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Discrete & Continuous Dynamical Systems - B

August 2019 , Volume 24 , Issue 8

Special issue in honor of Prof. Dr. Peter E. Kloeden on the occasion of his 70th birthday

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Preface
Tomas Caraballo, Xiaoying Han and Arnulf Jentzen
2019, 24(8): ⅰ-ⅱ doi: 10.3934/dcdsb.2019184 +[Abstract](257) +[HTML](82) +[PDF](95.7KB)
Abstract:
Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises
Wenqiang Zhao
2019, 24(8): 3453-3474 doi: 10.3934/dcdsb.2018251 +[Abstract](1480) +[HTML](763) +[PDF](481.15KB)
Abstract:

In this article, we study the dynamical behaviour of solutions of the non-autonomous stochastic Fitzhugh-Nagumo system on \begin{document}$\mathbb{R}^N$\end{document} with both multiplicative noises and non-autonomous forces, where the nonlinearity is a polynomial-like growth function of arbitrary order. An asymptotic smoothing effect of this system is demonstrated, namely, that the random pullback attractor in the initial space \begin{document}$L^2(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$\end{document} is actually a compact, measurable and attracting set in \begin{document}$H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$\end{document}. A difference estimates method, rather than the usual truncation estimate and spectrum decomposition technique, is employed to overcome the lack of Sobolev compact embedding in \begin{document}$H^1(\mathbb{R}^N)\times L^2(\mathbb{R}^N)$\end{document}, despite of the loss of the high-order integrability of the difference of solutions for this system.

A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients
Raphael Kruse and Yue Wu
2019, 24(8): 3475-3502 doi: 10.3934/dcdsb.2018253 +[Abstract](1448) +[HTML](690) +[PDF](614.96KB)
Abstract:

In this paper a drift-randomized Milstein method is introduced for the numerical solution of non-autonomous stochastic differential equations with non-differentiable drift coefficient functions. Compared to standard Milstein-type methods we obtain higher order convergence rates in the \begin{document}$ L^p(Ω) $\end{document} and almost sure sense. An important ingredient in the error analysis are randomized quadrature rules for Hölder continuous stochastic processes. By this we avoid the use of standard arguments based on the Itō-Taylor expansion which are typically applied in error estimates of the classical Milstein method but require additional smoothness of the drift and diffusion coefficient functions. We also discuss the optimality of our convergence rates. Finally, the question of implementation is addressed in a numerical experiment.

Numerical methods for PDE models related to pricing and expected lifetime of an extraction project under uncertainty
María Suárez-Taboada and Carlos Vázquez
2019, 24(8): 3503-3523 doi: 10.3934/dcdsb.2018254 +[Abstract](1321) +[HTML](722) +[PDF](669.08KB)
Abstract:

Numerical techniques for solving some mathematical models related to a mining extraction project under uncertainty are proposed. The mine valuation is formulated as a complementarity problem associated to a degenerate second order partial differential equation (PDE), which incorporates the option to abandon the project. The probability of completion and the expected lifetime of the project are the respective solutions of problems governed by similar degenerated PDE operators. In all models, the underlying stochastic factors are the commodity price and the remaining resource. After justifying the required boundary conditions on the computational bounded domain, the proposed numerical techniques mainly consist of a Crank-Nicolson characteristics method for the time discretization to cope with the convection dominating setting and Lagrange finite elements for the discretization in the commodity and resource variables, with the additional use of an augmented Lagrangian active set method for the complementarity problem. Some numerical examples are discussed to illustrate the performance of the methods and models.

Convergences of asymptotically autonomous pullback attractors towards semigroup attractors
Hongyong Cui
2019, 24(8): 3525-3535 doi: 10.3934/dcdsb.2018276 +[Abstract](937) +[HTML](506) +[PDF](522.07KB)
Abstract:

For pullback attractors of asymptotically autonomous dynamical systems we study the convergences of their components towards the global attractors of the limiting semigroups. We use some conditions of uniform boundedness of pullback attractors, instead of uniform compactness conditions used in the literature. Both forward convergence and backward convergence are studied.

Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems
Victor Kozyakin
2019, 24(8): 3537-3556 doi: 10.3934/dcdsb.2018277 +[Abstract](1127) +[HTML](637) +[PDF](537.43KB)
Abstract:

To estimate the growth rate of matrix products \begin{document}$A_{n}··· A_{1}$\end{document} with factors from some set of matrices \begin{document}$\mathscr{A}$\end{document}, such numeric quantities as the joint spectral radius \begin{document}$ρ(\mathscr{A})$\end{document} and the lower spectral radius \begin{document}$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})$\end{document} are traditionally used. The first of these quantities characterizes the maximum growth rate of the norms of the corresponding products, while the second one characterizes the minimal growth rate. In the theory of discrete-time linear switching systems, the inequality \begin{document}$ρ(\mathscr{A})<1$\end{document} serves as a criterion for the stability of a system, and the inequality \begin{document}$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})<1 $\end{document} as a criterion for stabilizability.

Given a set \begin{document}$\mathscr{A}$\end{document} of \begin{document}$N×M$\end{document} matrices and a set \begin{document}$\mathscr{B}$\end{document} of \begin{document}$M×N$\end{document} matrices. Then, for matrix products \begin{document}$A_{n}B_{n}··· A_{1}B_{1}$\end{document} with factors \begin{document}$A_{i}∈\mathscr{A}$\end{document} and \begin{document}$B_{i}∈\mathscr{B}$\end{document}, we introduce the quantities \begin{document}$μ(\mathscr{A},\mathscr{B})$\end{document} and \begin{document}$η(\mathscr{A},\mathscr{B})$\end{document}, called the lower and upper minimax joint spectral radius of the pair \begin{document}$\{\mathscr{A},\mathscr{B}\}$\end{document}, respectively, which characterize the maximum growth rate of the matrix products \begin{document}$A_{n}B_{n}··· A_{1}B_{1}$\end{document} over all sets of matrices \begin{document}$A_{i}∈\mathscr{A}$\end{document} and the minimal growth rate over all sets of matrices \begin{document}$B_{i}∈\mathscr{B}$\end{document}. In this sense, the minimax joint spectral radii can be considered as generalizations of both the joint and lower spectral radii. As an application of the minimax joint spectral radii, it is shown how these quantities can be used to analyze the stabilizability of discrete-time linear switching control systems in the presence of uncontrolled external disturbances of the plant.

On asymptotically autonomous dynamics for multivalued evolution problems
Jacson Simsen and Mariza Stefanello Simsen
2019, 24(8): 3557-3567 doi: 10.3934/dcdsb.2018278 +[Abstract](918) +[HTML](573) +[PDF](356.68KB)
Abstract:

In this work we improve the result presented by Kloeden-Simsen-Stefanello Simsen in [8] by reducing uniform conditions. We prove theoretical results in order to establish convergence in the Hausdorff semi-distance of the component subsets of the pullback attractor of a non-autonomous multivalued problem to the global attractor of the corresponding autonomous multivalued problem.

Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping
Daniel Pardo, José Valero and Ángel Giménez
2019, 24(8): 3569-3590 doi: 10.3934/dcdsb.2018279 +[Abstract](1046) +[HTML](619) +[PDF](4101.58KB)
Abstract:

In this paper we obtain the existence of global attractors for the dynamicalsystems generated by weak solution of the three-dimensional Navier-Stokesequations with damping. We consider two cases, depending on the values of the parameter β controlling the damping term. First, we prove that for β≥4 weaksolutions are unique and establish the existence of the global attractor forthe corresponding semigroup. Second, for 3≤β<4 we define amultivalued dynamical systems and prove the existence of the global attractoras well. Finally, some numerical simulations are performed.

Modeling and analysis of random and stochastic input flows in the chemostat model
Tomás Caraballo, Maria-José Garrido-Atienza, Javier López-de-la-Cruz and Alain Rapaport
2019, 24(8): 3591-3614 doi: 10.3934/dcdsb.2018280 +[Abstract](1056) +[HTML](551) +[PDF](577.51KB)
Abstract:

In this paper we study a new way to model noisy input flows in the chemostat model, based on the Ornstein-Uhlenbeck process. We introduce a parameter β as drift in the Langevin equation, that allows to bridge a gap between a pure Wiener process, which is a common way to model random disturbances, and no noise at all. The value of the parameter β is related to the amplitude of the deviations observed on the realizations. We show that this modeling approach is well suited to represent noise on an input variable that has to take non-negative values for almost any time.

Smoothness of density for stochastic differential equations with Markovian switching
Yaozhong Hu, David Nualart, Xiaobin Sun and Yingchao Xie
2019, 24(8): 3615-3631 doi: 10.3934/dcdsb.2018307 +[Abstract](959) +[HTML](787) +[PDF](392.85KB)
Abstract:

This paper is concerned with a class of stochastic differential equations with Markovian switching. The Malliavin calculus is used to study the smoothness of the density of the solution under a Hörmander type condition. Furthermore, we obtain a Bismut type formula which is used to establish the strong Feller property.

On the Cahn-Hilliard/Allen-Cahn equations with singular potentials
Alain Miranville, Wafa Saoud and Raafat Talhouk
2019, 24(8): 3633-3651 doi: 10.3934/dcdsb.2018308 +[Abstract](950) +[HTML](618) +[PDF](445.91KB)
Abstract:

The purpose of this work is to prove the existence and uniqueness of the solution for a Cahn-Hilliard/Allen-Cahn system with singular potentials (and, in particular, the thermodynamically relevant logarithmic potentials). We also prove the existence of the global attractor. Finally, we show further regularity results and we prove a strict separation property (from the pure states) in one space dimension.

Invariance principle in the singular perturbations limit
Zvi Artstein
2019, 24(8): 3653-3666 doi: 10.3934/dcdsb.2018309 +[Abstract](986) +[HTML](610) +[PDF](351.63KB)
Abstract:

We examine the invariance principle in the stability theory of differential equations, within a general singularly perturbed system. The limit dynamics of such a system is depicted by the evolution of a Young measure whose values are invariant measures of the fast equation. We establish an invariance principle for the limit dynamics, and examine the relations, at times subtle, with the singularly perturbed system itself.

Pollution control for switching diffusion models: Approximation methods and numerical results
Caojin Zhang, George Yin, Qing Zhang and Le Yi Wang
2019, 24(8): 3667-3687 doi: 10.3934/dcdsb.2018310 +[Abstract](973) +[HTML](545) +[PDF](4331.4KB)
Abstract:

This work focuses on optimal pollution controls. The main effort is devoted to obtaining approximation methods for optimal pollution control. To take into consideration of random environment and other random factors, the control system is formulated as a controlled switching diffusion. Markov chain approximation techniques are used to design the computational schemes. Convergence of the algorithms are obtained. To demonstrate, numerical experimental results are presented. A particular feature is that computation using real data sets is provided.

Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient
Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo and Antonio Suárez
2019, 24(8): 3689-3711 doi: 10.3934/dcdsb.2018311 +[Abstract](960) +[HTML](638) +[PDF](451.15KB)
Abstract:

In this paper we study a stationary problem arising from population dynamics with a local and nonlocal variable diffusion coefficient. We show the existence of an unbounded continuum of positive solutions that bifurcates from the trivial solution. The global structure of this continuum depends on the value of the nonlocal diffusion at infinity and the relative position of the refuge of the species and of the sets where it diffuses locally and not locally, respectively.

Mild solutions to the time fractional Navier-Stokes delay differential inclusions
Yejuan Wang and Tongtong Liang
2019, 24(8): 3713-3740 doi: 10.3934/dcdsb.2018312 +[Abstract](1174) +[HTML](627) +[PDF](559.95KB)
Abstract:

In this paper, we study a Navier-Stokes delay differential inclusion with time fractional derivative of order \begin{document} $\alpha\in(0,1)$ \end{document}. We first prove the local and global existence, decay and regularity properties of mild solutions when the initial data belongs to \begin{document} $C([-h,0];D(A_r^\varepsilon))$ \end{document}. The fractional resolvent operator theory and some techniques of measure of noncompactness are successfully applied to obtain the results.

The Vlasov-Navier-Stokes equations as a mean field limit
Franco Flandoli, Marta Leocata and Cristiano Ricci
2019, 24(8): 3741-3753 doi: 10.3934/dcdsb.2018313 +[Abstract](994) +[HTML](515) +[PDF](517.67KB)
Abstract:

Convergence of particle systems to the Vlasov-Navier-Stokes equations is a difficult topic with only fragmentary results. Under a suitable modification of the classical Stokes drag force interaction, here a partial result in this direction is proven. A particle system is introduced, its interaction with the fluid is modelled and tightness is proved, in a suitable topology, for the family of laws of the pair composed by solution of Navier-Stokes equations and empirical measure of the particles. Moreover, it is proved that every limit law is supported on weak solutions of the Vlasov-Navier-Stokes system. Open problems, like weak-strong uniqueness for this system and its relevance for the convergence of the particle system, are outlined.

A new proof of the competitive exclusion principle in the chemostat
Alain Rapaport and Mario Veruete
2019, 24(8): 3755-3764 doi: 10.3934/dcdsb.2018314 +[Abstract](991) +[HTML](521) +[PDF](345.58KB)
Abstract:

We give an new proof of the well-known competitive exclusion principle in the chemostat model with \begin{document} $N$ \end{document} species competing for a single resource, for any set of increasing growth functions. The proof is constructed by induction on the number of the species, after being ordered. It uses elementary analysis and comparisons of solutions of ordinary differential equations.

Stochastic one layer shallow water equations with Lévy noise
Justin Cyr, Phuong Nguyen and Roger Temam
2019, 24(8): 3765-3818 doi: 10.3934/dcdsb.2018331 +[Abstract](812) +[HTML](377) +[PDF](924.15KB)
Abstract:

This work investigates the existence of both martingale and pathwise solutions of the single layer shallow water equations on a bounded domain \begin{document}$ \mathcal{M} \subset \mathbb{R}^2 $\end{document} perturbed by a Lévy noise which may represent bursts of surface winds. The construction of both solutions are based on some truncation, the classical Faedo-Galerkin approximation scheme, a modified version of the Skorokhod representation theorem, stopping time arguments and anisotropic estimates.

The Magneto–Hydrodynamic equations: Local theory and blow-up of solutions
Jens Lorenz, Wilberclay G. Melo and Natã Firmino Rocha
2019, 24(8): 3819-3841 doi: 10.3934/dcdsb.2018332 +[Abstract](1036) +[HTML](419) +[PDF](515.92KB)
Abstract:

This work establishes local existence and uniqueness as well as blow-up criteria for solutions \begin{document}$ (u,b)(x,t) $\end{document} of the Magneto–Hydrodynamic equations in Sobolev–Gevrey spaces \begin{document}$ \dot{H}^s_{a,\sigma}(\mathbb{R}^3) $\end{document}. More precisely, we prove that there is a time \begin{document}$ T>0 $\end{document} such that \begin{document}$ (u,b)\in C([0,T];\dot{H}_{a,\sigma}^s(\mathbb{R}^3)) $\end{document} for \begin{document}$ a>0, \sigma\geq1 $\end{document} and \begin{document}$ \frac{1}{2}<s<\frac{3}{2} $\end{document}. If the maximal time interval of existence is finite, \begin{document}$ 0\leq t < T^* $\end{document}, then the blow–up inequality

holds for \begin{document}$ 0\leq t<T^*, \frac{1}{2}<s<\frac{3}{2} $\end{document}, \begin{document}$ a>0 $\end{document}, \begin{document}$ \sigma> 1 $\end{document} (\begin{document}$ 2\sigma_0 $\end{document} is the integer part of \begin{document}$ 2\sigma $\end{document}).

Construction of a contraction metric by meshless collocation
Peter Giesl and Holger Wendland
2019, 24(8): 3843-3863 doi: 10.3934/dcdsb.2018333 +[Abstract](746) +[HTML](428) +[PDF](627.43KB)
Abstract:

A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of an equilibrium and it is robust to small perturbations of the system, including those varying the position of the equilibrium.

The contraction metric is described by a matrix-valued function \begin{document}$ M(x) $\end{document} such that \begin{document}$ M(x) $\end{document} is positive definite and \begin{document}$ F(M)(x) $\end{document} is negative definite, where \begin{document}$ F $\end{document} denotes a certain first-order differential operator. In this paper, we show existence, uniqueness and continuous dependence on the right-hand side of the matrix-valued partial differential equation \begin{document}$ F(M)(x) = -C(x) $\end{document}. We then use a construction method based on meshless collocation, developed in the companion paper [12], to approximate the solution of the matrix-valued PDE. In this paper, we justify error estimates showing that the approximate solution itself is a contraction metric. The method is applied to several examples.

The Mandelbrot-van Ness fractional Brownian motion is infinitely differentiable with respect to its Hurst parameter
Stefan Koch and Andreas Neuenkirch
2019, 24(8): 3865-3880 doi: 10.3934/dcdsb.2018334 +[Abstract](834) +[HTML](403) +[PDF](391.89KB)
Abstract:

We study the Mandelbrot-van Ness representation of fractional Brownian motion \begin{document}$ B^H = (B^H_t)_{t \geq 0} $\end{document} with Hurst parameter \begin{document}$ H \in (0,1) $\end{document} and show that for arbitrary fixed \begin{document}$ t \geq 0 $\end{document} the mapping \begin{document}$ (0,1) \ni H \mapsto B_t^H \in \mathbb{R} $\end{document} is almost surely infinitely differentiable. Thus, the sample paths of fractional Brownian motion are smooth with respect to \begin{document}$ H $\end{document}. As a byproduct we obtain that scalar stochastic differential equations are differentiable with respect to the Hurst parameter of the driving fractional Brownian motion.

Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation
Michael B. Giles, Kristian Debrabant and Andreas Rössler
2019, 24(8): 3881-3903 doi: 10.3934/dcdsb.2018335 +[Abstract](1077) +[HTML](418) +[PDF](578.64KB)
Abstract:

The multilevel Monte Carlo path simulation method introduced by Giles (Operations Research, 56(3):607-617, 2008) exploits strong convergence properties to improve the computational complexity by combining simulations with different levels of resolution. In this paper we analyse its efficiency when using the Milstein discretisation; this has an improved order of strong convergence compared to the standard Euler-Maruyama method, and it is proved that this leads to an improved order of convergence of the variance of the multilevel estimator. Numerical results are also given for basket options to illustrate the relevance of the analysis.

Multiobjective model predictive control for stabilizing cost criteria
Lars Grüne and Marleen Stieler
2019, 24(8): 3905-3928 doi: 10.3934/dcdsb.2018336 +[Abstract](824) +[HTML](400) +[PDF](548.62KB)
Abstract:

In this paper we demonstrate how multiobjective optimal control problems can be solved by means of model predictive control. For our analysis we restrict ourselves to finite-dimensional control systems in discrete time. We show that convergence of the MPC closed-loop trajectory as well as upper bounds on the closed-loop performance for all objectives can be established if the ‘right’ Pareto-optimal control sequence is chosen in the iterations. It turns out that approximating the whole Pareto front is not necessary for that choice. Moreover, we provide statements on the relation of the MPC performance to the values of Pareto-optimal solutions on the infinite horizon, i.e. we investigate on the inifinite-horizon optimality of our MPC controller.

Some regularity results for a double time-delayed 2D-Navier-Stokes model
Julia García-Luengo, Pedro Marín-Rubio and Gabriela Planas
2019, 24(8): 3929-3946 doi: 10.3934/dcdsb.2018337 +[Abstract](718) +[HTML](380) +[PDF](417.49KB)
Abstract:

In this paper we analyze some regularity properties of a double time-delayed 2D-Navier-Stokes model, that includes not only a delay force but also a delay in the convective term. The interesting feature of the model -from the mathematical point of view- is that being in dimension two, it behaves similarly as a 3D-model without delay, and extra conditions in order to have uniqueness were required for well-posedness. This model was previously studied in several papers, being the existence of attractor in the \begin{document}$ L^2 $\end{document}-framework obtained by the authors [Discrete Contin. Dyn. Syst. 34 (2014), 4085-4105]. Here regularization properties of the solutions and existence of (regular) attractors for several associated dynamical systems are established. Moreover, relationships among these objects are also provided.

Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay
Rafael Obaya and Ana M. Sanz
2019, 24(8): 3947-3970 doi: 10.3934/dcdsb.2018338 +[Abstract](650) +[HTML](360) +[PDF](497.28KB)
Abstract:

This paper provides a dynamical frame to study non-autonomous parabolic partial differential equations with finite delay. Assuming monotonicity of the linearized semiflow, conditions for the existence of a continuous separation of type Ⅱ over a minimal set are given. Then, practical criteria for the uniform or strict persistence of the systems above a minimal set are obtained.

Stochastic dynamics of cell lineage in tissue homeostasis
Yuchi Qiu, Weitao Chen and Qing Nie
2019, 24(8): 3971-3994 doi: 10.3934/dcdsb.2018339 +[Abstract](875) +[HTML](472) +[PDF](1437.68KB)
Abstract:

During epithelium tissue maintenance, lineages of cells differentiate and proliferate in a coordinated way to provide the desirable size and spatial organization of different types of cells. While mathematical models through deterministic description have been used to dissect role of feedback regulations on tissue layer size and stratification, how the stochastic effects influence tissue maintenance remains largely unknown. Here we present a stochastic continuum model for cell lineages to investigate how both layer thickness and layer stratification are affected by noise. We find that the cell-intrinsic noise often causes reduction and oscillation of layer size whereas the cell-extrinsic noise increases the thickness, and sometimes, leads to uncontrollable growth of the tissue layer. The layer stratification usually deteriorates as the noise level increases in the cell lineage systems. Interestingly, the morphogen noise, which mixes both cell-intrinsic noise and cell-extrinsic noise, can lead to larger size of layer with little impact on the layer stratification. By investigating different combinations of the three types of noise, we find the layer thickness variability is reduced when cell-extrinsic noise level is high or morphogen noise level is low. Interestingly, there exists a tradeoff between low thickness variability and strong layer stratification due to competition among the three types of noise, suggesting robust layer homeostasis requires balanced levels of different types of noise in the cell lineage systems.

Trajectory and global attractors for generalized processes
Rodrigo Samprogna, Cláudia B. Gentile Moussa, Tomás Caraballo and Karina Schiabel
2019, 24(8): 3995-4020 doi: 10.3934/dcdsb.2019047 +[Abstract](673) +[HTML](286) +[PDF](490.81KB)
Abstract:

In this work the theory of generalized processes is used to describe the dynamics of a nonautonomous multivalued problem and, through this approach, some conditions for the existence of trajectory attractors are proved. By projecting the trajectory attractor on the phase space, the uniform attractor for the multivalued process associated to the problem is obtained and some conditions to guarantee the invariance of the uniform attractor are given. Furthermore, the existence of the uniform attractor for a class of \begin{document}$ p $\end{document}-Laplacian non-autonomous problems with dynamical boundary conditions is established.

A remark on global solutions to random 3D vorticity equations for small initial data
Michael Röckner, Rongchan Zhu and Xiangchan Zhu
2019, 24(8): 4021-4030 doi: 10.3934/dcdsb.2019048 +[Abstract](708) +[HTML](262) +[PDF](352.28KB)
Abstract:

In this paper, we prove that the solution constructed in [2] satisfies the stochastic vorticity equations with the stochastic integration being understood in the sense of the integration of controlled rough path introduced in [8]. As a result, we obtain the existence and uniqueness of the global solutions to the stochastic vorticity equations in 3D case for the small initial data independent of time, which can be viewed as a stochastic version of the Kato-Fujita result (see [10]).

Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation
Jean-Daniel Djida, Juan J. Nieto and Iván Area
2019, 24(8): 4031-4053 doi: 10.3934/dcdsb.2019049 +[Abstract](817) +[HTML](388) +[PDF](509.6KB)
Abstract:

We study a fractional time porous medium equation with fractional potential pressure. The initial data is assumed to be a bounded function with compact support and fast decay at infinity. We establish existence of weak solutions for which we determine whether the property of compact support is conserved in time depending on some parameters of the problem. Special attention is paid to the property of finite propagation for specific values of the parameters.

Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions
Klemens Fellner, Stefanie Sonner, Bao Quoc Tang and Do Duc Thuan
2019, 24(8): 4055-4078 doi: 10.3934/dcdsb.2019050 +[Abstract](897) +[HTML](286) +[PDF](452.15KB)
Abstract:

The stabilisation by noise on the boundary of the Chafee-Infante equation with dynamical boundary conditions subject to a multiplicative Itô noise is studied. In particular, we show that there exists a finite range of noise intensities that imply the exponential stability of the trivial steady state. This differs from previous works on the stabilisation by noise of parabolic PDEs, where the noise acts inside the domain and stabilisation typically occurs for an infinite range of noise intensities. To the best of our knowledge, this is the first result on the stabilisation of PDEs by boundary noise.

Detecting coupling directions with transcript mutual information: A comparative study
José M. Amigó, Beata Graff, Grzegorz Graff, Roberto Monetti and Katarzyna Tessmer
2019, 24(8): 4079-4097 doi: 10.3934/dcdsb.2019051 +[Abstract](617) +[HTML](318) +[PDF](1345.87KB)
Abstract:

Causal relationships are important to understand the dynamics of coupled processes and, moreover, to influence or control the effects by acting on the causes. Among the different approaches to determine cause-effect relationships and, in particular, coupling directions in interacting random or deterministic processes, we focus in this paper on information-theoretic measures. So, we study in the theoretical part the difference between directionality indicators based on transfer entropy as well as on its dimensional reduction via transcripts in algebraic time series representations. In the applications we consider specifically the lowest dimensional case, i.e., 3-dimensional transfer entropy, which is currently one of the most popular causality indicators, and the (2-dimensional) mutual information of transcripts. Needless to say, the lower dimensionality of the transcript-based indicator can make a difference in practice, where datasets are usually small. To compare numerically the performance of both directionality indicators, synthetic data (obtained with random processes) and real world data (in the form of biomedical recordings) are used. As happened in previous related work, we found again that the transcript mutual information performs as good as, and in some cases even better than, the lowest dimensional binned and symbolic transfer entropy, the symbols being ordinal patterns.

Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations
Zhenyu Lu, Junhao Hu and Xuerong Mao
2019, 24(8): 4099-4116 doi: 10.3934/dcdsb.2019052 +[Abstract](886) +[HTML](324) +[PDF](761.8KB)
Abstract:

Given an unstable hybrid stochastic differential equation (SDE, also known as an SDE with Markovian switching), can we design a delay feedback control to make the controlled hybrid SDE become asymptotically stable? The paper [14] by Mao et al. was the first to study the stabilisation by delay feedback controls for hybrid SDEs, though the stabilization by non-delay feedback controls had been well studied. A critical condition imposed in [14] is that both drift and diffusion coefficients of the given hybrid SDE need to satisfy the linear growth condition. However, many hybrid SDE models in the real world do not fulfill this condition (namely, they are highly nonlinear) and hence there is a need to develop a new theory for these highly nonlinear SDE models. The aim of this paper is to design delay feedback controls in order to stabilise a class of highly nonlinear hybrid SDEs whose coefficients satisfy the polynomial growth condition.

Attractors for A sup-cubic weakly damped wave equation in $ \mathbb{R}^{3} $
Xinyu Mei and Chunyou Sun
2019, 24(8): 4117-4143 doi: 10.3934/dcdsb.2019053 +[Abstract](833) +[HTML](312) +[PDF](565.08KB)
Abstract:

In this paper, the dynamical behavior of weakly damped wave equations with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then we obtain the existence of the \begin{document}$ \big(H_{lu}^{1}(\mathbb{R}^{3})\times L_{lu}^{2}(\mathbb{R}^{3}),H_{\rho}^{1}(\mathbb{R}^{3})\times L_{\rho}^{2}(\mathbb{R}^{3})\big) $\end{document}-global attractor for the Shatah-Struwe solutions semigroup of this equation. The results are crucially based on the recent extension of Strichartz estimates to the case of bounded domains.

Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients
Renhai Wang and Yangrong Li
2019, 24(8): 4145-4167 doi: 10.3934/dcdsb.2019054 +[Abstract](792) +[HTML](288) +[PDF](490.87KB)
Abstract:

We study some time-related properties of the random attractor for the stochastic wave equation on an unbounded domain with time-varying coefficient and force. We assume that the coefficient is bounded and the time-dependent force is backward tempered, backward complement-small, backward tail-small, and then prove both existence and backward compactness of a random attractor on the universe of all backward tempered sets. By using the Egoroff and Lusin theorems, we show the measurability of the absorbing set although it is the union of some random sets over an uncountable index set. Moreover, we obtain the backward compactness of the attractor if the force is periodic, and obtain the periodicity of the attractor if both force and coefficient are periodic.

Analysis of some splitting schemes for the stochastic Allen-Cahn equation
Charles-Edouard Bréhier and Ludovic Goudenège
2019, 24(8): 4169-4190 doi: 10.3934/dcdsb.2019077 +[Abstract](451) +[HTML](199) +[PDF](548.06KB)
Abstract:

We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution.

We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, \begin{document}$L^2(\Omega)$\end{document}-convergence of order almost \begin{document}$1/4$\end{document}, localized on an event of arbitrarily large probability, then convergence in probability of order almost \begin{document}$1/4$\end{document}.

The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.

Applications of optimal control of a nonconvex sweeping process to optimization of the planar crowd motion model
Tan H. Cao and Boris S. Mordukhovich
2019, 24(8): 4191-4216 doi: 10.3934/dcdsb.2019078 +[Abstract](409) +[HTML](201) +[PDF](527.96KB)
Abstract:

This paper concerns optimal control of a nonconvex perturbed sweeping process and its applications to optimization of the planar crowd motion model of traffic equilibria. The obtained theoretical results allow us to investigate a dynamic optimization problem for the microscopic planar crown motion model with finitely many participants and completely solve it analytically in the case of two participants.

Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary
José M. Arrieta, Ariadne Nogueira and Marcone C. Pereira
2019, 24(8): 4217-4246 doi: 10.3934/dcdsb.2019079 +[Abstract](464) +[HTML](204) +[PDF](862.41KB)
Abstract:

In this paper we analyze the asymptotic behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating region with reaction terms concentrated in a neighborhood of the oscillatory boundary $\theta_\varepsilon \subset\Omega_{\varepsilon }\subset \mathbb{R}^2$ when a small parameter $\varepsilon >0$ goes to zero. Our main result is concerned with the upper and lower semicontinuity of the set of solutions in $H^1$. We show that the solutions of our perturbed equation can be approximated with one defined in a fixed limit domain, which also captures the effects of reaction terms that take place in the original problem as a flux condition on the boundary of the limit domain.

Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function
Hjörtur Björnsson, Sigurdur Hafstein, Peter Giesl, Enrico Scalas and Skuli Gudmundsson
2019, 24(8): 4247-4269 doi: 10.3934/dcdsb.2019080 +[Abstract](429) +[HTML](185) +[PDF](633.36KB)
Abstract:

The γ-basin of attraction of the zero solution of a nonlinear stochastic differential equation can be determined through a pair of a local and a non-local Lyapunov function. In this paper, we construct a non-local Lyapunov function by solving a second-order PDE using meshless collocation. We provide a-posteriori error estimates which guarantee that the constructed function is indeed a non-local Lyapunov function. Combining this method with the computation of a local Lyapunov function for the linearisation around an equilibrium of the stochastic differential equation in question, a problem which is much more manageable than computing a Lyapunov function in a large area containing the equilibrium, we provide a rigorous estimate of the stochastic γ-basin of attraction of the equilibrium.

Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise
Adam Andersson and Felix Lindner
2019, 24(8): 4271-4294 doi: 10.3934/dcdsb.2019081 +[Abstract](428) +[HTML](198) +[PDF](544.02KB)
Abstract:

We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable Lévy noise. To this end, the Malliavin regularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting. For a class of path-dependent test functions, we obtain that the weak rate of convergence is twice the strong rate.

Convergence analysis of a symplectic semi-discretization for stochastic nls equation with quadratic potential
Jialin Hong, Lijun Miao and Liying Zhang
2019, 24(8): 4295-4315 doi: 10.3934/dcdsb.2019120 +[Abstract](174) +[HTML](89) +[PDF](1875.46KB)
Abstract:

In this paper, we investigate the convergence in probability of a stochastic symplectic scheme for stochastic nonlinear Schrödinger equation with quadratic potential and an additive noise. Theoretical analysis shows that our symplectic semi-discretization is of order one in probability under appropriate regularity conditions for the initial value and noise. Numerical experiments are given to simulate the long time behavior of the discrete averaged charge and energy as well as the influence of the external potential and noise, and to test the convergence order.

Approximation of the interface condition for stochastic Stefan-type problems
Marvin S. Müller
2019, 24(8): 4317-4339 doi: 10.3934/dcdsb.2019121 +[Abstract](181) +[HTML](104) +[PDF](485.29KB)
Abstract:

We consider approximations of the Stefan-type condition by imbalances of volume closely around the inner interface and study convergence of the solutions of the corresponding semilinear stochastic moving boundary problems. After a coordinate transformation, the problems can be reformulated as stochastic evolution equations on fractional power domains of linear operators. Here, the coefficients might fail to have linear growths and might be Lipschitz continuous only on bounded sets. We show continuity properties of the mild solution map in the coefficients and initial data, also incorporating the possibility of explosion of the solutions.

Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production
Felix X.-F. Ye and Hong Qian
2019, 24(8): 4341-4366 doi: 10.3934/dcdsb.2019122 +[Abstract](190) +[HTML](95) +[PDF](651.66KB)
Abstract:

We study finite state random dynamical systems (RDS) and their induced Markov chains (MC) as stochastic models for complex dynamics. The linear representation of deterministic maps in RDS is a matrix-valued random variable whose expectation corresponds to the transition matrix of the MC. The instantaneous Gibbs entropy, Shannon-Khinchin entropy of a step, and the entropy production rate of the MC are discussed. These three concepts, as key anchoring points in applications of stochastic dynamics, characterize respectively the uncertainties of a system at instant time \begin{document}$ t $\end{document}, the randomness generated in a step in the dynamics, and the dynamical asymmetry with respect to time reversal. The stationary entropy production rate, expressed in terms of the cycle distributions, has found an expression in terms of the probability of the deterministic maps with single attractor in the maximum entropy RDS. For finite RDS with invertible transformations, the non-negative entropy production rate of its MC is bounded above by the Kullback-Leibler divergence of the probability of the deterministic maps with respect to its time-reversal dual probability.

On a discrete three-dimensional Leslie-Gower competition model
Yunshyong Chow and Kenneth Palmer
2019, 24(8): 4367-4377 doi: 10.3934/dcdsb.2019123 +[Abstract](178) +[HTML](122) +[PDF](375.91KB)
Abstract:

We consider a special discrete time Leslie-Gower competition models for three species: \begin{document}$ x_i(t+1) = \frac{a_ix_i(t)}{1+x_i(t) +c \sum_{j\not = i} x_j(t)} $\end{document}   for \begin{document}$ 1\leq i \leq 3 $\end{document} and \begin{document}$ t \geq 0 $\end{document}. Here \begin{document}$ c $\end{document} is the interspecific coefficient among different species. Assume \begin{document}$ a_1>a_2>a_3>1 $\end{document}. It is shown that when \begin{document}$ 0<c< c_0: = (a_3-1)/(a_1+a_2-a_3-1) $\end{document}, a unique interior equilibrium \begin{document}$ E^* $\end{document} exists and is locally stable. Then from a general theorem in Balreira, Elaydi and Luis (2017), it follows that \begin{document}$ E^* $\end{document} is globally asymptotically stable. Using a result of Ruiz-Herrera [11], it is shown that the unique positive equilibrium in the \begin{document}$ x_1x_2 $\end{document}-plane is globally asymptotically stable for \begin{document}$ c_0<c<\beta_{21} = (a_2-1)/(a_1-1) $\end{document}. Then it is shown that \begin{document}$ (a_1-1, 0, 0) $\end{document} is globally asymptotically stable for \begin{document}$ \beta_{21} <c<\beta_{12} = (a_1-1)/(a_2-1) $\end{document}. This partially generalizes a result in Chow and Hsieh (2013) and Ackleh, Sacker and Salceanu (2014). For \begin{document}$ c>\beta_{12} $\end{document}, it is shown that there are multiple asymptotically stable equilibria.

The hipster effect: When anti-conformists all look the same
Jonathan D. Touboul
2019, 24(8): 4379-4415 doi: 10.3934/dcdsb.2019124 +[Abstract](165) +[HTML](124) +[PDF](11940.9KB)
Abstract:

In such different domains as statistical physics, neurosciences, spin glasses, social science, economics and finance, large ensemble of interacting individuals evolving following (mainstream) or against (hipsters) the majority are ubiquitous. Moreover, in a variety of applications, interactions between agents occur after specific delays that depends on the time needed to transport, transmit or take into account information. This paper focuses on the role of opposition to majority and delays in the emerging dynamics in a population composed of mainstream and anti-conformist individuals. To this purpose, we introduce a class of simple statistical system of interacting agents taking into account (ⅰ) the presence of mainstream and anti-conformist individuals and (ⅱ) delays, possibly heterogeneous, in the transmission of information. In this simple model, each agent can be in one of two states, and can change state in continuous time with a rate depending on the state of others in the past. We express the thermodynamic limit of these systems as the number of agents diverge, and investigate the solutions of the limit equation, with a particular focus on synchronized oscillations induced by delayed interactions. We show that when hipsters are too slow in detecting the trends, they will consistently make the same choice, and realizing this too late, they will switch, all together to another state where they remain alike. Another modality synchronizing hipsters are asymmetric interactions, particularly when the cross-interaction between hipsters and mainstreams aree prominent, i.e. when hipsters radically oppose to mainstream and mainstreams wish to follow the majority, even when led by hipsters. We demonstrate this phenomenon analytically using bifurcation theory and reduction to normal form. We find that, in the case of asymmetric interactions, the level of randomness in the decisions themselves also leads to synchronization of the hipsters. Beyond the choice of the best suit to wear this winter, this study may have important implications in understanding synchronization of nerve cells, investment strategies in finance, or emergent dynamics in social science, domains in which delays of communication and the geometry of information accessibility are prominent.

Asymptotic behavior of gene expression with complete memory and two-time scales based on the chemical Langevin equations
Yun Li, Fuke Wu and George Yin
2019, 24(8): 4417-4443 doi: 10.3934/dcdsb.2019125 +[Abstract](185) +[HTML](135) +[PDF](509.25KB)
Abstract:

Gene regulatory networks, which are complex high-dimensional stochastic dynamical systems, are often subject to evident intrinsic fluctuations. It is deemed reasonable to model the systems by the chemical Langevin equations. Since the mRNA dynamics are faster than the protein dynamics, we have a two-time scales system. In general, the process of protein degradation involves time delays. In this paper, we take the system memory into consideration in which we consider a model with a complete memory represented by an integral delay from \begin{document}$ 0 $\end{document} to \begin{document}$ t $\end{document}. Based on the averaging principle and perturbed test function method, this work examines the weak convergence of the slow-varying process. By treating the fast-varying process as a random noise, under appropriate conditions, it is shown that the slow-varying process converges weakly to the solution of a stochastic differential delay equation whose coefficients are the average of those of the original slow-varying process with respect to the invariant measure of the fast-varying process.

Some monotone properties for solutions to a reaction-diffusion model
Rui Li and Yuan Lou
2019, 24(8): 4445-4455 doi: 10.3934/dcdsb.2019126 +[Abstract](207) +[HTML](95) +[PDF](393.14KB)
Abstract:

Motivated by the recent investigation of a predator-prey model in heterogeneous environments [20], we show that the maximum of the unique positive solution of the scalar equation

\begin{document}$ \begin{equation} \begin{cases} \mu\Delta\theta+(m(x)-\theta)\theta = 0 \quad &\text{in} \quad \Omega,\\ \frac{\partial \theta}{\partial n} = 0 \quad &\text{on} \quad \partial\Omega \end{cases} \end{equation} $\end{document}

is a strictly monotone decreasing function of the diffusion rate \begin{document}$ \mu $\end{document} for several classes of function \begin{document}$ m $\end{document}, which substantially improves a result in [20]. However, the minimum of the positive solution of (1) is not always monotone increasing in the diffusion rate [15].

On the eventual stability of asymptotically autonomous systems with constraints
Jinlong Bai, Xuewei Ju, Desheng Li and Xiulian Wang
2019, 24(8): 4457-4473 doi: 10.3934/dcdsb.2019127 +[Abstract](196) +[HTML](95) +[PDF](420.57KB)
Abstract:

In this paper we first give a criterion on stability of equilibrium solutions for autonomous systems with constraints. Then we discuss the relationship between asymptotic behaviors of an asymptotically autonomous system with constraint and its limit system. Finally as an example, we revisit an extreme ideology model proposed in the literature and give a more detailed description on the dynamics of the system.

On the Alekseev-Gröbner formula in Banach spaces
Arnulf Jentzen, Felix Lindner and Primož Pušnik
2019, 24(8): 4475-4511 doi: 10.3934/dcdsb.2019128 +[Abstract](182) +[HTML](90) +[PDF](617.09KB)
Abstract:

The Alekseev-Gröbner formula is a well known tool in numerical analysis for describing the effect that a perturbation of an ordinary differential equation (ODE) has on its solution. In this article we provide an extension of the Alekseev-Gröbner formula for Banach space valued ODEs under, loosely speaking, mild conditions on the perturbation of the considered ODEs.

Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients
Ziheng Chen, Siqing Gan and Xiaojie Wang
2019, 24(8): 4513-4545 doi: 10.3934/dcdsb.2019154 +[Abstract](234) +[HTML](99) +[PDF](671.55KB)
Abstract:

This paper first establishes a fundamental mean-square convergence theorem for general one-step numerical approximations of Lévy noise driven stochastic differential equations with non-globally Lipschitz coefficients. Then two novel explicit schemes are designed and their convergence rates are exactly identified via the fundamental theorem. Different from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate appropriate assumptions to allow for its super-linear growth. However, we require that the Lévy measure is finite. New arguments are developed to handle essential difficulties in the convergence analysis, caused by the super-linear growth of the jump coefficient and the fact that higher moment bounds of the Poisson increments \begin{document}$ \int_t^{t+h} \int_Z \,\bar{N}(\mbox{d}s,\mbox{d}z), t \geq 0, h >0 $\end{document} contribute to magnitude not more than \begin{document}$ O(h) $\end{document}. Numerical results are finally reported to confirm the theoretical findings.

Nonlocal hyperbolic population models structured by size and spatial position: Well-posedness
Thomas Lorenz
2019, 24(8): 4547-4628 doi: 10.3934/dcdsb.2019156 +[Abstract](188) +[HTML](119) +[PDF](1117.63KB)
Abstract:

Detailed models of structured populations are spatial and involve nonlocal effects. These features lead to a broad class of population models structured by a physiological parameter and space. Our focus of interest is on the well-posedness of their initial value problems. In more detail, we specify sufficient conditions on the coefficient functions for existence, positivity, uniqueness of weak solutions and their continuous dependence on the given data. The solutions considered here have their values in \begin{document}$ L^p $\end{document} and, all conclusions about convergence and sequential compactness use an adaptation of the KANTOROVICH-RUBINSTEIN metric for this \begin{document}$ L^p $\end{document} space.

SDE-driven modeling of phenotypically heterogeneous tumors: The influence of cancer cell stemness
Julia M. Kroos, Christian Stinner, Christina Surulescu and Nico Surulescu
2019, 24(8): 4629-4663 doi: 10.3934/dcdsb.2019157 +[Abstract](216) +[HTML](97) +[PDF](2555.04KB)
Abstract:

We deduce cell population models describing the evolution of a tumor (possibly interacting with its environment of healthy cells) with the aid of differential equations. Thereby, different subpopulations of cancer cells allow accounting for the tumor heterogeneity. In our settings these include cancer stem cells known to be less sensitive to treatment and differentiated cancer cells having a higher sensitivity towards chemo- and radiotherapy. Our approach relies on stochastic differential equations in order to account for randomness in the system, arising e.g., due to the therapy-induced decreasing number of clonogens, which renders a pure deterministic model arguable. The equations are deduced relying on transition probabilities characterizing innovations of the two cancer cell subpopulations, and similarly extended to also account for the evolution of normal tissue. Several therapy approaches are introduced and compared by way of tumor control probability (TCP) and uncomplicated tumor control probability (UTCP). A PDE approach allows to assess the evolution of tumor and normal tissue with respect to time and to cell population densities which can vary continuously in a given set of states. Analytical approximations of solutions to the obtained PDE system are provided as well.

2018  Impact Factor: 1.008

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