American Institute of Mathematical Sciences

A linearized implicit local energy-preserving (LEP) scheme is proposed for the KPI equation by discretizing its multi-symplectic Hamiltonian form with the Kahan's method in time and symplectic Euler-box rule in space. It can be implemented easily, and also it is less storage-consuming and more efficient than the fully implicit methods. Several numerical experiments, including simulations of evolution of the line-soliton and lump-type soliton and interaction of the two lumps, are carried out to show the good performance of the scheme.

We consider a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects which is introduced in [2]. It is comprised of phase-field equation to describe tumor growth, which is coupled to a reaction-diffusion type equation for generic nutrient for the tumor. An additional equation couples the concentration of prostate-specific antigen (PSA) in the prostatic tissue and it obeys a linear reaction-diffusion equation. The system completes with homogeneous Dirichlet boundary conditions for the tumor variable and Neumann boundary condition for the nutrient and the concentration of PSA. Here we investigate the long time dynamics of the model. We first prove that the initial-boundary value problem generates a strongly continuous semigroup on a suitable phase space that admits the global attractor in a proper phase space. Moreover, we also discuss the convergence of a solution to a single stationary state and obtain a convergence rate estimate under some conditions on the coefficients.

We provide a new multiplicity result for a weighted \begin{document}$ p(x) $\end{document}-biharmonic problem on a bounded domain \begin{document}$ \Omega $\end{document} of \begin{document}$ \mathbb R^n $\end{document} with Navier conditions on \begin{document}$ \partial\Omega $\end{document}. Our approach, of variational nature, requires a suitable oscillating behavior of the nonlinearity and the associated weight to be compactly supported in \begin{document}$ \Omega $\end{document}.

The aim of this paper is to develop a center manifold theory for a class of stochastic partial differential equations with a non-dense domain through the Lyapunov-Perron method. We construct a novel variation of constants formula by the resolvent operator to formulate the integrated solutions. Moreover, we impose an additional condition involving a non-decreasing map to deduce the required estimate since Young's convolution inequality is not applicable. As an application, we present a stochastic parabolic equation to illustrate the obtained results.

We consider the susceptible-infected-removed (SIR) epidemic model and apply optimal control to it successfully. Here two control inputs are considered, so that the infection rate is decreased and infected individuals are removed. Our approach is to reduce computation of the optimal control input to that of the stable manifold of an invariant manifold in a Hamiltonian system. The situation in which the target equilibrium has a center direction is very different from similar previous work. Some numerical examples in which the computer software AUTO is used to numerically compute the stable manifold are given to demonstrate the usefulness of our approach for the optimal control in the SIR model. Our study suggests how we can decrease the number of infected individuals quickly before a critical situation occurs while keeping social and economic burdens small.

This paper is devoted to the study of the global well-posedness for a non-local-in-time Navier-Stokes equation. Our results recover in particular other existing well-posedness results for the Navier-Stokes equations and their time-fractional version. We show the appropriate manner to apply Kato's strategy and this context, with initial conditions in the divergence-free Lebesgue space \begin{document}$ L^\sigma_d(\mathbb{R}^d) $\end{document}. Temporal decay at \begin{document}$ 0 $\end{document} and \begin{document}$ \infty $\end{document} are obtained for the solution and its gradient.

This work deals with the Cauchy problem and the asymptotic behavior of the solution of the fermion equation in the Sobolev spaces with a polynomial weight in the torus. We first investigate the linearized equation and obtain the optimal exponential decay rate for the associated semigroup. Our strategy is taking advantage of quantitative spectral gap estimates in smaller reference Hilbert space, the factorization method and the enlargement of the functional space. We then turn to the nonlinear equation and prove the global existence and uniqueness of solutions in a close-to-equilibrium regime. Moreover, we prove an exponential stability for such a solution with the optimal decay rate given by the semigroup decay of the linearized equation.

We perturb with an additive noise the Hamiltonian system associated to a cubic anharmonic oscillator. This gives rise to a system of stochastic differential equations with quadratic drift and degenerate diffusion matrix. Firstly, we show that such systems possess explosive solutions for certain initial conditions. Then, we carry a small noise expansion's analysis of the stochastic system which is assumed to start from initial conditions that guarantee the existence of a periodic solution for the unperturbed equation. We then investigate the probabilistic properties of the sequence of coefficients which turn out to be the unique strong solutions of stochastic perturbations of the well-known Lamé's equation. We also obtain explicit expressions of these in terms of Jacobi elliptic functions. Furthermore, we prove, in the case of Brownian noise, a lower bound for the probability that the truncated expansion stays close to the solution of the deterministic problem. Lastly, when the noise is bounded, we provide conditions for the almost sure convergence of the global expansion.

In this paper we develop a time splitting combined with exponential wave integrator (EWI) Fourier pseudospectral (FP) method for the quantum Zakharov system (QZS), i.e. using the FP method for spatial derivatives, a time splitting technique and an EWI method for temporal derivatives in the Schrödinger-like equation and wave-type equations, respectively. The scheme is fully explicit and efficient due to fast Fourier transform. Numerical experiments for the QZS are presented to illustrate the accuracy and capability of the method, including accuracy tests, convergence of the QZS to the classical Zakharov system in the semi-classical limit, soliton-soliton collisions and pattern dynamics of the QZS in one-dimension, as well as the blow-up phenomena of QZS in two-dimension.

In this article a model of tumor growth is considered. The model is based on the reaction-diffusion equation that describes the distribution of nutrients within the tissue. Our aim was to predict the influence of nutrients on the tumor development. In the tissue the nutrients are transformed into energy, which supports the transfer of chemical and electrical signals and also transfer and copy the information in the tumor cells. We investigate, from a mathematical point of view, under which conditions this process takes place and how it affects the evolution of the tumor.

We study a normal form of the subcritical Hopf bifurcation subjected to time-delayed feedback. An unstable periodic orbit is born at the bifurcation in the normal form without the delay and it can be stabilized by the time-delayed feedback. We show that there exist finite time blow-up solutions for small initial functions, near the bifurcation point, when the feedback gains are small. This can happen even if the origin is stable or the unstable periodic orbit of the normal form is stabilized by the delay feedback. We give numerical examples to illustrate the theoretical result.

This paper is concerned with the wave phenomena in a compartmental epidemic model with nonlocal dispersal and relapse. We first show the well-posedness of solutions for such a problem. Then, in terms of the basic reproduction number and the wave speed, we establish a threshold result which reveals the existence and non-existence of the strong traveling waves accounting for phase transitions between the disease-free equilibrium and the endemic steady state. Further, we clarify and characterize the minimal wave speed of traveling waves. Finally, numerical simulations and discussions are also given to illustrate the analytical results. Our result indicates that the relapse can encourage the spread of the disease.

We consider a hyperbolic free boundary problem by means of minimizing time discretized functionals of Crank-Nicolson type. The feature of this functional is that it enjoys energy conservation in the absence of free boundaries, which is an essential property for numerical calculations. The existence and regularity of minimizers is shown and an energy estimate is derived. These results are then used to show the existence of a weak solution to the free boundary problem in the 1-dimensional setting.

We introduce a Gaussian measure formally preserved by the 2-dimensional Primitive Equations driven by additive Gaussian noise. Under such measure the stochastic equations under consideration are singular: we propose a solution theory based on the techniques developed by Gubinelli and Jara in [15] for a hyperviscous version of the equations.

In this paper, we investigate a weighted Dirichlet eigenvalue problem for a class of degenerate operators related to the \begin{document}$ h $\end{document} degree homogeneous \begin{document}$ p $\end{document}-Laplacian

\begin{document}$ \begin{equation} \nonumber \left \{ \begin{array}{ll} {|Du|^{h-1}}\Delta_p^N u+ \lambda a(x)|u|^{h-1}u = 0, \quad\quad \rm{in}\quad \Omega, \\ u = 0, \quad\quad \quad \quad \rm{on} \quad\partial\Omega. \end{array}\right. \end{equation} $\end{document}

Here \begin{document}$ a(x) $\end{document} is a positive continuous bounded function in the closure of \begin{document}$ \Omega\subset \mathbb{R}^n(n\geq 2), $\end{document} \begin{document}$ h>1, $\end{document} \begin{document}$ 2< p<\infty, $\end{document} and \begin{document}$ \Delta_p^N u = \frac{1}{p}|Du|^{2-p} {\rm div}\left(|Du|^{p-2}Du\right) $\end{document} is the normalized version of the \begin{document}$ p $\end{document}-Laplacian arising from a stochastic game named Tug-of-War with noise. We prove the existence of the principal eigenvalue \begin{document}$ \lambda_\Omega $\end{document}, which is positive and has a corresponding positive eigenfunction for \begin{document}$ p>n $\end{document}. The method is based on the maximum principle and approach analysis to the weighted eigenvalue problem. When a parameter \begin{document}$ \lambda<\lambda_\Omega $\end{document}, we establish some existence and uniqueness results related to this problem. During this procedure, we also prove some regularity estimates including Hölder continuity and Harnack inequality.

We consider the fourth-order Schrödinger equation

\begin{document}$ i\partial_tu+\Delta^2 u+\mu\Delta u+\lambda|u|^\alpha u = 0, $\end{document}

where \begin{document}$ \alpha>0, \mu = \pm1 $\end{document} or \begin{document}$ 0 $\end{document} and \begin{document}$ \lambda\in\mathbb{C} $\end{document}. Firstly, we prove local well-posedness in \begin{document}$ H^4\left( {\mathbb R}^N\right) $\end{document} in both \begin{document}$ H^4 $\end{document} subcritical and critical case: \begin{document}$ \alpha>0 $\end{document}, \begin{document}$ (N-8)\alpha\leq8 $\end{document}. Then, for any given compact set \begin{document}$ K\subset\mathbb{R}^N $\end{document}, we construct \begin{document}$ H^4( {\mathbb R}^N) $\end{document} solutions that are defined on \begin{document}$ (-T, 0) $\end{document} for some \begin{document}$ T>0 $\end{document}, and blow up exactly on \begin{document}$ K $\end{document} at \begin{document}$ t = 0 $\end{document}.

In this paper, using the Poincaré compactification technique we classify the topological phase portraits of a special kind of quadratic differential system, the Abel quadratic equations of third kind. In [1] where such investigation was presented for the first time some phase portraits were not correct and some were missing. Here we provide the complete list of non equivalent phase portraits that the Abel quadratic equations of third kind can exhibit and the bifurcation diagram of a \begin{document}$ 3 $\end{document}-parametric subfamily of it.

In this study, we develop a diffusive HIV-1 infection model with intracellular invasion, production and latent infection distributed delays, nonlinear incidence rate and nonlinear CTL immune response. The well-posedness, local and global stability for the model proposed are carefully investigated in spite of its strong nonlinearity and high dimension. It is revealed that its threshold dynamics are fully determined by the viral infection reproduction number \begin{document}$ \mathfrak{R}_0 $\end{document} and the reproduction number of CTL immune response \begin{document}$ \mathfrak{R}_1 $\end{document}. We also observe that the viral load at steady state (SS) fails to decrease even if \begin{document}$ \mathfrak{R}_1 $\end{document} increases through unit to lead to a stability switch from immune-inactivated infected SS to immune-activated infected SS. Finally, some simulations are performed to verify the analytical conclusions and we explore the significant impact of delays and CTL immune response on the spatiotemporal dynamics of HIV-1 infection.

The manuscript aims to investigate the qualitative analysis of a plankton-fish interaction with food limited growth rate of plankton population and non-constant harvesting of fish population. The ecological feasibility of population densities of both plankton and fish in terms of positivity and boundedness of solutions is shown. The conditions for the existence of various equilibrium points and their stability are derived thoroughly. This study mainly focuses on how the harvesting affects equilibrium points, their stability, periodic solutions and bifurcations in the proposed system. It is shown that the system exhibits saddle-node bifurcation in the form of a collision of two interior equilibrium points. Existence conditions for the occurrence of Hopf-bifurcation around interior equilibrium points are discussed. Lyapunov coefficients are examined to check the stability properties of these periodic solutions. We have also plotted the bifurcation diagrams for saddle-node, transcritical and Hopf bifurcations. A detailed algorithm for the occurrence of Bogdanov-Takens bifurcation is derived and finally some numerical simulations are also carried out to validate the theoretical results. This work suggests that the harvesting of fish population can change the dynamics of the system, which may be useful for the ecological management.

We consider the Cournot-Theocharis oligopoly model, where firms make their choices under adaptive expectations. Following [2], we assume that quantities cannot be negative, which implies that the model is nonlinear. The stability of the equilibrium point in the general case is analyzed. We focus on the conditions for which the number of competitors is reduced to a monopoly. In particular, we find necessary and sufficient conditions giving an analytic proof of the convergence to oligopoly to monopoly.

We consider the logistic family \begin{document}$ f_{a} $\end{document} and a family of homeomorphisms \begin{document}$ \phi _{q} $\end{document}. The \begin{document}$ q $\end{document}-deformed system is given by the composition map \begin{document}$ f_{a}\circ \phi _{q} $\end{document}. We study when this system has non zero fixed points which are LAS and GAS. We also give an alternative approach to study the dynamics of the \begin{document}$ q $\end{document}-deformed system with special emphasis on the so-called Parrondo's paradox finding parameter values \begin{document}$ a $\end{document} for which \begin{document}$ f_{a} $\end{document} is simple while \begin{document}$ f_{a}\circ \phi _{q} $\end{document} is dynamically complicated. We explore the dynamics when several \begin{document}$ q $\end{document}-deformations are applied.

In this paper, a stabilized extended finite element method is proposed for Stokes interface problems on unfitted triangulation elements which do not require the interface align with the triangulation. The problem is written on mixed form using nonconforming \begin{document}$ P_1 $\end{document} velocity and elementwise \begin{document}$ P_0 $\end{document} pressure. Extra stabilization terms involving velocity and pressure are added in the discrete bilinear form. An inf-sup stability result is derived, which is uniform with respect to mesh size \begin{document}$ h $\end{document}, the viscosity and the position of the interface. An optimal priori error estimates are obtained. Moreover, the errors in energy norm for velocity and in \begin{document}$ L^2 $\end{document} norm for pressure are uniform to the viscosity and the location of the interface. Results of numerical experiments are presented to support the theoretical analysis.

In this note, we consider generalizations of the Cucker-Smale dynamical system and we derive rigorously in Wasserstein's type topologies the mean-field limit (and propagation of chaos) to the Vlasov-type equation introduced in [13]. Unlike previous results on the Cucker-Smale model, our approach is not based on the empirical measures, but, using an Eulerian point of view introduced in [9] in the Hamiltonian setting, we show the limit providing explicit constants. Moreover, for non strictly Cucker-Smale particles dynamics, we also give an insight on what induces a flocking behavior of the solution to the Vlasov equation to the - unknown a priori - flocking properties of the original particle system.

Based on the method of compression and pull forming mechanism (CAP), the authors in a well-known paper proposed and analyzed the Lü-like system: \begin{document}$ \dot{x} = a(y - x) + dxz $\end{document}, \begin{document}$ \dot{y} = - xz + fy $\end{document}, \begin{document}$ \dot{z} = -ex^{2} + xy + cz $\end{document}, which was thought to display an interesting three-scroll chaotic attractors (called as Pan-A attractor) when \begin{document}$ (a, d, f, e, c) = (40, 0.5, 20, 0.65, \frac{5}{6}) $\end{document}. Unfortunately, by further analysis and Matlab simulation, we show that the Pan-A attractor found is actually a stable torus. Accordingly, we find a new true three-scroll chaotic attractor coexisting with a single saddle-node \begin{document}$ (0, 0, 0) $\end{document} for the case with \begin{document}$ (a, d, f, e, c) = (168, 0.4, 100, 0.70, 11) $\end{document}. Interestingly, the forming mechanism of singularly degenerate heteroclinic cycles of that system is bidirectional, rather than unilateral in the case of most other Lorenz-like systems. This further motivates us to revisit in detail its other complicated dynamical behaviors, i.e., the ultimate bound sets, the globally exponentially attractive sets, Hopf bifurcation, limit cycles coexisting attractors and so on. Numerical simulations not only are consistent with the results of theoretical analysis, but also illustrate that collapse of infinitely many singularly degenerate heteroclinic cycles and explosions of normally hyperbolic stable nodes or foci generate the aforementioned three-scroll attractor. In particular, four or two unstable limit cycles coexisting one chaotic attractor, the saddle \begin{document}$ E_{0} $\end{document} and the stable \begin{document}$ E_{\pm} $\end{document} are located in two globally exponentially attractive sets. These results together indicate that this system deserves further exploration in chaos-based applications.

The paper studies the pattern formation dynamics of a discrete in time and space model with nonlocal resource competition and dispersal. Our model is generalized from the metapopulation model proposed by Doebeli and Killingback [2003. Theor. Popul. Biol. 64, 397-416] in which competition for resources occurs only between neighboring populations. Our study uses symmetric discrete probability kernels to model nonlocal interaction and dispersal. A linear stability analysis of the model shows that solutions to this equation exhibits pattern formation when the dispersal rate is sufficiently small and the discrete interaction kernel satisfies certain conditions. Moreover, a weakly nonlinear analysis is used to approximate stationary patterns arising from the model. Numerical solutions to the model and the approximations obtained through the weakly nonlinear analysis are compared.