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Discrete & Continuous Dynamical Systems - B

November 2018 , Volume 23 , Issue 9

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Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays
Shihe Xu, Meng Bai and Fangwei Zhang
2018, 23(9): 3535-3551 doi: 10.3934/dcdsb.2017213 +[Abstract](61916) +[HTML](847) +[PDF](416.76KB)
Abstract:

In this paper we study a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. It is assumed that the process of proliferation is delayed compared with apoptosis. The delay represents the time taken for cells to undergo mitosis. By employing stability theory for functional differential equations, comparison principle and some meticulous mathematical analysis, we mainly study the asymptotic behavior of the solution, and prove that in the case \begin{document} $c$ \end{document} (the ratio of the diffusion time scale to the tumor doubling time scale) is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to one of two dormant states as \begin{document} $t\to ∞$ \end{document}. The results show that dynamical behavior of solutions of the model are similar to that of solutions for corresponding nonretarded problems under some conditions.

Pullback attractors for a class of non-autonomous thermoelastic plate systems
Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento and Karina Schiabel
2018, 23(9): 3553-3571 doi: 10.3934/dcdsb.2017214 +[Abstract](1273) +[HTML](686) +[PDF](461.59KB)
Abstract:

In this article we study the asymptotic behavior of solutions, in the sense of pullback attractors, of the evolution system

subject to boundary conditions

where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $N≥ 2$, which boundary $\partialΩ$ is assumed to be a $\mathcal{C}^4$-hypersurface, $κ>0$ is constant, $a$ is an Hölder continuous function and $f$ is a dissipative nonlinearity locally Lipschitz in the second variable. Using the theory of uniform sectorial operators, in the sense of P. Sobolevskiǐ ([23]), we give a partial description of the fractional power spaces scale for the thermoelastic plate operator and we show the local and global well-posedness of this non-autonomous problem. Furthermore we prove existence and uniform boundedness of pullback attractors.

A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential
Stefano Pasquali
2018, 23(9): 3573-3594 doi: 10.3934/dcdsb.2017215 +[Abstract](1052) +[HTML](650) +[PDF](513.85KB)
Abstract:

We study the one-dimensional nonlinear Klein-Gordon (NLKG) equation with a convolution potential, and we prove that solutions with small analytic norm remain small for exponentially long times. The result is uniform with respect to $c ≥ 1$, which however has to belong to a set of large measure.

A space-time discontinuous Galerkin spectral element method for the Stefan problem
Chaoxu Pei, Mark Sussman and M. Yousuff Hussaini
2018, 23(9): 3595-3622 doi: 10.3934/dcdsb.2017216 +[Abstract](1418) +[HTML](760) +[PDF](947.01KB)
Abstract:

A novel space-time discontinuous Galerkin (DG) spectral element method is presented to solve the one dimensional Stefan problem in an Eulerian coordinate system. This method employs the level set procedure to describe the time-evolving interface. To deal with the prior unknown interface, a backward transformation and a forward transformation are introduced in the space-time mesh. By combining an Eulerian description, i.e., a fixed frame of reference, with a Lagrangian description, i.e., a moving frame of reference, the issue of dealing with implicitly defined arbitrary shaped space-time elements is avoided. The backward transformation maps the unknown time-varying interface in the fixed frame of reference to a known stationary interface in the moving frame of reference. In the moving frame of reference, the transformed governing equations, written in the space-time framework, are discretized by a DG spectral element method in each space-time slab. The forward transformation is used to update the level set function and then to project the solution in each phase back from the moving frame of reference to the fixed Eulerian grid. Two options for calculating the interface velocity are presented, and both options exhibit spectral accuracy. Benchmark tests indicate that the method converges with spectral accuracy in both space and time for the temperature distribution and the interface velocity. A Picard iteration algorithm is introduced in order to solve the nonlinear algebraic system of equations and it is found that just a few iterations lead to convergence.

Transient growth in stochastic Burgers flows
Diogo Poças and Bartosz Protas
2018, 23(9): 3623-3643 doi: 10.3934/dcdsb.2018052 +[Abstract](757) +[HTML](526) +[PDF](1207.59KB)
Abstract:

This study considers the problem of the extreme behavior exhibited by solutions to Burgers equation subject to stochastic forcing. More specifically, we are interested in the maximum growth achieved by the "enstrophy" (the Sobolev \begin{document}$H^1$\end{document} seminorm of the solution) as a function of the initial enstrophy \begin{document}$\mathcal{E}_0$\end{document}, in particular, whether in the stochastic setting this growth is different than in the deterministic case considered by Ayala & Protas (2011). This problem is motivated by questions about the effect of noise on the possible singularity formation in hydrodynamic models. The main quantities of interest in the stochastic problem are the expected value of the enstrophy and the enstrophy of the expected value of the solution. The stochastic Burgers equation is solved numerically with a Monte Carlo sampling approach. By studying solutions obtained for a range of optimal initial data and different noise magnitudes, we reveal different solution behaviors and it is demonstrated that the two quantities always bracket the enstrophy of the deterministic solution. The key finding is that the expected values of the enstrophy exhibit the same power-law dependence on the initial enstrophy \begin{document}$\mathcal{E}_0$\end{document} as reported in the deterministic case. This indicates that the stochastic excitation does not increase the extreme enstrophy growth beyond what is already observed in the deterministic case.

A stochastic SIRI epidemic model with Lévy noise
Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo and Roger Pettersson
2018, 23(9): 3645-3661 doi: 10.3934/dcdsb.2018057 +[Abstract](1578) +[HTML](767) +[PDF](2371.8KB)
Abstract:

Some diseases such as herpes, bovine and human tuberculosis exhibit relapse in which the recovered individuals do not acquit permanent immunity but return to infectious class. Such diseases are modeled by SIRI models. In this paper, we establish the existence of a unique global positive solution for a stochastic epidemic model with relapse and jumps. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we present some numerical results to support the theoretical work.

Qualitative analysis of kinetic-based models for tumor-immune system interaction
Martina Conte, Maria Groppi and Giampiero Spiga
2018, 23(9): 3663-3684 doi: 10.3934/dcdsb.2018060 +[Abstract](881) +[HTML](514) +[PDF](1032.82KB)
Abstract:

A mathematical model, based on a mesoscopic approach, describing the competition between tumor cells and immune system in terms of kinetic integro-differential equations is presented. Four interacting components are considered, representing, respectively, tumors cells, cells of the host environment, cells of the immune system, and interleukins, which are capable to modify the tumor-immune system interaction and to contribute to destroy tumor cells. The internal state variable (activity) measures the capability of a cell of prevailing in a binary interaction. Under suitable assumptions, a closed set of autonomous ordinary differential equations is then derived by a moment procedure and two three-dimensional reduced systems are obtained in some partial quasi-steady state approximations. Their qualitative analysis is finally performed, with particular attention to equilibria and their stability, bifurcations, and their meaning. Results are obtained on asymptotically autonomous dynamical systems, and also on the occurrence of a particular backward bifurcation.

On a coupled SDE-PDE system modeling acid-mediated tumor invasion
Sandesh Athni Hiremath, Christina Surulescu, Anna Zhigun and Stefanie Sonner
2018, 23(9): 3685-3715 doi: 10.3934/dcdsb.2018071 +[Abstract](1211) +[HTML](701) +[PDF](4097.16KB)
Abstract:

We propose and analyze a multiscale model for acid-mediated tumor invasion accounting for stochastic effects on the subcellular level. The setting involves a PDE of reaction-diffusion-taxis type describing the evolution of the tumor cell density, the movement being directed towards pH gradients in the local microenvironment, which is coupled to a PDE-SDE system characterizing the dynamics of extracellular and intracellular proton concentrations, respectively. The global well-posedness of the model is shown and numerical simulations are performed in order to illustrate the solution behavior.

A new criterion to a two-chemical substances chemotaxis system with critical dimension
Xueli Bai and Suying Liu
2018, 23(9): 3717-3721 doi: 10.3934/dcdsb.2018074 +[Abstract](917) +[HTML](750) +[PDF](350.32KB)
Abstract:

We mainly investigate the global boundedness of the solution to the following system,

under homogeneous Neumann boundary conditions with nonnegative smooth initial data in a smooth bounded domain $Ω\subset \mathbb{R}^n$ with critical space dimension $n = 4$. This problem has been considered by K. Fujie and T. Senba in [5]. They proved that for the symmetric case the condition $\int_\Omega {u_0 < \frac{(8π)^2}{χ}} $ yields global boundedness, where $u_0$ is the instal data for $u$. In this paper, inspired by some new techniques established in [3], we give a new criterion for global boundedness of the solution. As a byproduct, we obtain a simplified proof for one of the main results in [5].

Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations
Mickaël D. Chekroun
2018, 23(9): 3723-3753 doi: 10.3934/dcdsb.2018075 +[Abstract](991) +[HTML](640) +[PDF](727.95KB)
Abstract:

Semilinear parabolic problems are considered for which we prove their topological sensitivity to arbitrarily small perturbations of the nonlinear term. This instability result is a consequence of the sensitivity of the multiplicity of solutions of the corresponding nonlinear elliptic problems. As shown here, it is indeed always possible (in dimension \begin{document} $d = 1$ \end{document} or \begin{document} $d = 2$ \end{document}) to find an arbitrary small perturbation that e.g. generates locally an S on the global bifurcation diagram, substituting thus a single solution by several ones. Such an increase in the local multiplicity of the solutions to the elliptic problem results then into a topological instability for the corresponding parabolic problem.

The rigorous proof of this instability result requires though to revisit the classical concept of topological equivalence to encompass important cases for applications such as semi-linear parabolic problems for which the semigroup may exhibit non-global dissipative properties, allowing for the coexistence of blow-up regions and local attractors in the phase space; cases that arise e.g. in combustion theory. A revised framework of topological robustness is thus introduced in that respect within which the main topological instability result is then proved for continuous, locally Lipschitz but not necessarily \begin{document} $C^1$ \end{document} nonlinear terms, that prevent in particular the use of linearization techniques.

Extinction and coexistence of species for a diffusive intraguild predation model with B-D functional response
Guohong Zhang and Xiaoli Wang
2018, 23(9): 3755-3786 doi: 10.3934/dcdsb.2018076 +[Abstract](567) +[HTML](529) +[PDF](714.73KB)
Abstract:

Extinction and coexistence of species are two fundamental issues in systems with IGP. In this paper, we constructed a mathematical model with IGP by introducing heterogeneous environment and B-D functional response between the predator and prey. First, some sufficient conditions for the extinction and permanence of the time-dependent system were obtained by using comparison principle and upper and lower solution method. Second, we got some necessary and sufficient conditions for the existence of coexistence states by means of the fixed point index theory. In addition, we discussed the uniqueness and stability of coexistence state under some conditions. Finally, we studied the effects of the parameters in system on the spatial distribution of species and obtained some interesting results about the extinction and coexistence of species by using numerical simulations.

Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type
B. Ambrosio, M. A. Aziz-Alaoui and V. L. E. Phan
2018, 23(9): 3787-3797 doi: 10.3934/dcdsb.2018077 +[Abstract](729) +[HTML](416) +[PDF](366.31KB)
Abstract:

We focus on the long time behavior of complex networks of reaction-diffusion systems. We prove the existence of the global attractor and the $L^{∞}$-bound for networks of $n$ reaction-diffusion systems that belong to a class that generalizes the FitzHugh-Nagumo reaction-diffusion equations.

Coexistence and extinction in Time-Periodic Volterra-Lotka type systems with nonlocal dispersal
Tung Nguyen and Nar Rawal
2018, 23(9): 3799-3816 doi: 10.3934/dcdsb.2018080 +[Abstract](571) +[HTML](432) +[PDF](480.63KB)
Abstract:

This paper deals with coexistence and extinction of time periodic Volterra-Lotka type competing systems with nonlocal dispersal. Such issues have already been studied for time independent systems with nonlocal dispersal and time periodic systems with random dispersal, but have not been studied yet for time periodic systems with nonlocal dispersal. In this paper, the relations between the coefficients representing Malthusian growths, self regulations and competitions of the two species have been obtained which ensure coexistence and extinction for the time periodic Volterra-Lotka type system with nonlocal dispersal. The underlying environment of the Volterra-Lotka type system under consideration has either hostile surroundings, or non-flux boundary, or is spatially periodic.

A non-autonomous predator-prey model with infected prey
Yang Lu, Xia Wang and Shengqiang Liu
2018, 23(9): 3817-3836 doi: 10.3934/dcdsb.2018082 +[Abstract](704) +[HTML](453) +[PDF](769.98KB)
Abstract:

A non-constant eco-epidemiological model with SIS-type infectious disease in prey is formulated and investigated, it is assumed that the disease is endemic in prey before the invasion of predator and that predation is more likely on infected prey than on the uninfected. Sufficient conditions for both permanence and extinction of the infected prey, and the necessary conditions for the permanence of the infected prey are established. It is shown that the predation preference to infected prey may even increase the possibility of disease endemic, and that the introduction of new resource for predator could be helpful for it to eradicate the infected prey. Numerical simulations have been performed to verify/extend our analytical results.

The impact of releasing sterile mosquitoes on malaria transmission
Hongyan Yin, Cuihong Yang, Xin'an Zhang and Jia Li
2018, 23(9): 3837-3853 doi: 10.3934/dcdsb.2018113 +[Abstract](657) +[HTML](451) +[PDF](506.03KB)
Abstract:

The sterile mosquitoes technique in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population has been used in preventing the malaria transmission. To study the impact of releasing sterile mosquitoes on the malaria transmission, we first formulate a simple SEIR (susceptible-exposed-infected-recovered) malaria transmission model as our baseline model, derive a formula for the reproductive number of infection, and determine the existence of endemic equilibria. We then include sterile mosquitoes in the baseline model and consider the case of constant releases of sterile mosquitoes. We examine how the releases affect the reproductive numbers and endemic equilibria for the model with interactive mosquitoes and investigate the impact of releasing sterile mosquitoes on the malaria transmission.

On one problem of viscoelastic fluid dynamics with memory on an infinite time interval
Victor Zvyagin and Vladimir Orlov
2018, 23(9): 3855-3877 doi: 10.3934/dcdsb.2018114 +[Abstract](616) +[HTML](367) +[PDF](437.22KB)
Abstract:

In the present paper we establish the existence of weak solutions of one boundary value problem for one model of a viscoelastic fluid with memory along the trajectories of the velocity field on an infinite time interval. We use solvability of related approximating initial-boundary value problems on finite time intervals and responding pass to the limit.

On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals
Yuning Liu and Wei Wang
2018, 23(9): 3879-3899 doi: 10.3934/dcdsb.2018115 +[Abstract](541) +[HTML](359) +[PDF](496.5KB)
Abstract:

This work is concerned with the solvability of a Navier-Stokes/Q-tensor coupled system modeling the nematic liquid crystal flow on a bounded domain in three dimensional Euclidian space with strong anchoring boundary condition for the order parameter. We prove the existence and uniqueness of local in time strong solutions to the system with an anisotropic elastic energy. The proof is based on mainly two ingredients: first, we show that the Euler-Lagrange operator corresponding to the Landau-de Gennes free energy with general elastic coefficients fulfills the strong Legendre condition. This result together with a higher order energy estimate leads to the well-posedness of the linearized system, and then a local in time solution of the original system which is regular in temporal variable follows via a fixed point argument. Secondly, the hydrodynamic part of the coupled system can be reformulated into a quasi-stationary Stokes type equation to which the regularity theory of the generalized Stokes system, and then a bootstrap argument can be applied to enhance the spatial regularity of the local in time solution.

Asymptotic spreading of time periodic competition diffusion systems
Wei-Jian Bo and Guo Lin
2018, 23(9): 3901-3914 doi: 10.3934/dcdsb.2018116 +[Abstract](589) +[HTML](387) +[PDF](404.48KB)
Abstract:

This paper deals with the asymptotic spreading of time periodic Lotka-Volterra competition diffusion systems, which formulates the coinvasion-coexistence process. By combining auxiliary systems with comparison principle, some results on asymptotic spreading are established. Our conclusions indicate that the coinvasions of two competitors may be successful, and the interspecific competitions slow the invasion speed of one species.

Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives
Kai Liu
2018, 23(9): 3915-3934 doi: 10.3934/dcdsb.2018117 +[Abstract](489) +[HTML](384) +[PDF](467.12KB)
Abstract:

In this work, we shall consider the existence and uniqueness of stationary solutions to stochastic partial functional differential equations with additive noise in which a neutral type of delay is explicitly presented. We are especially concerned about those delays appearing in both spatial and temporal derivative terms in which the coefficient operator under spatial variables may take the same form as the infinitesimal generator of the equation. We establish the stationary property of the neutral system under investigation by focusing on distributed delays. In the end, an illustrative example is analyzed to explain the theory in this work.

Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain
Yiqiu Mao
2018, 23(9): 3935-3947 doi: 10.3934/dcdsb.2018118 +[Abstract](587) +[HTML](354) +[PDF](327.88KB)
Abstract:

The main objective of this article is to study the dynamic transitions of the FitzHugh-Nagumo equations on a finite domain with the Neumann boundary conditions and with uniformly injected current. We show that when certain parameter conditions are satisfied, the system undergoes a continuous dynamic transition to a limit cycle. A mixed type transition is also found when other conditions are imposed on the parameters. The main method used here is Ma & Wang's dynamic transition theory, which can be used generally on different set-ups for the FitzHugh-Nagumo equations.

Improved extensibility criteria and global well-posedness of a coupled chemotaxis-fluid model on bounded domains
Jishan Fan and Kun Zhao
2018, 23(9): 3949-3967 doi: 10.3934/dcdsb.2018119 +[Abstract](475) +[HTML](388) +[PDF](452.29KB)
Abstract:

This paper is contributed to the qualitative analysis of a coupled chemotaxis-fluid model on bounded domains in multiple spatial dimensions. Based on scaling-invariant argument and energy method, several optimal extensibility criteria for local classical solutions are established. As a by-product, a global well-posedness result is obtained in the two-dimensional case for general initial data.

A multiscale model of the CD8 T cell immune response structured by intracellular content
Loïc Barbarroux, Philippe Michel, Mostafa Adimy and Fabien Crauste
2018, 23(9): 3969-4002 doi: 10.3934/dcdsb.2018120 +[Abstract](797) +[HTML](585) +[PDF](1742.29KB)
Abstract:

During the primary CD8 T cell immune response, CD8 T cells undergo proliferation and continuous differentiation, acquiring cytotoxic abilities to address the infection and generate an immune memory. At the end of the response, the remaining CD8 T cells are antigen-specific memory cells that will respond stronger and faster in case they are presented this very same antigen again. We propose a nonlinear multiscale mathematical model of the CD8 T cell immune response describing dynamics of two inter-connected physical scales. At the intracellular scale, the level of expression of key proteins involved in proliferation, death, and differentiation of CD8 T cells is modeled by a delay differential system whose dynamics define maturation velocities of CD8 T cells. At the population scale, the amount of CD8 T cells is represented by a discrete density and cell fate depends on their intracellular content. We introduce the model, then show essential mathematical properties (existence, uniqueness, positivity) of solutions and analyse their asymptotic behavior based on the behavior of the intracellular regulatory network. We numerically illustrate the model's ability to qualitatively reproduce both primary and secondary responses, providing a preliminary tool for investigating the generation of long-lived CD8 memory T cells and vaccine design.

Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions
Xi Wang, Zuhan Liu and Ling Zhou
2018, 23(9): 4003-4020 doi: 10.3934/dcdsb.2018121 +[Abstract](610) +[HTML](428) +[PDF](490.06KB)
Abstract:

In this paper, we study the generalized chemotaxis system with fractional Laplacian. The existence and the uniqueness of global classical solution are proved under the assumption that the initial data are small enough. During the proof, with the help of the fixed point theorem, the asymptotic decay behaviors of \begin{document}$ u $\end{document} and \begin{document}$ \nabla{v} $\end{document} are also shown.

Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations
Caidi Zhao, Gang Xue and Grzegorz Łukaszewicz
2018, 23(9): 4021-4044 doi: 10.3934/dcdsb.2018122 +[Abstract](589) +[HTML](401) +[PDF](580.83KB)
Abstract:

In this article, we first provide a sufficient and necessary condition for the existence of a pullback-\begin{document}$ {\mathcal D} $\end{document} attractor for the process defined on a Hilbert space of infinite sequences. As an application, we investigate the non-autonomous discrete Klein-Gordon-Schrödinger system of equations, prove the existence of the pullback-\begin{document}$ {\mathcal D} $\end{document} attractor and then the existence of a unique family of invariant Borel probability measures associated with the considered system.

Analysis of a stage-structured dengue model
Jinping Fang, Guang Lin and Hui Wan
2018, 23(9): 4045-4061 doi: 10.3934/dcdsb.2018125 +[Abstract](560) +[HTML](570) +[PDF](770.76KB)
Abstract:

In order to study the impact of control measures and limited resource on dengue transmission dynamics, we formulate a stage-structured dengue model. The basic investigation of the model, such as the existence of equilibria and their stability, have been proved. It is also shown that this model may undergo backward bifurcation, where the stable disease-free equilibrium co-exists with an endemic equilibrium. The backward bifurcation property can be removed by ignoring the disease-induced death in human population and the global stability of the unique endemic equilibrium has been proved. Sensitivity analysis with respect to \begin{document}$R_0$\end{document} has been carried out to explore the impact of model parameters. In addition, numerical analysis manifests that the more intensive control measures in targeting immature and adult mosquitoes are both effective in preventing dengue outbreaks. It is also shown that the earlier the control intervention begins, the less people would be infected and the earlier dengue would be eradicated. Even later epidemic prevention and control can also effectively reduce the severity of pandemic. Moreover, comprehensive control measures are more effective than a single measure.

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