The main aim of this paper is to study hyponormal and dissipative correct restrictions and extensions as well as their applications to differential operators.

A Riemann-Liouville fractional Robin boundary-value problem is proposed to describe the fast heat transfer law both within isotropic materials and through the boundary of the materials in high temperature environment. The variational formulation of the fractional model is given, and further the energy estimation of the weak solution is deduced. The uniqueness theorem of weak solution is proved. A valid finite difference scheme is developed for the fractional model and numerical experiment is implemented. Numerical results indicate that the fractional model is applicable to discover the thermal superdiffusion in the thermal protective clothing(TPC) system and numerical algorithms are effective to improve the intelligence of TPC design.

In this paper, we consider the chemotaxis model u_t&=\Delta u-\nabla\cdot(u\nabla v),& \qquad x\in\Omega,\,t>0,v_t&=\Delta v-vw,& \qquad x\in\Omega,\,t>0,w_t&=-\delta w+u,& \qquad x\in\Omega,\,t>0,under homogeneous Neumann boundary conditions in a bounded and convex domain $\Om\subset \mathbb{R}^3$ with smooth boundary, where $\delta>0$ is a given parameter. It is shown that for arbitrarily large initial data, this problem admits at least one global weak solution for which there exists $T>0$ such that the solution $(u,v,w)$ is bounded and smooth in $\Om\times(T,\infty)$. Furthermore, it is asserted that such solutions approach spatially constant equilibria in the large time limit.

Population dynamics with complex biological interactions, accounting for uncertainty quantification, are critical for many application areas. However, due to the complexity of biological systems, the mathematical formulation of the corresponding problems faces the challenge that the corresponding stochastic processes should, in most cases, be considered in bounded domains. We propose a model based on a coupled system of reflecting Skorokhod-type stochastic differential equations with jump-like exit from a boundary. The setting describes the population dynamics of active and passive populations. As main working techniques, we use compactness methods and Skorokhod's representation of solutions to SDEs posed in bounded domains to prove the well-posedness of the system. This functional setting is a new point of view in the field of modelling and simulation of population dynamics. We provide the details of the model, as well as representative numerical examples, and discuss the applications of a Wilson-Cowan-type system, modelling the dynamics of two interacting populations of excitatory and inhibitory neurons. Furthermore, the presence of random input current, reflecting factors together with Poisson jumps, increases firing activity in neuronal systems.

Geometrically, a complex dynamical network (CDN) can be regarded as the interconnected system composed of the node subsystem (NS) and the link subsystem (LS) coupled with each other. Guided by this idea, in order to achieve the goal of each node following asymptotically its own reference target in a CDN, this paper investigates the model following adaptive control (MFAC) problem of NS via the dynamics of links, which implies that the LS plays the important dynamic auxiliary role in the MFAC realization of nodes. Meanwhile, we focus on the condition that the links state information is unavailable, due to sensor practical application and measurement cost constraints. To obviate this restriction, we construct the asymptotical state observer for the LS. Next, to achieve the control goal of this paper, an appropriate Lyapunov candidate function is selected, by which the MFAC scheme for NS is synthesized based on the state observer of LS. Finally, the simulation example is performed to demonstrate the theoretical results in this paper.

In this article, we study the global well-posedness and large-time behaviors of solutions to the two-dimensional tropical climate system with zero thermal diffusion for a small initial data in the whole space. The main approaches include high and low frequency decomposition method and exploiting the structure of system (1) to obtain the estimates of thermal dissipation. We utilize the time decay properties of the kernels to a linear differential equation to obtain the decay rates of solutions of the low frequency part and the decay property of exponential operator for the high frequency part. The key ingredient here is the explicit large-time decay rate of solutions.

A class of time-varying delay distributed parameter systems with input saturation is investigated in this paper. The periodic intermittent control method is adopted to make the system stable in finite time, improve the control performance of the system, and save on control cost. A periodic intermittent controller combined saturated input is designed to ensure the stability of the proposed system in finite time. Lyapunov--Krasoviskii stability theory and Matrix inequality techniques are used to analyze the finite-time stability of the system, and sufficient conditions for the system to be stable in finite time are obtained. Finally, the correctness of the theorems is verified by simulation experiments.

The backward Euler method is employed to approximate the invariant measure of a class of stochastic differential equations(SDEs) driven by α-stable processes. The existence and uniqueness of the numerical invariant measure is proved. Then the numerical invariant measure is shown to converge to the underlying invariant measure. Numerical examples are provided to demonstrate the theoretical results.

The existence of decomposition solutions of the well-known nonlinear BKP hierarchy is analyzed. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy. Further, some special decomposition solutions display a rare property: they can be linearly superposed. With the emphasis on the case of the fifth BKP equation, the structure characteristic having linear superposition solutions is analyzed. Finally, we obtain similar superposed solutions in the dispersionless BKP hierarchy.

We consider the global well-posedness and asymptotic behavior of compressible viscous, heat-conductive, and non-resistive magnetohydrodynamics (MHD) fluid in a field of external forces over three-dimensional periodic thin domain $\Omega=\mathbb{T}^2\times(0,\delta)$. The unique existence of the stationary solution is shown under the adhesion and the adiabatic boundary conditions. Then, it is shown that a solution to the initial boundary value problem with the same boundary and periodic conditions uniquely exists globally in time and converges to the stationary solution as time tends to infinity. Moreover, if the external forces are small or disappeared in an appropriate Sobolev space, then $\delta$ can be a general constant. Our proof relies on the two-tier energy method for the reformulated system in Lagrangian coordinates and the background magnetic field which is perpendicular to the flat layer. Compared to the work of Tan and Wang (SIAM J. Math. Anal. 50:1432–1470, 2018), we not only overcome the difficulties caused by temperature, but also consider the big external forces.

n this work, we investigate the poroelastic waves by solving the time-domain Biot-JKD equation with an efficient numerical method. The viscous dissipation occurring in the pores depends on the square root of the frequency and is described by the Johnson-Koplik-Dashen (JKD) dynamic tortuosity/permeability model. The temporal convolutions of order 1/2 shifted fractional derivatives are involved in the time-domain Biot-JKD model, causing the problem to be stiff and challenging to be implemented numerically. Based on the best relative approximation of the square-root function, we design an efficient algorithm to approximate and localize the convolution kernel by introducing a finite number of auxiliary variables that satisfy a local system of ordinary differential equations. The imperfect hydraulic contact condition is used to describe the interface boundary conditions and the Runge-Kutta discontinuous Galerkin (RKDG) method together with the splitting method is applied to compute the numerical solutions. Several numerical examples are presented to show the accuracy and efficiency of our approach.

This paper is devoted to dealing with the following nonlinear Kirchhoff type problem with general convolution nonlinearity and variable potential: $$\left\{\begin{array}{ll} -({a + b\int_{{\R^3}} | \nabla u{|^2}dx})\Delta u + V(x)u =(I_{\alpha}\ast F(u))f(u),\quad \text{in}\ \ \R^3,\\ u \in H^{1} (\R^{3}), \end{array}\right.$$ where $a>0$, $b\geq0$ are constants, $V\in C^1(\R^3,[0,+\infty))$, $f\in C(\R,\R)$, $F(t)=\int_{0}^{t}f(s)ds$, $I_{\alpha}:\R^{3}\rightarrow \R$ is the Riesz potential, $\alpha\in(0,3)$. By applying some new analytical tricks introduced by [X.H. Tang, S.T. Chen, Adv. Nonlinear Anal. 9 (2020) 413-437], the existence results of ground state solutions of Poho\v{z}aev type for the above Kirchhoff type problem are obtained under some mild assumptions on $V$ and the general “Berestycki-Lions assumptions” on the nonlinearity $f$. Our results generalize and improve the ones in [P. Chen, X.C. Liu, J. Math. Anal. Appl. 473 (2019) 587-608.] and other related results in the literature.

This paper investigates the spatial behavior of the solutions of the double-diffusive Darcy plane flow in a semi-infinite channel. Using the energy estimate method and the differential inequality technology, a differential inequality about the solutions is derived. By solving this differential inequality, it is proved that the solutions grow polynomially or decay exponentially with spatial variable. In the case of decay, we obtain the upper bound for the total energy. We also give some remarks to generalize the results of this paper.

The information detection of complex systems from data is currently undergoing a revolution,driven by the emergence of big data and machine learning methodology. Discovering governingequations and quantifying dynamical properties of complex systems are among central challenges. Inthis work, we devise a nonparametric approach to learn the relative entropy rate from observationsof stochastic differential equations with different drift functions. The estimator corresponding tothe relative entropy rate then is presented via the Gaussian process kernel theory. Meanwhile, thisapproach enables to extract the governing equations. We illustrate our approach in several examples.Numerical experiments show the proposed approach performs well for rational drift functions, notonly polynomial drift functions.

We consider a nonautonomous eco-epidemiological model with general functions for predation on infected and uninfected preys as well as general functions associated to the vital dynamics of the susceptible prey and predator populations. We obtain persistence and extinction results for the infected prey based on assumptions on auxiliary systems constructed from the disease-free system. We moreover consider an iterative process that can improve the extinction results. We apply our results to general eco-epidemiological models that include several examples existent in the literature.

A theorem on the maximum regularity of solutions of the nonlinear Sturm-Liouville equation with greatly growing and rapidly oscillating potential in the space $L_2(R)\,(R=(-\infty,\infty))$ is proved in this paper. Two-sided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear Sturm-Liouville equation are also obtained. As is known, the obtained estimates given the opportunity to choose approximation apparatus that guarantees the maximum possible error.

This paper investigates the theoretical modeling and coupled free vibration behaviors of a rotating double-bladed shaft assembly resting on elastic supports in a spacecraft system. According to the Kirchhoff plate theory and the Euler-Bernoulli beam theory, the theoretical model is established. The studied rotor is considered to be made of porous foam metal matrix and graphene nanoplatelet (GPL) reinforcement. Non-uniform distributions of porosity and graphene nanoplatelets (GPLs) are taken into account and lead to functionally graded (FG) structures. The effective material properties of the double-bladed shaft are varying along the radius and thickness direction of the shaft and blade, respectively. Moreover, the rule of mixture, the Halpin-Tsai model, and the open-cell scheme are used to determine its material properties. Considering the gyroscopic effect, the Lagrange equation is utilized to derive the coupled equations of motion. Then the traveling wave frequencies of the double-bladed shaft assembly is obtained by employing the assumed modes method and substructure modal synthesis method. A detailed parametric analysis is conducted to examine the effects of the rotating speed, GPL weight fraction, GPL distribution pattern, GPL length-to-thickness ratio, GPL length-to-width ratio, porosity coefficient, porosity distribution pattern, shaft length-to-radius ratio, blade length-to-thickness ratio, support stiffness and support location on the free vibration behaviors of the double-bladed shaft assembly.

In this paper, we study the stability of a viscous shock wave for the isentropic Navier-Stokes-Korteweg equations under space-periodic perturbation. It is shown that if the initial perturbation around the shock and the amplitude of the shock are small, then the solution of the N-S-K equations tends to the viscous shock.