy and the local dislocation line energy, and enables the study of low angle grain boundaries with non-equilibrium dislocation structures that involves the long-range elastic interaction. Using this new energy formulation, we show that the orthogonal network of two arrays of screw dislocations on a planar twist low angle grain boundary is always stable subject to both perturbations of the constituent dislocations within the grain boundary and the perturbations of the grain boundary itself.

2018, 23(8): 3023-3045 doi: 10.3934/dcdsb.2017199 +[Abstract](8288) +[HTML](699) +[PDF](1150.37KB)
Abstract:

In this paper we study the chemotaxis-system

defined in a convex smooth and bounded domain \begin{document}$Ω$\end{document} of \begin{document}$\mathbb{R}^n$\end{document}, \begin{document}$n≥ 1$\end{document}, with \begin{document}$χ>0$\end{document} and endowed with homogeneous Neumann boundary conditions. The source \begin{document}$g$\end{document} behaves similarly to the logistic function and satisfies \begin{document}$g(s)≤ a -bs^α$\end{document}, for \begin{document}$s≥ 0$\end{document}, with \begin{document}$a≥ 0$\end{document}, \begin{document}$b>0$\end{document} and \begin{document}$α>1$\end{document}. Continuing the research initiated in [33], where for appropriate \begin{document}$1 < p < α < 2$\end{document} and \begin{document}$(u_0,v_0) ∈ C^0(\bar{Ω})× C^2(\bar{Ω})$\end{document} the global existence of very weak solutions \begin{document}$(u,v)$\end{document} to the system (for any \begin{document}$n≥ 1$\end{document}) is shown, we principally study boundedness and regularity of these solutions after some time. More precisely, when \begin{document}$n=3$\end{document}, we establish that

-for all \begin{document}$τ>0$\end{document} an upper bound for \begin{document}$\frac{a}{b}, ||u_0||_{L^1(Ω)}, ||v_0||_{W^{2,α}(Ω)}$\end{document} can be prescribed in a such a way that \begin{document}$(u,v)$\end{document} is bounded and Hölder continuous beyond \begin{document}$τ$\end{document};

-for all \begin{document}$(u_0,v_0)$\end{document}, and sufficiently small ratio \begin{document}$\frac{a}{b}$\end{document}, there exists a \begin{document}$T>0$\end{document} such that \begin{document}$(u,v)$\end{document} is bounded and Hölder continuous beyond \begin{document}$T$\end{document}.

Finally, we illustrate the range of dynamics present within the chemotaxis system in one, two and three dimensions by means of numerical simulations.

2018, 23(8): 3047-3070 doi: 10.3934/dcdsb.2017196 +[Abstract](3986) +[HTML](698) +[PDF](1387.75KB)
Abstract:

This study examines the dynamics of tropical cyclone (TC) development in a TC scale framework. It is shown that this TC-scale dynamics contains the maximum potential intensity (MPI) limit as an asymptotically stable point for which the Coriolis force and the tropospheric stratification are two key parameters responsible for the bifurcation of TC development. In particular, it is found that the Coriolis force breaks the symmetry of the TC development and results in a larger basin of attraction toward the cyclonic (anticyclonic) stable point in the Northern (Southern) Hemisphere. Despite the sensitive dependence of intensity bifurcation on these two parameters, the structurally stable property of the MPI critical point is maintained for a wide range of parameters.

2018, 23(8): 3071-3085 doi: 10.3934/dcdsb.2017197 +[Abstract](7118) +[HTML](708) +[PDF](462.02KB)
Abstract:

We consider the following attraction-repulsion Keller-Segel system:

with homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n(n>2)$ with smooth boundary. Here all the parameters \begin{document}$χ, ξ, α, β, γ$\end{document} and \begin{document} $δ$\end{document} are positive. The smooth diffusion \begin{document}$D(u)$\end{document} satisfies \begin{document}$D(u)≥ d u^θ, u>0$\end{document} for some \begin{document}$d>0, θ∈\mathbb{R}$\end{document}. It is recently known from [25] that boundedness of solutions is ensured whenever \begin{document}$θ>1-\frac{2}{n}$\end{document}. Here, it is shown, if repulsion dominates or cancels attraction in the sense either \begin{document}$\{ξγ> χα\}$\end{document} or \begin{document}$\{ξγ=χα, β≥ δ\}$\end{document}, the corresponding initial-boundary value problem possesses a unique global classical solution which is uniformly-in-time bounded for large initial data provided \begin{document}$θ>1-\frac{4}{n+2}$\end{document}. In this way, the range of \begin{document}$θ>1-\frac{2}{n}$\end{document} of boundedness is enlarged and thus the repulsion effect on boundedness is exhibited.

2018, 23(8): 3087-3107 doi: 10.3934/dcdsb.2017209 +[Abstract](7750) +[HTML](734) +[PDF](602.62KB)
Abstract:

In this paper, we investigate the rich dynamics of a diffusive Holling type-Ⅱ predator-prey model with density-dependent death rate for the predator under homogeneous Neumann boundary condition. The value of this study lies in two-aspects. Mathematically, we show the stability of the constant positive steady state solution, the existence and nonexistence, the local and global structure of nonconstant positive steady state solutions. And biologically, we find that Turing instability is induced by the density-dependent death rate, and both the general stationary pattern and Turing pattern can be observed as a result of diffusion.

2018, 23(8): 3109-3135 doi: 10.3934/dcdsb.2017210 +[Abstract](7484) +[HTML](676) +[PDF](624.84KB)
Abstract:

In this paper, we study 1D autonomous fractional ODEs \begin{document} $D_c^{γ}u=f(u), 0< γ <1$ \end{document}, where \begin{document} $u: [0,∞) \to \mathbb{R}$ \end{document} is the unknown function and \begin{document} $D_c^{γ}$ \end{document} is the generalized Caputo derivative introduced by Li and Liu (arXiv:1612.05103). Based on the existence and uniqueness theorem and regularity results in previous work, we show the monotonicity of solutions to the autonomous fractional ODEs and several versions of comparison principles. We also perform a detailed discussion of the asymptotic behavior for \begin{document} $f(u)=Au^p$ \end{document}. In particular, based on an Osgood type blow-up criteria, we find relatively sharp bounds of the blow-up time in the case \begin{document} $A>0, p>1$ \end{document}. These bounds indicate that as the memory effect becomes stronger (\begin{document} $γ \to 0$ \end{document}), if the initial value is big, the blow-up time tends to zero while if the initial value is small, the blow-up time tends to infinity. In the case \begin{document} $A<0, p>1$ \end{document}, we show that the solution decays to zero more slowly compared with the usual derivative. Lastly, we show several comparison principles and Grönwall inequalities for discretized equations, and perform some numerical simulations to confirm our analysis.

2018, 23(8): 3137-3151 doi: 10.3934/dcdsb.2017211 +[Abstract](7188) +[HTML](715) +[PDF](387.29KB)
Abstract:

The paper deals with a stochastic SEIR model with nonlinear incidence rate and limited resources for a treatment. We focus on a long term study of two measures for the severity of an epidemic: the total number of cases of infection and the maximum of individuals simultaneously infected during an outbreak of the communicable disease. Theoretical and computational results are numerically illustrated.

2018, 23(8): 3153-3165 doi: 10.3934/dcdsb.2017212 +[Abstract](7563) +[HTML](644) +[PDF](393.68KB)
Abstract:

By applying the method of sub-super solutions, we obtain the existence of weak solutions to fractional Laplacian

\begin{document}$\left\{ \begin{array}{*{35}{l}} {{(-\Delta )}^{s}}u=f(x,u),&\text{in}\ \Omega , \\ u=0,&\text{in}\ {{\mathbb{R}}^{N}}\backslash \Omega , \\\end{array} \right.$\end{document}

where \begin{document}$f:\Omega \text{ }\!\!\times\!\!\text{ }\mathbb{R}\to \mathbb{R}$\end{document} is a Caratheódory function.

Let \begin{document}$ν$\end{document} be a Radon measure. Based on the existence result in (1), we derive the existence of weak solutions for the semilinear fractional elliptic equation with measure data

Some results in[7] are extended.

In addition, we generalize some results to systems of fractional Laplacian equations by constructing subsolutions and supersolutions.

2018, 23(8): 3167-3194 doi: 10.3934/dcdsb.2018224 +[Abstract](4940) +[HTML](224) +[PDF](4520.34KB)
Abstract:

Gene transcription is a stochastic process, as the mRNA copies of the same gene in a population of isogeneic cells are often distributed unevenly. The fluctuation has been attributed to the random transition of system states and random production or degradation of transcripts, as characterized by the prevailing two-state model. In addition, as cells live in heterogeneous environments, noisy signals provide a further source of randomness for transcription activation. In this paper, we study how the coupling of random environmental signals and the core transcription system coordinates transcriptional dynamics and noise by extending the two-state model. One of our major concerns is whether noisy signals activate noisier transcription. We find the exact forms for the steady-states of the mean mRNA level and its noise and clarify their dynamical behavior. Our numerical examples strongly suggest that the randomness of the signals inducing a positive or negative regulation does not make significant impact on transcription. Corresponding to each noisy signal, there is a deterministic signal such that the two signals generate nearly identical temporal profiles for the mean and the noise. When transcription is regulated by pulsatile signals, the mean and the noise exhibit damped but almost synchronized oscillations, indicating that noisy pulsatile signals may even reduce transcription noise at some time intervals. Our further analysis reveals that the transition rates in the core transcription system make more notable impacts on creating transcription noise than what the randomness in external signals may contribute.

2018, 23(8): 3195-3212 doi: 10.3934/dcdsb.2018281 +[Abstract](4048) +[HTML](128) +[PDF](515.07KB)
Abstract:

For solving the Helmholtz transmission eigenvalue problem, we use the mixed formulation of Cakoni et al. to construct a new nonconforming element discretization. Based on the discretization, this paper first discuss the nonconforming element methods of class \begin{document}$L^2$\end{document}, and prove the error estimates of the discrete eigenvalues obtained by the cubic tetrahedron element, incomplete cubic tetrahedral element and Morley element et al. We report some numerical examples using the nonconforming elements mixed with linear Lagrange element to show that our discretization can obtain the transmission eigenvalues of higher accuracy in 3D domains than the nonconforming element discretization in the existing literature.

2018, 23(8): 3213-3235 doi: 10.3934/dcdsb.2018242 +[Abstract](4952) +[HTML](173) +[PDF](542.75KB)
Abstract:

We consider a class of non-cooperative reaction-diffusion system, which includes different types of incidence rates for virus dynamical models with nonlocal infections. Threshold dynamics are expressed by basic reproduction number \begin{document}$R_0$\end{document} in the following sense, if \begin{document}$R_0 < 1$\end{document}, the infection-free steady state is globally attractive, implying infection becomes extinct; while if \begin{document}$R_0 > 1$\end{document}, virus will persist. To study the invasion speed of virus, the existence of travelling wave solutions is studied by employing Schauder's fixed point theorem. The method of constructing super-solutions and sub-solutions is very technical. The mathematical difficulty is the problem constructing a bounded cone to apply the Schauder's fixed point theorem. As compared to previous mathematical studies for diffusive virus dynamical models, the novelty here is that we successfully establish the general existence result of travelling wave solutions for a class of virus dynamical models with complex nonlinear transmissions and nonlocal infections.

2018, 23(8): 3237-3274 doi: 10.3934/dcdsb.2018241 +[Abstract](3720) +[HTML](149) +[PDF](6707.18KB)
Abstract:

The presence of a nonattractive chaotic set, also called chaotic saddle, in phase space implies the appearance of a finite time kind of chaos that is known as transient chaos. For a given dynamical system in a certain region of phase space with transient chaos, trajectories eventually abandon the chaotic region escaping to an external attractor, if no external intervention is done on the system. In some situations, this attractor may involve an undesirable behavior, so the application of a control in the system is necessary to avoid it. Both, the nonattractive nature of transient chaos and eventually the presence of noise may hinder this task. Recently, a new method to control chaos called partial control has been developed. The method is based on the existence of a set, called the safe set, that allows to sustain transient chaos by only using a small amount of control. The surprising result is that the trajectories can be controlled by using an amount of control smaller than the amount of noise affecting it. We present here a broad survey of results of this control method applied to a wide variety of dynamical systems. We also review here all the variations of the partial control method that have been developed so far. In all the cases various systems of different dimensionality are treated in order to see the potential of this method. We believe that this method is a step forward in controlling chaos in presence of disturbances.

2018, 23(8): 3275-3296 doi: 10.3934/dcdsb.2018244 +[Abstract](4063) +[HTML](178) +[PDF](621.15KB)
Abstract:

The objective of this article is to study the significance of dynamical properties of non-autonomous deterministic as well as stochastic prey-predator model with Holling type-Ⅲ functional response. Firstly, uniform persistence of the deterministic model has been demonstrated. Secondly, stochastic non-autonomous prey-predator system with Holling type-Ⅲ functional response is proposed. The existence of a global positive solution has been derived. Sufficient conditions for non-persistence in mean, weakly persistence in mean, extinction have been derived. Moreover the sufficient conditions for permanence of the system have been established. The analytical results are verified by numerical simulation.

2018, 23(8): 3297-3308 doi: 10.3934/dcdsb.2018252 +[Abstract](3842) +[HTML](139) +[PDF](386.14KB)
Abstract:

The paper deals with the nonlinear differential equation

in the case when the weight \begin{document}$b$\end{document} has indefinite sign. In particular, the problem of the existence of the so-called globally positive Kneser solutions, that is solutions \begin{document}$x$\end{document} such that \begin{document}$x(t)>0, {{x}'}(t)<0$\end{document} on the whole closed interval \begin{document}$[1,\infty )$\end{document}, is considered. Moreover, conditions assuring that these solutions tend to zero as \begin{document}$t\rightarrow\infty$\end{document} are investigated by a Schauder's half-linearization device jointly with some properties of the principal solution of an associated half-linear differential equation. The results cover also the case in which the weight \begin{document}$b$\end{document} is a periodic function or it is unbounded from below.

2018, 23(8): 3309-3345 doi: 10.3934/dcdsb.2018282 +[Abstract](4323) +[HTML](130) +[PDF](513.87KB)
Abstract:

In this paper, we introduce and study the concepts and properties of Poisson Stepanov-like almost automorphy (or Poisson \begin{document}$S^2$\end{document}-almost automorphy) for stochastic processes. With appropriate conditions, we apply the results obtained to investigate the asymptotic behavior of the soulutions to SPDEs driven by Lévy noise under \begin{document}$S^2$\end{document}-almost automorphic coefficients without global Lipschitz conditions. Moreover, the local asymptotic stability of the solutions under local Lipschitz condition is discussed and the attractive domain is also given. Finally, an illustrative example is provided to justify the practical usefulness of the established theoretical results.

2018, 23(8): 3347-3360 doi: 10.3934/dcdsb.2018248 +[Abstract](4336) +[HTML](168) +[PDF](398.15KB)
Abstract:

In this article, we discuss a class of impulsive stochastic function differential equations driven by \begin{document}$G$\end{document}-Brownian motion with delayed impulsive effects (\begin{document}$G$\end{document}-DISFDEs, in short). Some sufficient conditions for \begin{document}$p$\end{document}-th moment exponential stability of \begin{document}$G$\end{document}-DISFDEs are derived by means of \begin{document}$G$\end{document}-Lyapunov function method, average impulsive interval approach and Razumikhin-type conditions. An example is provided to show the effectiveness of the theoretical results.

2018, 23(8): 3361-3386 doi: 10.3934/dcdsb.2018246 +[Abstract](3590) +[HTML](134) +[PDF](640.9KB)
Abstract:

The sweeping process was proposed by J. J. Moreau as a general mathematical formalism for quasistatic processes in elastoplastic bodies. This formalism deals with connected Prandtl's elastic-ideal plastic springs, which can form a system with an arbitrarily complex topology. The model describes the complex relationship between stresses and elongations of the springs as a multi-dimensional differential inclusion (variational inequality). On the other hand, the Prandtl-Ishlinskii model assumes a very simple connection of springs. This model results in an input-output operator, which has many good mathematical properties and admits an explicit solution for an arbitrary input. It turns out that the sweeping processes can be reducible to the Prandtl-Ishlinskii operator even if the topology of the system of springs is complex. In this work, we analyze the conditions for such reducibility.

2018, 23(8): 3387-3413 doi: 10.3934/dcdsb.2018239 +[Abstract](4481) +[HTML](1203) +[PDF](1087.46KB)
Abstract:

Tuberculosis (TB) is a leading cause of death from infectious disease. TB is caused mainly by a bacterium called Mycobacterium tuberculosis which often initiates in the respiratory tract. The interaction of macrophages and T cells plays an important role in the immune response during TB infection. Recent experimental results support that active TB infection may be induced by the dysfunction of Treg cell regulation that provides a balance between anti-TB T cell responses and pathology. To better understand the dynamics of TB infection and Treg cell regulation, we build a mathematical model using a system of differential equations that qualitatively and quantitatively characterizes the dynamics of macrophages, Th1 and Treg cells during TB infection. For sufficiently analyzing the interaction between immune response and bacterial infection, we separate our model into several simple subsystems for further steady state and stability studies. Using this system, we explore the conditions of parameters for three situations, recovery, latent disease and active disease, during TB infection. Our numerical simulations support that Th1 cells and Treg cells play critical roles in TB infection: Th1 cells inhibit the number of infected macrophages to reduce the chance of active disease; Treg cell regulation reduces the immune response to stabilize the dynamics of the system.

2018, 23(8): 3415-3426 doi: 10.3934/dcdsb.2018283 +[Abstract](3853) +[HTML](171) +[PDF](371.87KB)
Abstract:

We show that for a system of transport equations defined on an infinite network, the semigroup generated is hypercyclic if and only if the adjacency matrix of the line graph is also hypercyclic. We further show that there is a range of parameters for which a transport equation on an infinite network with no loops is chaotic on a subspace \begin{document}$X_e$\end{document} of the weighted Banach space \begin{document}$\ell^1_s$\end{document}. We relate these results to Banach-space birth-and-death models in literature by showing that when there is no proliferation, the birth-and-death model is also chaotic in the same subspace \begin{document}$X_e$\end{document} of \begin{document}$\ell^1_s$\end{document}. We do this by noting that the eigenvalue problem for the birth-and-death model is in fact an eigenvalue problem for the adjacency matrix of the line graph (of the network on which the transport problem is defined) which controls the dynamics of the the transport problem.

2018, 23(8): 3427-3460 doi: 10.3934/dcdsb.2018284 +[Abstract](4909) +[HTML](228) +[PDF](628.66KB)
Abstract:

In this article, we consider the Cauchy problem to chemotaxis model coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish the global-in-time existence of the solution when the gravitational potential ϕ and the small initial data \begin{document}$(u_{0}, n_{0}, c_{0})$\end{document} in critical Besov spaces under certain conditions. Moreover, we prove that there exist two positive constants σ0 and \begin{document}$C_{0}$\end{document} such that if the gravitational potential \begin{document}$\phi \in \dot B_{p,1}^{3/p}({\mathbb{R}^3})$\end{document} and the initial data \begin{document}$(u_{0}, n_{0}, c_{0}): = (u_{0}^{1}, u_{0}^{2}, u_{0}^{3}, n_{0}, c_{0}): = (u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$\end{document} satisfies

for some \begin{document}$p, q$\end{document} with \begin{document}$1<p, q<6,\frac{1}{p}+\frac{1}{q}>\frac{2}{3}$\end{document} and \begin{document}$\frac{1}{\min\{p, q\}}-\frac{1}{\max\{p, q\}} \le \frac{1}{3}$\end{document}, then the global existence results can be extended to the global solutions without any small conditions imposed on the third component of the initial velocity field \begin{document}$u_{0}^{3}$\end{document} in critical Besov spaces with the aid of continuity argument. Our initial data class is larger than that of some known results. Our results are completely new even for three-dimensional chemotaxis-Navier-Stokes system.

2018, 23(8): 3461-3482 doi: 10.3934/dcdsb.2018234 +[Abstract](4293) +[HTML](138) +[PDF](779.14KB)
Abstract:

We investigate the stability of synchronization in networks of dynamical systems with strongly delayed connections. We obtain strict conditions for synchronization of periodic and equilibrium solutions. In particular, we show the existence of a critical coupling strength \begin{document}$κ_{c}$\end{document}, depending only on the network structure, isolated dynamics and coupling function, such that for large delay and coupling strength \begin{document}$κ<κ_{c}$\end{document}, the network possesses stable synchronization. The critical coupling \begin{document}$κ_{c}$\end{document} can be chosen independently of the delay for the case of equilibria, while for the periodic solution, $κ_{c}$ depends essentially on the delay and vanishes as the delay increases. We observe that, for random networks, the synchronization interval is maximal when the network is close to the connectivity threshold. We also derive scaling of the coupling parameter that allows for a synchronization of large networks for different network topologies.

2018, 23(8): 3483-3501 doi: 10.3934/dcdsb.2018250 +[Abstract](5164) +[HTML](192) +[PDF](873.67KB)
Abstract:

This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results.

2018, 23(8): 3503-3534 doi: 10.3934/dcdsb.2018285 +[Abstract](4563) +[HTML](126) +[PDF](695.25KB)
Abstract:

In this paper we study the process of phase separation from directional quenching, considered as an externally triggered variation in parameters that changes the system from monostable to bistable across an interface (quenching front); in our case the interface moves with speed \begin{document}$c$ \end{document} in such a way that the bistable region grows. According to results from [9,10], several patterns exist when \begin{document}$c\underset{\tilde{\ }}{\mathop{>}}\,0$ \end{document}, and here we investigate their persistence for finite \begin{document}$c>0$ \end{document}. We find existence and nonexistence results of multidimensional horizontal stripped patterns, clarifying the selection mechanism relating their existence to the speed \begin{document}$c$ \end{document} of the quenching front. We further illustrate our results by allying them to those of [9], hence obtaining the existence of a family of single interface patterns displaying different contact angles between their nodal lines and the quenching front; the existence of these patterns was known for small speeds \begin{document}$c> 0$ \end{document} and here we show that they also exist in the range \begin{document}$0 < c < 2$ \end{document}.

2020 Impact Factor: 1.327
5 Year Impact Factor: 1.492
2020 CiteScore: 2.2