# American Institute of Mathematical Sciences

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## Electronic Research Archive

November 2021 , Volume 29 , Issue 5

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2021, 29(5): 2829-2839 doi: 10.3934/era.2021016 +[Abstract](834) +[HTML](353) +[PDF](358.71KB)
Abstract:

We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation \begin{document}$u_t = \Delta u+|u|^{p-1}u$\end{document} which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent \begin{document}$p$\end{document} is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution.

2021, 29(5): 2841-2876 doi: 10.3934/era.2021017 +[Abstract](844) +[HTML](361) +[PDF](509.76KB)
Abstract:

In this work, we study the behavior of the solutions of following three-dimensional system of difference equations

where \begin{document}$n\in \mathbb{N}_{0}$\end{document}, the initial values \begin{document}$x_{-1}$\end{document}, \begin{document}$x_{0}$\end{document}, \begin{document}$y_{-1}$\end{document}, \begin{document}$y_{0}$\end{document} \begin{document}$z_{-1}$\end{document}, \begin{document}$z_{0}$\end{document} are positive real numbers, the functions \begin{document}$f, \, g, \, h:\, \left(0, +\infty\right)^{2}\rightarrow\left(0, +\infty\right)$\end{document} are continuous and homogeneous of degree zero. By proving some general convergence theorems, we have established conditions for the global stability of the corresponding unique equilibrium point. We give necessary and sufficient conditions on existence of prime period two solutions of the above mentioned system. Also, we prove a result on oscillatory solutions. As applications of the obtained results, some particular systems of difference equations defined by homogeneous functions of degree zero are investigated. Our results generalize some existing ones in the literature.

2021, 29(5): 2877-2913 doi: 10.3934/era.2021018 +[Abstract](800) +[HTML](348) +[PDF](551.56KB)
Abstract:

We launch a systematic study of the refined Wilf-equivalences by the statistics \begin{document}${\mathsf{comp}}$\end{document} and \begin{document}${\mathsf{iar}}$\end{document}, where \begin{document}${\mathsf{comp}}(\pi)$\end{document} and \begin{document}${\mathsf{iar}}(\pi)$\end{document} are the number of components and the length of the initial ascending run of a permutation \begin{document}$\pi$\end{document}, respectively. As Comtet was the first one to consider the statistic \begin{document}${\mathsf{comp}}$\end{document} in his book Analyse combinatoire, any statistic equidistributed with \begin{document}${\mathsf{comp}}$\end{document} over a class of permutations is called by us a Comtet statistic over such class. This work is motivated by a triple equidistribution result of Rubey on \begin{document}$321$\end{document}-avoiding permutations, and a recent result of the first and third authors that \begin{document}${\mathsf{iar}}$\end{document} is a Comtet statistic over separable permutations. Some highlights of our results are:

● Bijective proofs of the symmetry of the joint distribution \begin{document}$({\mathsf{comp}}, {\mathsf{iar}})$\end{document} over several Catalan and Schröder classes, preserving the values of the left-to-right maxima.

● A complete classification of \begin{document}${\mathsf{comp}}$\end{document}- and \begin{document}${\mathsf{iar}}$\end{document}-Wilf-equivalences for length \begin{document}$3$\end{document} patterns and pairs of length \begin{document}$3$\end{document} patterns. Calculations of the \begin{document}$({\mathsf{des}}, {\mathsf{iar}}, {\mathsf{comp}})$\end{document} generating functions over these pattern avoiding classes and separable permutations.

● A further refinement of Wang's descent-double descent-Wilf equivalence between separable permutations and \begin{document}$(2413, 4213)$\end{document}-avoiding permutations by the Comtet statistic \begin{document}${\mathsf{iar}}$\end{document}.

2021, 29(5): 2915-2944 doi: 10.3934/era.2021019 +[Abstract](835) +[HTML](353) +[PDF](513.06KB)
Abstract:

The stability and convergence of the Fourier pseudo-spectral method are analyzed for the three dimensional incompressible Navier-Stokes equation, coupled with a variety of time-stepping methods, of up to fourth order temporal accuracy. An aliasing error control technique is applied in the error estimate for the nonlinear convection term, while an a-priori assumption for the numerical solution at the previous time steps will also play an important role in the analysis. In addition, a few multi-step temporal discretization is applied to achieve higher order temporal accuracy, while the numerical stability is preserved. These semi-implicit numerical schemes use a combination of explicit Adams-Bashforth extrapolation for the nonlinear convection term, as well as the pressure gradient term, and implicit Adams-Moulton interpolation for the viscous diffusion term, up to the fourth order accuracy in time. Optimal rate convergence analysis and error estimates are established in details. It is proved that, the Fourier pseudo-spectral method coupled with the carefully designed time-discretization is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. Some numerical results are also presented to verify the established convergence rates of the proposed schemes.

2021, 29(5): 2945-2957 doi: 10.3934/era.2021020 +[Abstract](723) +[HTML](374) +[PDF](327.2KB)
Abstract:

Sympathetic Lie superalgebras are defined and some classical properties of sympathetic Lie superalgebras are given. Among the main results, we prove that any Lie superalgebra \begin{document}$L$\end{document} contains a maximal sympathetic graded ideal and we obtain some properties about sympathetic decomposition. More specifically, we study a general sympathetic Lie superalgebra \begin{document}$L$\end{document} with graded ideals \begin{document}$I$\end{document}, \begin{document}$J$\end{document} and \begin{document}$S$\end{document} such that \begin{document}$L = I\oplus J$\end{document} and \begin{document}$L/S$\end{document} is a sympathetic Lie superalgebra, and we obtain some properties of \begin{document}$L/S$\end{document}. Furthermore, under certain assumptions on \begin{document}$L$\end{document} we prove that the derivation algebra \begin{document}$\mathrm{Der}(L)$\end{document} is sympathetic and that if in addition \begin{document}$L$\end{document} is indecomposable, then \begin{document}$\mathrm{Der}(L)$\end{document} is simply sympathetic.

Fang Li and
2021, 29(5): 2959-2972 doi: 10.3934/era.2021021 +[Abstract](706) +[HTML](342) +[PDF](331.87KB)
Abstract:

The main aim of this article is to characterize inner Poisson structure on a quantum cluster algebra without coefficients. Mainly, we prove that inner Poisson structure on a quantum cluster algebra without coefficients is always a standard Poisson structure. We introduce the concept of so-called locally inner Poisson structure on a quantum cluster algebra and then show it is equivalent to locally standard Poisson structure in the case without coefficients. Based on the result from [7] we obtain finally the equivalence between locally inner Poisson structure and compatible Poisson structure in this case.

2021, 29(5): 2973-2985 doi: 10.3934/era.2021022 +[Abstract](747) +[HTML](381) +[PDF](3713.1KB)
Abstract:

Coupled networks are common in diverse real-world systems and the dynamical properties are crucial for their function and application. This paper focuses on the behaviors of a network consisting of mutually coupled neural groups and time-delayed interactions. These interacting groups can include different sets of nodes and topological architecture, respectively. The local and global stability of the system are analyzed and the stable regions and bifurcation curves in parameter planes are obtained. Different patterns of bifurcated solutions arising from trivial and non-trivial equilibrium points are given, such as the coexistence of non-trivial equilibrium points and periodic responses and multiple coexisting periodic orbits. The bifurcation diagrams are shown and plenty of complex dynamic phenomena are observed, such as multi-period oscillations and multiple coexisting attractors.

2021, 29(5): 2987-3015 doi: 10.3934/era.2021023 +[Abstract](778) +[HTML](454) +[PDF](2479.36KB)
Abstract:

Post-inhibitory rebound (PIR) spike induced by the negative stimulation, which plays important roles and presents counterintuitive nonlinear phenomenon in the nervous system, is mainly related to the Hopf bifurcation and hyperpolarization-active caution (\begin{document}$I_h$\end{document}) current. In the present paper, the emerging condition for the PIR spike is extended to the bifurcation of the big homoclinic (BHom) orbit in a model without \begin{document}$I_h$\end{document} current. The threshold curve for a spike evoked from a mono-stable or coexisting steady state surrounds the steady state from left, to below, and to right, because the BHom orbit is big enough to surround the steady state. The right part of the threshold curve coincides with the stable manifold of the saddle and acts the threshold for the spike induced by the positive stimulation, resembling that of the saddle-node bifurcation on an invariant cycle, and the left part acts the threshold for the PIR spike, resembling that of the Hopf bifurcation. The bifurcation curve and a codimension-2 bifurcation point related to the BHom orbit are acquired in the two-parameter plane. The results present a comprehensive viewpoint to the dynamics near the BHom orbit bifurcation, which presents a novel threshold curve and extends the conditions for the PIR spike.

2021, 29(5): 3017-3030 doi: 10.3934/era.2021024 +[Abstract](722) +[HTML](317) +[PDF](165.45KB)
Abstract:

This paper is devoted to studying the Cauchy problem corresponding to the nonlocal bistable reaction diffusion equation. It is the first attempt to use the method of comparison principle to study the well-posedness for the nonlocal bistable reaction-diffusion equation. We show that the problem has a unique solution for any non-negative bounded initial value by using Gronwall's inequality. Moreover, the boundedness of the solution is obtained by means of the auxiliary problem. Finally, in the case that the initial data with compactly supported, we analyze the asymptotic behavior of the solution.

2021, 29(5): 3031-3068 doi: 10.3934/era.2021025 +[Abstract](822) +[HTML](369) +[PDF](551.73KB)
Abstract:

We show that picture groups are directly related to maximal green sequences for valued Dynkin quivers of finite type. Namely, there is a bijection between maximal green sequences and positive expressions (words in the generators without inverses) for the Coxeter element of the picture group. We actually prove the theorem for the more general set up of finite "vertically and laterally ordered" sets of positive real Schur roots for any hereditary algebra (not necessarily of finite type).

Furthermore, we show that every picture for such a set of positive roots is a linear combination of "atoms" and we give a precise description of atoms as special semi-invariant pictures.

2021, 29(5): 3069-3079 doi: 10.3934/era.2021026 +[Abstract](809) +[HTML](324) +[PDF](831.64KB)
Abstract:

In this paper, we focus on a linear cooperative system with periodic coefficients proposed by Mierczyński [SIAM Review 59(2017), 649-670]. By introducing a switching strategy parameter \begin{document}$\lambda$\end{document} in the periodic coefficients, the bifurcation of instability and the optimization of the switching strategy are investigated. The critical value of unstable branches is determined by appealing to the theory of monotone dynamical system. A bifurcation diagram is presented and numerical examples are given to illustrate the effectiveness of our theoretical result.

2021, 29(5): 3081-3096 doi: 10.3934/era.2021027 +[Abstract](726) +[HTML](342) +[PDF](542.59KB)
Abstract:

This paper is concerned with a practical inverse problem of simultaneously reconstructing the surface heat flux and the thickness of a solid structure from the associated ultrasonic measurements. In a thermoacoustic coupling model, the thermal boundary condition and the thickness of a solid structure are both unknown, while the measurements of the propagation time by ultrasonic sensors are given. We reformulate the inverse problem as a PDE-constrained optimization problem by constructing a proper objective functional. We then develop an alternating iteration scheme which combines the conjugate gradient method and the deepest decent method to solve the optimization problem. Rigorous convergence analysis is provided for the proposed numerical scheme. By using experimental real data from the lab, we conduct extensive numerical experiments to verify several promising features of the newly developed method.

2021, 29(5): 3097-3119 doi: 10.3934/era.2021028 +[Abstract](694) +[HTML](351) +[PDF](653.5KB)
Abstract:

The global well-posedness and long-time mean random dynamics are studied for a high-dimensional non-autonomous stochastic nonlinear lattice pseudo-parabolic equation with locally Lipschitz drift and diffusion terms. The existence and uniqueness of three different types of weak pullback mean random attractors as well as their relations are established for the mean random dynamical systems generated by the solution operators. This is the first paper to study the well-posedness and dynamics of the stochastic lattice pseudo-parabolic equation even when the nonlinear noise reduces to the linear one.

2021, 29(5): 3121-3139 doi: 10.3934/era.2021029 +[Abstract](653) +[HTML](300) +[PDF](414.47KB)
Abstract:

The global asymptotic stability of the unique positive equilibrium point and the rate of convergence of positive solutions of the system of two recursive sequences has been studied recently. Here we generalize this study to the system of \begin{document}$p$\end{document} recursive sequences \begin{document}$x_{n+1}^{(j)}=A+\left(x_{n-m}^{(j+1) mod (p)} \;\;/ x_{n}^{(j+1) mod (p)}\;\;\;\right)$\end{document}, \begin{document}$n = 0,1,\ldots,$\end{document} \begin{document}$m,p\in \mathbb{N}$\end{document}, where \begin{document}$A\in(0,+\infty)$\end{document}, \begin{document}$x_{-i}^{(j)}$\end{document} are arbitrary positive numbers for \begin{document}$i = 1,2,\ldots,m$\end{document} and \begin{document}$j = 1,2,\ldots,p.$\end{document} We also give some numerical examples to demonstrate the effectiveness of the results obtained.

2021, 29(5): 3141-3170 doi: 10.3934/era.2021031 +[Abstract](753) +[HTML](394) +[PDF](18688.3KB)
Abstract:

In this paper, a new Cartesian grid finite difference method is introduced to solve two-dimensional parabolic interface problems with second order accuracy achieved in both temporal and spatial discretization. Corrected central difference and the Matched Interface and Boundary (MIB) method are adopted to restore second order spatial accuracy across the interface, while the standard Crank-Nicolson scheme is employed for the implicit time stepping. In the proposed augmented MIB (AMIB) method, an augmented system is formulated with auxiliary variables introduced so that the central difference discretization of the Laplacian could be disassociated with the interface corrections. A simple geometric multigrid method is constructed to efficiently invert the discrete Laplacian in the Schur complement solution of the augmented system. This leads a significant improvement in computational efficiency in comparing with the original MIB method. Being free of a stability constraint, the implicit AMIB method could be asymptotically faster than explicit schemes. Extensive numerical results are carried out to validate the accuracy, efficiency, and stability of the proposed AMIB method.

2021, 29(5): 3171-3191 doi: 10.3934/era.2021032 +[Abstract](796) +[HTML](336) +[PDF](1857.98KB)
Abstract:

In this article, we develop a new mixed immersed finite element discretization for two-dimensional unsteady Stokes interface problems with unfitted meshes. The proposed IFE spaces use conforming linear elements for one velocity component and non-conforming linear elements for the other velocity component. The pressure is approximated by piecewise constant. Unisolvency, among other fundamental properties of the new vector-valued IFE functions, is analyzed. Based on the new IFE spaces, semi-discrete and full-discrete schemes are developed for solving the unsteady Stokes equations with a stationary or a moving interface. Re-meshing is not required in our numerical scheme for solving the moving-interface problem. Numerical experiments are carried out to demonstrate the performance of this new IFE method.

2021, 29(5): 3193-3203 doi: 10.3934/era.2021033 +[Abstract](781) +[HTML](337) +[PDF](324.94KB)
Abstract:

We consider the minimal model program for varieties that are not \begin{document}$\mathbb{Q}$\end{document}-factorial. We show that, in many cases, its steps are simpler than expected. The main applications are to log terminal singularities, removing the earlier \begin{document}$\mathbb{Q}$\end{document}-factoriality assumption from several theorems of Hacon-Witaszek and de Fernex-Kollár-Xu.

2021, 29(5): 3205-3226 doi: 10.3934/era.2021034 +[Abstract](796) +[HTML](434) +[PDF](6556.42KB)
Abstract:

Based on the three-dimensional endocrine neuron model, a four-dimensional endocrine neuron model was constructed by introducing the magnetic flux variable and induced current according to the law of electromagnetic induction. Firstly, the codimension-one bifurcation and Interspike Intervals (ISIs) analysis were applied to study the bifurcation structure with respect to external stimuli and parameter \begin{document}$k_0$\end{document}, and two dynamical behaviors were found: period-adding and period-doubling bifurcation leading to chaos. Besides, Hopf bifurcation was specially discussed corresponding to the transformation of the state. Secondly, the different firing patterns such as regular bursting, subthreshold oscillations, fast spiking, mixed-mode oscillations (MMOs) etc. can be observed by changing the external stimuli and the induced current. The neuron model presented more firing activities under strong coupling strength. Finally, the codimension-two bifurcation analysis shown more details of bifurcation. At the same time, the Bogdanov-Takens bifurcation point was also analyzed and three bifurcation curves were derived.

2021, 29(5): 3227-3241 doi: 10.3934/era.2021035 +[Abstract](774) +[HTML](357) +[PDF](1061.53KB)
Abstract:

Experimental observations suggest that gamma oscillations are enhanced by the increase of the difference between the components of external stimuli. To explain these experimental observations, we firstly construct a small excitatory/inhibitory (E/I) neural network of IAF neurons with external current input to E-neuron population differing from that to I-neuron population. Simulation results show that the greater the difference between the external inputs to excitatory and inhibitory neurons, the stronger gamma oscillations in the small E/I neural network. Furthermore, we construct a large-scale complicated neural network with multi-layer columns to explore gamma oscillations regulated by external stimuli which are simulated by using a novel CUDA-based algorithm. It is further found that gamma oscillations can be caused and enhanced by the difference between the external inputs in a large-scale neural network with a complicated structure. These results are consistent with the existing experimental findings well.

2021, 29(5): 3243-3260 doi: 10.3934/era.2021036 +[Abstract](593) +[HTML](274) +[PDF](1957.3KB)
Abstract:

This study examines the existence and multiplicity of non-negative solutions of the following fractional \begin{document}$p$\end{document}-sub-Laplacian problem

where \begin{document}$\Omega$\end{document} is an open bounded in homogeneous Lie group \begin{document}$\mathbb{G}$\end{document} with smooth boundary, \begin{document}$p>1$\end{document}, \begin{document}$s\in(0,1)$\end{document}, \begin{document}$(-\Delta_{p,g})^{s}$\end{document} is the fractional \begin{document}$p$\end{document}-sub-Laplacian operator with respect to the quasi-norm \begin{document}$g$\end{document}, \begin{document}$\lambda>0$\end{document}, \begin{document}$1< \alpha<p <\beta < p^*_{s}$\end{document}, \begin{document}$p^*_{s}: = \frac{Qp}{Q-sp}$\end{document} is the fractional critical Sobolev exponents, \begin{document}$Q$\end{document} is the homogeneous dimensions of the homogeneous Lie group \begin{document}$\mathbb{G}$\end{document} with \begin{document}$Q> sp$\end{document}, and \begin{document}$f$\end{document}, \begin{document}$h$\end{document} are sign-changing smooth functions. With the help of the Nehari manifold, we prove that the nonlocal problem on homogeneous group has at least two nontrivial solutions when the parameter \begin{document}$\lambda$\end{document} belong to a center subset of \begin{document}$(0,+\infty)$\end{document}.

2021, 29(5): 3261-3279 doi: 10.3934/era.2021037 +[Abstract](555) +[HTML](265) +[PDF](368.11KB)
Abstract:

In this work, the fully parabolic chemotaxis-competition system with loop

is considered under the homogeneous Neumann boundary condition, where \begin{document}$x\in\Omega, t>0$\end{document}, \begin{document}$\Omega\subset \mathbb{R}^{n} (n\leq 3)$\end{document} is a bounded domain with smooth boundary. For any regular nonnegative initial data, it is proved that if the parameters \begin{document}$\mu_1, \mu_2$\end{document} are sufficiently large, then the system possesses a unique and global classical solution for \begin{document}$n\leq 3$\end{document}. Specifically, when \begin{document}$n = 2$\end{document}, the global boundedness can be attained without any constraints on \begin{document}$\mu_1, \mu_2$\end{document}.

2021, 29(5): 3281-3295 doi: 10.3934/era.2021038 +[Abstract](605) +[HTML](261) +[PDF](370.73KB)
Abstract:

In this paper, we consider the existence of least energy nodal solution and ground state solution, energy doubling property for the following fractional critical problem

where \begin{document}$k$\end{document} is a positive parameter, \begin{document}$\mathcal{L}_K$\end{document} stands for a nonlocal fractional operator which is defined with the kernel function \begin{document}$K$\end{document}. By using the nodal Nehari manifold method, we obtain a least energy nodal solution \begin{document}$u$\end{document} and a ground state solution \begin{document}$v$\end{document} to this problem when \begin{document}$k\gg1$\end{document}, where the nonlinear function \begin{document}$f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow \mathbb{R}$\end{document} is a Carathéodory function.

2021, 29(5): 3297-3308 doi: 10.3934/era.2021039 +[Abstract](585) +[HTML](283) +[PDF](350.58KB)
Abstract:

We prove the Nonvanishing conjecture for uniruled projective log canonical pairs of dimension \begin{document}$n$\end{document}, assuming the Nonvanishing conjecture for smooth projective varieties in dimension \begin{document}$n-1$\end{document}. We also show that the existence of good minimal models for non-uniruled projective klt pairs in dimension \begin{document}$n$\end{document} implies the existence of good minimal models for projective log canonical pairs in dimension \begin{document}$n$\end{document}.

2021, 29(5): 3309-3321 doi: 10.3934/era.2021040 +[Abstract](575) +[HTML](319) +[PDF](339.8KB)
Abstract:

We show that, for any nonsingular projective 4-fold \begin{document}$V$\end{document} of general type with geometric genus \begin{document}$p_g\geq 2$\end{document}, the pluricanonical map \begin{document}$\varphi_{33}$\end{document} is birational onto the image and the canonical volume \begin{document}${\rm Vol}(V)$\end{document} has the lower bound \begin{document}$\frac{1}{480}$\end{document}, which improves a previous theorem by Chen and Chen.

2021, 29(5): 3323-3340 doi: 10.3934/era.2021041 +[Abstract](618) +[HTML](306) +[PDF](325.94KB)
Abstract:

In this paper, the synchronization problem of complex-valued memristive competitive neural networks(CMCNNs) with different time scales is investigated. Based on differential inclusions and inequality techniques, some novel sufficient conditions are derived to ensure synchronization of the drive-response systems by designing a proper controller. Finally, a numerical example is provided to illustrate the usefulness and feasibility of our results.

2021, 29(5): 3341-3359 doi: 10.3934/era.2021042 +[Abstract](534) +[HTML](260) +[PDF](433.03KB)
Abstract:

Let \begin{document}$(\mathcal{C}, \mathbb{E}, \mathfrak{s})$\end{document} be an extriangulated category with a proper class \begin{document}$\xi$\end{document} of \begin{document}$\mathbb{E}$\end{document}-triangles. In this paper, we study the balance of complete cohomology in \begin{document}$(\mathcal{C}, \mathbb{E}, \mathfrak{s})$\end{document}, which is motivated by a result of Nucinkis that complete cohomology of modules is not balanced in the way the absolute cohomology Ext is balanced. As an application, we give some criteria for identifying a triangulated catgory to be Gorenstein and an Artin algebra to be \begin{document}$F$\end{document}-Gorenstein.

2021, 29(5): 3361-3382 doi: 10.3934/era.2021043 +[Abstract](560) +[HTML](268) +[PDF](489.02KB)
Abstract:

We propose an immersed hybrid difference method for elliptic boundary value problems by artificial interface conditions. The artificial interface condition is derived by imposing the given boundary condition weakly with the penalty parameter as in the Nitsche trick and it maintains ellipticity. Then, the derived interface problems can be solved by the hybrid difference approach together with a proper virtual to real transformation. Therefore, the boundary value problems can be solved on a fixed mesh independently of geometric shapes of boundaries. Numerical tests on several types of boundary interfaces are presented to demonstrate efficiency of the suggested method.

2021, 29(5): 3383-3403 doi: 10.3934/era.2021044 +[Abstract](645) +[HTML](275) +[PDF](837.87KB)
Abstract:

In Bayesian inverse problems, using the Markov Chain Monte Carlo method to sample from the posterior space of unknown parameters is a formidable challenge due to the requirement of evaluating the forward model a large number of times. For the purpose of accelerating the inference of the Bayesian inverse problems, in this work, we present a proper orthogonal decomposition (POD) based data-driven compressive sensing (DCS) method and construct a low dimensional approximation to the stochastic surrogate model on the prior support. Specifically, we first use POD to generate a reduced order model. Then we construct a compressed polynomial approximation by using a stochastic collocation method based on the generalized polynomial chaos expansion and solving an \begin{document}$l_1$\end{document}-minimization problem. Rigorous error analysis and coefficient estimation was provided. Numerical experiments on stochastic elliptic inverse problem were performed to verify the effectiveness of our POD-DCS method.

2021, 29(5): 3405-3427 doi: 10.3934/era.2021045 +[Abstract](462) +[HTML](291) +[PDF](470.73KB)
Abstract:

This paper formulates a general framework for a space-time finite element method for solving Richards Equation in one spatial dimension, where the spatial variable is discretized using the linear finite volume element and the temporal variable is discretized using a discontinuous Galerkin method. The actual implementation of a particular scheme is realized by imposing certain finite element space in temporal variable to the variational equation and appropriate "variational crime" in the form of numerical integrations for calculating integrations in the formulation. Once this is in place, adjoint-based error estimators for the approximate solution from the scheme is derived. The adjoint problem is obtained from an appropriate linearization of the nonlinear system. Numerical examples are presented to illustrate performance of the methods and the error estimators.

2021, 29(5): 3429-3447 doi: 10.3934/era.2021046 +[Abstract](619) +[HTML](275) +[PDF](987.71KB)
Abstract:

In this paper, we present a feedback design for numerical solution to optimal control problems, which is based on solving the corresponding Hamilton-Jacobi-Bellman (HJB) equation. An upwind finite-difference scheme is adopted to solve the HJB equation under the framework of the dynamic programming viscosity solution (DPVS) approach. Different from the usual existing algorithms, the numerical control function is interpolated in turn to gain the approximation of optimal feedback control-trajectory pair. Five simulations are executed and both of them, without exception, output the accurate numerical results. The design can avoid solving the HJB equation repeatedly, thus efficaciously promote the computation efficiency and save memory.

2021, 29(5): 3449-3469 doi: 10.3934/era.2021047 +[Abstract](578) +[HTML](227) +[PDF](5044.67KB)
Abstract:

In this paper, we establish the existence of ground state solutions for a fractional Schrödinger equation in the presence of a harmonic trapping potential. We also address the orbital stability of standing waves. Additionally, we provide interesting numerical results about the dynamics and compare them with other types of Schrödinger equations [11,18]. Our results explain the effect of each term of the Schrödinger equation: the fractional power, the power of the nonlinearity, and the harmonic potential.

2021, 29(5): 3471-3487 doi: 10.3934/era.2021048 +[Abstract](453) +[HTML](230) +[PDF](1218.31KB)
Abstract:

Deep brain stimulation (DBS) alleviates the symptoms of tremor, rigidity, and akinesia of the Parkinson's disease (PD). Over decades of the clinical experience, subthalamic nucleus (STN), globus pallidus externa (GPe) and globus pallidus internal (GPi) have been chosen as the common DBS target sites. However, how to design the DBS waveform is still a challenging problem. There is evidence that chronic high-frequency stimulation may cause long-term tissue damage and other side effects. In this paper, we apply a form of DBS with delayed rectangular waveform, denoted as pulse-delay-pulse (PDP) type DBS, on multiple-site based on a computational model of the basal ganglia-thalamus (BG-TH) network. We mainly investigate the effects of the stimulation frequency on relay reliability of the thalamus neurons, beta band oscillation of GPi nucleus and firing rate of the BG network. The results show that the PDP-type DBS at STN-GPe site results in better performance at lower frequencies, while the DBS at GPi-GPe site causes the number of spikes of STN to decline and deviate from the healthy status. Fairly good therapeutic effects can be achieved by PDP-type DBS at STN-GPi site only at higher frequencies. Thus, it is concluded that the application of multiple-site stimulation with PDP-type DBS at STN-GPe is of great significance in treating symptoms of neurological disorders in PD.

2021, 29(5): 3489-3507 doi: 10.3934/era.2021049 +[Abstract](460) +[HTML](217) +[PDF](333.06KB)
Abstract:

We present here a general rule of construction of identities for recursive sequences by using sequence transformation techniques developed in [16]. Numerous identities are constructed, and many well known identities can be proved readily by using this unified rule. Various Catalan-like and Cassini-like identities are given for recursive number sequences and recursive polynomial sequences. Sets of identities for Diophantine quadruple are shown.

2021, 29(5): 3509-3533 doi: 10.3934/era.2021050 +[Abstract](924) +[HTML](275) +[PDF](416.25KB)
Abstract:

We consider a two-species chemotaxis-Navier-Stokes system with \begin{document}$p$\end{document}-Laplacian in three-dimensional smooth bounded domains. It is proved that for any \begin{document}$p\geq2$\end{document}, the problem admits a global weak solution.

2021, 29(5): 3535-3550 doi: 10.3934/era.2021051 +[Abstract](420) +[HTML](208) +[PDF](209.92KB)
Abstract:

This paper is concerned with the traveling wave fronts for a lattice dynamical system with global interaction, which arises in a single species in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction by delay. We prove that all non-critical traveling wave fronts are globally exponentially stable in time, and the critical traveling wave fronts are globally algebraically stable by the weighted energy method combined with the comparison principle and the discrete Fourier transform.

2021, 29(5): 3551-3580 doi: 10.3934/era.2021083 +[Abstract](359) +[HTML](170) +[PDF](482.49KB)
Abstract:

This paper presents a survey for some recent research on the controllability of nonlinear fractional evolution systems (FESs) in Banach spaces. The prime focus is exact controllability and approximate controllability of several types of FESs, which include the basic systems with classical initial and nonlocal conditions, FESs with time delay or impulsive effect. In addition, controllability results via resolvent operator are reviewed in detail. At last, the conclusions of this work and the research prospect are presented, which provides a reference for further study.

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