American Institute of Mathematical Sciences

This paper is concerned with the Galerkin spectral approximation of an optimal control problem governed by the elliptic partial differential equations (PDEs). Its objective functional depends on the control variable governed by the \begin{document}$ L^2 $\end{document}-norm constraint. The optimality conditions for both the optimal control problem and its corresponding spectral approximation problem are given, successively. Thanks to some lemmas and the auxiliary systems, a priori error estimates of the Galerkin spectral approximation problem are established in detail. Moreover, a posteriori error estimates of the spectral approximation problem are also investigated, which include not only \begin{document}$ H^1 $\end{document}-norm error for the state and co-state but also \begin{document}$ L^2 $\end{document}-norm error for the control, state and costate. Finally, three numerical examples are executed to demonstrate the errors decay exponentially fast.

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation

\begin{document}$ iu_{t} +\Delta u = |x|^{-b} f(u), \;u(0)\in H^{s} (\mathbb R^{n} ), $\end{document}

where \begin{document}$ n\in \mathbb N $\end{document}, \begin{document}$ 0<s<\min \{ n, \; 1+n/2\} $\end{document}, \begin{document}$ 0<b<\min \{ 2, \;n-s, \;1+\frac{n-2s}{2} \} $\end{document} and \begin{document}$ f(u) $\end{document} is a nonlinear function that behaves like \begin{document}$ \lambda |u|^{\sigma } u $\end{document} with \begin{document}$ \sigma>0 $\end{document} and \begin{document}$ \lambda \in \mathbb C $\end{document}. Recently, the authors in [1] proved the local existence of solutions in \begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document} with \begin{document}$ 0\le s<\min \{ n, \; 1+n/2\} $\end{document}. However even though the solution is constructed by a fixed point technique, continuous dependence in the standard sense in \begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document} with \begin{document}$ 0< s<\min \{ n, \; 1+n/2\} $\end{document} doesn't follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial data in the standard sense in \begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document}, i.e. in the sense that the local solution flow is continuous \begin{document}$ H^{s}(\mathbb R^{n} )\to H^{s}(\mathbb R^{n} ) $\end{document}, if \begin{document}$ \sigma $\end{document} satisfies certain assumptions.

The main purpose of the present paper is to study the blow-up problem of a weakly coupled quasilinear parabolic system as follows:

\begin{document}$ \left\{ \begin{array}{ll} u_{t} = \nabla\cdot\left(r(u)\nabla u\right)+f(u,v,x,t), & \\ v_{t} = \nabla\cdot\left(s(v)\nabla v\right)+g(u,v,x,t) &{\rm in} \ \Omega\times(0,t^{*}), \\ \frac{\partial u}{\partial\nu} = h(u), \ \frac{\partial v}{\partial\nu} = k(v) &{\rm on} \ \partial\Omega\times(0,t^{*}), \\ u(x,0) = u_{0}(x), \ v(x,0) = v_{0}(x) &{\rm in} \ \overline{\Omega}. \end{array} \right. $\end{document}

Here \begin{document}$ \Omega $\end{document} is a spatial bounded region in \begin{document}$ \mathbb{R}^{n} \ (n\geq2) $\end{document} and the boundary \begin{document}$ \partial\Omega $\end{document} of the spatial region \begin{document}$ \Omega $\end{document} is smooth. We give a sufficient condition to guarantee that the positive solution \begin{document}$ (u,v) $\end{document} of the above problem must be a blow-up solution with a finite blow-up time \begin{document}$ t^* $\end{document}. In addition, an upper bound on \begin{document}$ t^* $\end{document} and an upper estimate of the blow-up rate on \begin{document}$ (u,v) $\end{document} are obtained.

This paper deals with the asymptotic behavior of the non-autonomous random dynamical systems generated by the wave equations with supercritical nonlinearity driven by colored noise defined on \begin{document}$ \mathbb{R}^n $\end{document} with \begin{document}$ n\le 6 $\end{document}. Based on the uniform Strichartz estimates, we prove the well-posedness of the equation in the natural energy space and define a continuous cocycle associated with the solution operator. We also establish the existence and uniqueness of tempered random attractors of the equation by showing the uniform smallness of the tails of the solutions outside a bounded domain in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.

This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when \begin{document}$ \alpha = 1 $\end{document}. Finally, several numerical tests are reported for verification of the theoretical findings.

We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type \begin{document}$ \varepsilon^2\left(A(x, u(x), \varepsilon)u'(x)\right)' = f(x, u(x), \varepsilon) $\end{document}. The new features are that we do not consider one scalar equation, but systems, that the systems are allowed to be quasilinear, and that the systems are spatially non-smooth. Although the results about existence, asymptotic behavior, local uniqueness and stability of boundary layer solutions are similar to those known for semilinear, scalar and smooth problems, there are at least three essential differences. First, the asymptotic convergence rates valid for smooth problems are not true anymore, in general, in the non-smooth case. Second, a specific local uniqueness condition from the scalar case is not sufficient anymore in the vectorial case. And third, the monotonicity condition, which is sufficient for stability of boundary layers in the scalar case, must be adjusted to the vectorial case.

In this paper we give a new sufficient condition for the existence of asymptotic periodicity of Frobenius–Perron operators corresponding to two–dimensional maps. Asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical system was already known. Our new result enables one to deal with systems having an eigenvalue smaller than one. The key idea for the proof is to use a function of bounded variation defined by line integration. Finally, we introduce a new two-dimensional dynamical system numerically exhibiting asymptotic periodicity with different periods depending on parameter values, and discuss the application of our theorem to the example.

We characterize the meromorphic Liouville integrability of the Hamiltonian systems with Hamiltonian \begin{document}$ H = \left(p_1^2+p_2^2\right)/2+1/P(q_1, q_2) $\end{document}, being \begin{document}$ P(q_1, q_2) $\end{document} a homogeneous polynomial of degree \begin{document}$ 4 $\end{document} of one of the following forms \begin{document}$ \pm q_1^4 $\end{document}, \begin{document}$ 4q_1^3q_2 $\end{document}, \begin{document}$ \pm 6q_1^2q_2^2 $\end{document}, \begin{document}$ \pm \left(q_1^2+q_2^2\right)^2 $\end{document}, \begin{document}$ \pm q_2^2\left(6q_1^2-q_2^2\right) $\end{document}, \begin{document}$ \pm q_2^2\left(6q_1^2+q_2^2\right) $\end{document}, \begin{document}$ q_1^4+6\mu q_1^2q_2^2-q_2^4 $\end{document}, \begin{document}$ -q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document} with \begin{document}$ \mu>-1/3 $\end{document} and \begin{document}$ \mu\neq 1/3 $\end{document}, and \begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document} with \begin{document}$ \mu \neq \pm 1/3 $\end{document}. We note that any homogeneous polynomial of degree \begin{document}$ 4 $\end{document} after a linear change of variables and a rescaling can be written as one of the previous polynomials. We remark that for the polynomial \begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document} when \begin{document}$ \mu\in\left\{-5/3, -2/3\right\} $\end{document} we only can prove that it has no a polynomial first integral.

In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change the gmCH equation for unknowns \begin{document}$ (u,m) $\end{document} into its Lagrangian dynamics for characteristics \begin{document}$ X(\xi,t) $\end{document}, where \begin{document}$ \xi\in\mathbb{R} $\end{document} is the Lagrangian label. When \begin{document}$ X_\xi(\xi,t)>0 $\end{document}, we use the solutions to the Lagrangian dynamics to recover the classical solutions with \begin{document}$ m(\cdot,t)\in C_0^k(\mathbb{R}) $\end{document} (\begin{document}$ k\in\mathbb{N},\; \; k\geq1 $\end{document}) to the gmCH equation. The classical solutions \begin{document}$ (u,m) $\end{document} to the gmCH equation will blow up if \begin{document}$ \inf_{\xi\in\mathbb{R}}X_\xi(\cdot,T_{\max}) = 0 $\end{document} for some \begin{document}$ T_{\max}>0 $\end{document}. After the blow-up time \begin{document}$ T_{\max} $\end{document}, we use a double mollification method to mollify the Lagrangian dynamics and construct global weak solutions (with \begin{document}$ m $\end{document} in space-time Radon measure space) to the gmCH equation by some space-time BV compactness arguments.

This paper examines the stabilization problem of the axially moving Kirchhoff beam. Under the nonlinear damping criterion established by the slope-restricted condition, the existence and uniqueness of solutions of the closed-loop system equipped with nonlinear time-delay disturbance at the boundary is investigated via the Faedo-Galerkin approximation method. Furthermore, the solution is continuously dependent on initial conditions. Then the exponential stability of the closed-loop system is established by the direct Lyapunov method, where a novel energy function is constructed.

In this paper, we obtain the uniform estimates with respect to the Knudsen number \begin{document}$ \varepsilon $\end{document} for the fluctuations \begin{document}$ g^{\pm}_{\varepsilon} $\end{document} to the two-species Vlasov-Poisson-Boltzmann (in briefly, VPB) system. Then, we prove the existence of the global-in-time classical solutions for two-species VPB with all \begin{document}$ \varepsilon \in (0,1] $\end{document} on the torus under small initial data and rigorously derive the convergence to the two-fluid incompressible Navier-Stokes-Fourier-Poisson (in briefly, NSFP) system as \begin{document}$ \varepsilon $\end{document} go to 0.

In this work, by combining the Feynman-Kac formula with an Itô-Taylor expansion, we propose a class of high order one-step schemes for backward stochastic differential equations, which can achieve at most six order rate of convergence and only need the terminal conditions on the last one step. Numerical experiments are carried out to show the efficiency and high order accuracy of the proposed schemes.

In this paper, we study an elliptic system arising from the U(1)\begin{document}$ \times $\end{document}U(1) Abelian Chern-Simons Model[25,37] of the form

\begin{document}$ \begin{equation} \left\{\begin{split} \Delta u = &\lambda \left(a(b-a)e^{u}-b(b-a)e^{v}+a^2e^{2u} -abe^{2v}+b(b-a)e^{u+v}\right)\\ & +4\pi \sum\limits_{j = 1}^{k_1}m_j\delta_{p_j}, \\ \Delta v = &\lambda \left(-b(b-a)e^{u}+a(b-a)e^{v}-abe^{2u} +a^2e^{2v}+b(b-a)e^{u+v}\right)\\ & +4\pi \sum\limits_{j = 1}^{k_2}n_j\delta_{q_j}, \end{split}\right. \quad\quad\quad\quad (1)\end{equation} $\end{document}

which are defined on a parallelogram \begin{document}$ \Omega $\end{document} in \begin{document}$ \mathbb{R}^2 $\end{document} with doubly periodic boundary conditions. Here, \begin{document}$ a $\end{document} and \begin{document}$ b $\end{document} are interaction constants, \begin{document}$ \lambda>0 $\end{document} is related to coupling constant, \begin{document}$ m_j>0(j = 1,\cdots,k_1) $\end{document}, \begin{document}$ n_j>0(j = 1,\cdots,k_2) $\end{document}, \begin{document}$ \delta_{p} $\end{document} is the Dirac measure, \begin{document}$ p $\end{document} is called vortex point. Concerning the existence results of this system over \begin{document}$ \Omega $\end{document}, only the cases \begin{document}$ (a,b) = (0,1) $\end{document}[28] and \begin{document}$ a>b>0 $\end{document}[14] were studied in the literature. The solvability of this system (1) is still an open problem as regards other parameters \begin{document}$ (a,b) $\end{document}. We show that the system (1) admits topological solutions provided \begin{document}$ \lambda $\end{document} is large and \begin{document}$ b>a>0 $\end{document} Our arguments are based on a iteration scheme and variational formulation.

Whether increasing biodiversity will lead to a promotion (amplification effect) or inhibition (dilution effect) in the transmission of infectious diseases remains to be discovered. In vector-borne infectious diseases, Lyme Disease (LD) and West Nile Virus (WNV) have become typical examples of the dilution effect of biodiversity. Thus, as a vector-borne disease, biodiversity may also play a positive role in the control of the Zika virus. We developed a Zika virus model affected by biodiversity through a competitive mechanism. Through the qualitative analysis of the model, the stability condition of the disease-free equilibrium point and the control threshold of the disease - the basic reproduction number is given. Not only has the numerical analysis verified the inference results, but also it has shown the regulatory effect of the competition mechanism on Zika virus transmission. As competition limits the size of the vector population, the number of final viral infections also decreases. Besides, we also find that under certain parameter conditions, the dilution effect may disappear because of the different initial values. Finally, we emphasized the impact of human activities on biological diversity, to indirectly dilute the abundance of diversity and make the virus continuously spread.

In this paper, we study the sufficient conditions for the existence of solutions of first-order Hamiltonian random impulsive differential equations under Dirichlet boundary value conditions. By using the variational method, we first obtain the corresponding energy functional. And by using Legendre transformation, we obtain the conjugation of the functional. Then the existence of critical point is obtained by mountain pass lemma. Finally, we assert that the critical point of the energy functional is the mild solution of the first order Hamiltonian random impulsive differential equation. Finally, an example is presented to illustrate the feasibility and effectiveness of our results.

This paper is concerned with a predator-prey model with stage structure for the predator, with a cross-diffusion term modeling the effect that mature predators move toward the direction of gradient of prey. It is first shown that the corresponding Neumann initial-boundary value problem in an \begin{document}$ n $\end{document}-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly-in-time bounded for the weak cross-diffusion. It is further shown that, in the presence of cross-diffusion, the model admits threshold-type dynamics in terms of the cross-diffusion coefficient; that is, the homogenous steady state keeps stability for weak attractive prey-taxis, while the stationary patterns will occur for strong attractive prey-taxis. This implies that such cross diffusion does contribute to the rich dynamics of predator-prey model with stage structure for predators.

HIV infects active uninfected CD4\begin{document}$ ^+ $\end{document} T cells, and the active CD4\begin{document}$ ^+ $\end{document} T cells are transformed from quiescent state in response to antigenic activation. Activation effect of the CD4\begin{document}$ ^+ $\end{document} T cells may play an important role in HIV infection. In this paper, we formulate a mathematical model to investigate the activation effect of CD4\begin{document}$ ^+ $\end{document} T cells on HIV dynamics. In the model, the uninfected CD4\begin{document}$ ^+ $\end{document} T cells are divided into two pools: quiescent and active, and the stimuli rate of quiescent cells by HIV is described by saturated form function. We derive the basic reproduction number \begin{document}$ R_0 $\end{document} and analyze the existence and the stability of equilibria. Numerical simulations confirm that the system may have backward bifurcation and Hopf bifurcation. The results imply that \begin{document}$ R_0 $\end{document} cannot completely determine the dynamics of the system and the system may have complex dynamics, which are quite different from the models without the activation effect of CD4\begin{document}$ ^+ $\end{document} T cells. Some numerical results are further presented to assess the activation parameters on HIV dynamics. The simulation results show that the changes of the activation parameters can cause the system periodic oscillation, and activation rate by HIV may induce the supercritical Hopf bifurcation and subcritical Hopf bifurcation. Finally, we proceed to investigate the effect of activation on steady-state viral loads during antiretroviral therapy. The results indicate that, viral load may exist and remain high level even if antiretroviral therapy is effective to reduce the basic reproduction number below 1.

In this paper, we propose a time-delayed West Nile virus (WNv) model with impulsive culling of mosquitoes. The mathematical difficulty lies in how to choose a suitable phase space and deal with the interaction of delay and impulse. By the recent theory developed in [3], we define the basic reproduction number \begin{document}$ \mathcal {R}_0 $\end{document} as the spectral radius of a linear integraloperator and show that \begin{document}$ \mathcal {R}_0 $\end{document} acts as a threshold parameter determining the persistence of the model. More precisely, it is proved that if \begin{document}$ \mathcal {R}_0<1 $\end{document}, then the disease-free periodic solution is globally attractive, while if \begin{document}$ \mathcal {R}_0>1 $\end{document}, then the disease is uniformly persistent.Numerical simulations suggest that culling frequency and culling rate are strongly influenced by the biting rate. We also find that prolonging the length of the incubation period in mosquitoes can reduce the risk of disease spreading.

This work considers a pursuit-evasion model

\begin{document}$\begin{equation} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+u(\mu-u+av),\\ &v_t = \Delta v+\xi\nabla\cdot(v\nabla z)+v(\lambda-v-bu),\\ &w_t = \Delta w-w+v,\\ &z_t = \Delta z-z+u\\ \end{split} \right. \ \ \ \ \ (1) \end{equation}$\end{document}

with positive parameters \begin{document}$ \chi $\end{document}, \begin{document}$ \xi $\end{document}, \begin{document}$ \mu $\end{document}, \begin{document}$ \lambda $\end{document}, \begin{document}$ a $\end{document} and \begin{document}$ b $\end{document} in a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} (\begin{document}$ N $\end{document} is the dimension of the space) with smooth boundary. We prove that if \begin{document}$ a<2 $\end{document} and \begin{document}$ \frac{N(2-a)}{2(C_{\frac{N}{2}+1})^{\frac{1}{\frac{N}{2}+1}}(N-2)_+}>\max\{\chi,\xi\} $\end{document}, (1) possesses a global bounded classical solution with a positive constant \begin{document}$ C_{\frac{N}{2}+1} $\end{document} corresponding to the maximal Sobolev regularity. Moreover, it is shown that if \begin{document}$ b\mu<\lambda $\end{document}, the solution (\begin{document}$ u,v,w,z $\end{document}) converges to a spatially homogeneous coexistence state with respect to the norm in \begin{document}$ L^\infty(\Omega) $\end{document} in the large time limit under some exact smallness conditions on \begin{document}$ \chi $\end{document} and \begin{document}$ \xi $\end{document}. If \begin{document}$ b\mu>\lambda $\end{document}, the solution converges to (\begin{document}$ \mu,0,0,\mu $\end{document}) with respect to the norm in \begin{document}$ L^\infty(\Omega) $\end{document} as \begin{document}$ t\rightarrow \infty $\end{document} under some smallness assumption on \begin{document}$ \chi $\end{document} with arbitrary \begin{document}$ \xi $\end{document}.

The bistable dynamics of a modified Nicholson's blowflies delay differential equation with Allee effect is analyzed. The stability and basins of attraction of multiple equilibria are studied by using Lyapunov-LaSalle invariance principle. The existence of multiple periodic solutions are shown using local and global Hopf bifurcations near positive equilibria, and these solutions generate long transient oscillatory patterns and asymptotic stable oscillatory patterns.

We study integrability and bifurcations of a three-dimensional circuit differential system. The emerging of periodic solutions under Hopf bifurcation and zero-Hopf bifurcation is investigated using the center manifolds and the averaging theory. The zero-Hopf equilibrium is non-isolated and lies on a line filled in with equilibria. A Lyapunov function is found and the global stability of the origin is proven in the case when it is a simple and locally asymptotically stable equilibrium. We also study the integrability of the model and the foliations of the phase space by invariant surfaces. It is shown that in an invariant foliation at most two limit cycles can bifurcate from a weak focus.

In the present study, a nonlinear model is formulated to demonstrate crop - weed interactions, when they both grow together on agricultural land and compete with each other for the same resources like sunlight, water, nutrients etc., under the aegis of herbicides. The developed model is mathematically analyzed through qualitative theory of differential equations to demonstrate rich dynamical characteristics of the system, which are important to be known for maximizing crop yield. The qualitative results reveal that the system not only exhibits stability of more than one equilibrium states, but also undergoes saddle - node, transcritical and Hopf bifurcations, however, depending on parametric combinations. The results of saddle - node and transcritical bifurcations help to plan strategies for maximum crop yield by putting check over the parameters responsible for the depletion of crops due to their interaction with weeds and herbicides. Hopf - bifurcation shows bifurcation of limit cycle through Hopf - bifurcation threshold, which supports that crop - weed interactions are not always of regular type, but they can also be periodic.

The optimal harvesting of biological resources, which is directly relevant to sustainable development, has attracted more attention. In this paper, we first prove the existence and uniqueness of generalized solution of a size-stage-structured population model while the optimal harvesting effort is discontinuous. Next, we demonstrate the existence of the optimal harvesting policy. Further, based on the idea of the Pontryagin's maximum principle of the optimal control problem in ordinary differential equations, we derive the maximum principle describing the optimal control. Finally, the dynamical behavior of the population is simulated by solving the corresponding optimality system numerically with an algorithm based on the method of backward Euler implicit finite-difference approximation. The numerical simulations indicate harvesting activity will reduce the quantity of the population and that increasing harvesting cost will result in less adult harvested. This provides guideline of implementing harvesting tactic to guarantee the persistence of the population.

In this paper, we propose some second-order stabilized semi-implicit methods for solving the Allen-Cahn-Ohta-Kawasaki and the Allen-Cahn-Ohta-Nakazawa equations. In the numerical methods, some nonlocal linear stabilizing terms are introduced and treated implicitly with other linear terms, while other nonlinear and nonlocal terms are treated explicitly. We consider two different forms of such stabilizers and compare the difference regarding the energy stability. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. Numerically, we verify the second order temporal convergence rate of the proposed schemes. In both binary and ternary systems, the coarsening dynamics is visualized as bubble assemblies in hexagonal or square patterns.

The nonlinear Rayleigh damping term that is introduced to the classical parametrically excited pendulum makes the parametrically excited pendulum more complex and interesting. The effect of the nonlinear damping term on the new excitable systems is investigated based on analytical techniques such as Melnikov theory. The threshold conditions for the occurrence of Smale-horseshoe chaos of this deterministic system are obtained. Compared with the existing conclusion, i.e. the smaller the damping term is, the easier the chaotic motions become when the damping term is linear, our analysis, however, finds that the smaller or the larger the damping term is, the easier the Smale-horseshoe heteroclinic chaotic motions become. Moreover, the bifurcation diagram and the patterns of attractors in Poincaré map are studied carefully. The results demonstrate the new system exhibits rich dynamical phenomena: periodic motions, quasi-periodic motions and even chaotic motions. Importantly, according to the property of transitive as well as the fractal layers for a chaotic attractor, we can verify whether a attractor is a quasi-periodic one or a chaotic one when the maximum lyapunov exponent method is difficult to distinguish. Numerical simulations confirm the analytical predictions and show that the transition from regular to chaotic motion.