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Abstract: Abstract This paper is concerned with the four-wave Riemann problem for a two-dimensional hyperbolic system of nonlinear conservation laws derived from a quasi-linear wave equation. The self-similar form of this system is of mixed type. The four-wave Riemann problem in the self-similar plane consists of interactions of four planar elementary waves (exterior waves), which contain rarefaction waves, shocks and contact discontinuities. The Riemann problem is classified into sixteen genuinely different nontrivial cases. The structures of solutions for four rarefaction waves, four shocks and two nonadjacent rarefaction waves plus two nonadjacent shocks are constructed completely. For the rest cases, the solutions are roughly analyzed. For each case, the corresponding numerical solutions are illustrated via contour plots. Comparing with the compressible Euler equations and related models, one of the highlights for this paper is that the interactions of two rarefaction waves, two shocks, as well as a rarefaction wave and a shock in hyperbolic domains are clarified. PubDate: 2021-10-13

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Abstract: Abstract In this paper, we study an inverse problem to determine an unknown source term in the time-fractional diffusion equation with variable coefficients in a general bound domain. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. We introduce the fractional Landweber iterative regularization method to solve inverse source problem. Based on an a conditional stability result, error convergent estimates between the exact solution and the regularization solution by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule are also given. Some numerical experiments prove that the fractional Landweber method provides better numerical result than the classical Landweber method under the same iterative steps. PubDate: 2021-10-12

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Abstract: Abstract Despite their relevance in mathematical biology, there are, as yet, few general results about the asymptotic behaviour of measure valued solutions of renewal equations on the basis of assumptions concerning the kernel. We characterise, via their kernels, a class of renewal equations whose measure-valued solution can be expressed in terms of the solution of a scalar renewal equation. The asymptotic behaviour of the solution of the scalar renewal equation, is studied via Feller’s classical renewal theorem and, from it, the large time behaviour of the solution of the original renewal equation is derived. PubDate: 2021-10-06

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Abstract: Abstract In this paper, we study the component configuration issue of the line-shaped wave networks which is made of two viscoelastic components and an elastic component and the viscoelastic parts produce the infinite memory and damping and distributed delay. The structural memory of viscoelastic component results in energy dissipative and the damping memory arouses the instability, and the elastic component is energy conservation, such a hybrid effects lead to complex dynamic behaviour of network. Our purpose of the present paper is to find out stability condition of such a network, in particular, the configuration condition of the wave network under which the network is exponentially stable. At first, using a resolvent family approach, we prove the well-posed of the wave network systems under suitable assumptions on the memory kernel \(g(s)\) , the damping coefficient \(\mu _{1}\) and delay distributed kernel \(\mu _{2}(s)\) . Next, using the Lyapunov function method, we seek for a structural condition of the wave networks under which the wave networks are exponentially stable. By constructing new functions we obtain the sufficient conditions for the exponential stability of the wave networks, the structural conditions are given as inequalities. PubDate: 2021-10-04

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Abstract: Abstract This paper studies a coupled chemotaxis-Navier-Stokes model with arbitrary superlinear dampening logistic term $$\begin{aligned} \left \{ \textstyle\begin{array}{l} n_{t}+u\cdot \nabla n=\Delta n-\nabla \cdot (n\nabla c)+ rn-\mu n^{ \alpha }, \\ c_{t}+u\cdot \nabla c=\Delta c-nc, \\ u_{t}+(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \Phi , \quad \nabla \cdot u=0 \end{array}\displaystyle \right .\quad \quad (\star ) \end{aligned}$$ in a bounded domain \(\Omega \subset \mathbb{R}^{3}\) with smooth boundary, under homogeneous Neumann boundary conditions for \(n\) and \(c\) , and homogeneous Dirichlet boundary condition for \(u\) . Here \(\Phi \in C^{1+\beta }(\overline{\Omega })\) , \(r \ge 0\) , \(\mu >0\) and \(\alpha >1\) . It is shown in Wang (Math. Models Methods Appl. Sci. 30(06):1217–1252, 2020) that (⋆) possesses at least one global weak solution for \(1<\alpha < 2\) and the solution becomes sufficiently smooth after finite time if we require \(\frac{6}{5}\le \alpha < 2\) . The present work proves the eventual smoothness property holds for \(1<\alpha <2\) . PubDate: 2021-09-27

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Abstract: Abstract This work deals with general linear conservative neutron transport semigroups without spectral gaps in \(L^{1}(\mathcal{T}^{n}\times \mathbb{R} ^{n})\) where \(\mathcal{T}^{n}\) is the \(n\) -dimensional torus. We study the mean ergodicity of such semigroups and their strong convergence to their ergodic projections as time goes to infinity. Systematic functional analytic results are given. PubDate: 2021-09-17

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Abstract: Abstract In this paper, we study the Cauchy problem of 1D non-resistive compressible magnetohydrodynamics (MHD) equations. We established the global existence and uniqueness of strong solutions for large initial data, where the initial density and initial magnetic field approach non-zero constants at infinity, but the initial vacuum of the density inside the region can be permitted. The analysis is based on the Caffarelli-Kohn-Nirenberg weighted inequality and the technique of mathematical frequency decomposition to get the upper bound of the density, and no more artificial conditions are needed to obtain the upper bound estimate of magnetic field \(b\) . PubDate: 2021-09-14

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Abstract: Abstract The purpose of this paper is to investigate the propagation of topological currents along magnetic interfaces (also known as magnetic walls) of a two-dimensional material. We consider tight-binding magnetic models associated to generic magnetic multi-interfaces and describe the \(K\) -theoretical setting in which a bulk-interface duality can be derived. Then, the (trivial) case of a localized magnetic field and the (non trivial) case of the Iwatsuka magnetic field are considered in full detail. This is a pedagogical preparatory work that aims to anticipate the study of more complicated multi-interface magnetic systems. PubDate: 2021-09-09

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Abstract: Abstract In this paper, we are concerned with the controllability for a class of impulsive fractional integro-differential evolution equation in a Banach space. Sufficient conditions of the existence of mild solutions and approximate controllability for the concern problem are presented by considering the term \(u'(\cdot )\) and finding a control \(v\) such that the mild solution satisfies \(u(b)=u_{b}\) and \(u'(b)=u'_{b}\) . The discussions are based on Mönch fixed point theorem as well as the theory of fractional calculus and \((\alpha ,\beta )\) -resolvent operator. Finally, an example is given to illustrate the feasibility of our results. PubDate: 2021-08-31

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Abstract: Abstract In this paper, for a class of nonlocal dispersal SIR epidemic models with nonlinear incidence, we study the existence of traveling waves connecting the disease-free equilibrium with endemic equilibrium. We obtain that the existence of traveling waves depends on the minimal wave speed \(c^{*}\) and basic reproduction number \(\mathcal{R}_{0}\) . That is, if \(\mathcal{R}_{0}>1\) and \(c> c^{*}\) then the model has a traveling wave connecting the disease-free equilibrium with endemic equilibrium. Otherwise, if \(\mathcal{R}_{0}>1\) and \(0< c< c^{*}\) , then there does not exist the traveling wave connecting the disease-free equilibrium with endemic equilibrium. The numerical simulations verify the theoretical results. Our results improve and generalize some known results. PubDate: 2021-08-31

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Abstract: Abstract In this paper, we study the existence of multiple solutions for the boundary value problem $$\begin{aligned} -\Delta _{\gamma } u =& f(x,u) + g(\theta ,x,u) \ \text{ in }\ \ \Omega , \\ u =&\ 0 \hspace{2.6cm}\text{ on }\ \partial \Omega , \end{aligned}$$ where \(\Omega \) is a bounded domain with smooth boundary in \(\mathbb{R}^{N} \ (N \ge 2)\) , where \(f(x,\cdot )\) is odd, \(g(\theta ,x, \xi )\) is a non-symmetric, perturbative term and \(\Delta _{\gamma } \) is the strongly degenerate elliptic operator of the type $$ \Delta _{\gamma }: =\sum _{j=1}^{N}\partial _{x_{j}} \left ( \gamma _{j}^{2} \partial _{x_{j}} \right ), \quad \partial _{x_{j}}: = \frac{\partial }{\partial x_{j}},\quad \gamma : = (\gamma _{1}, \gamma _{2}, \ldots, \gamma _{N}). $$ By using a perturbation method introduced by Bolle (J. Differ. Equ. 152:274-288, 1999), we prove the existence of multiple solutions in spite of the lack of symmetry of the problem. PubDate: 2021-08-29

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Abstract: Abstract In this paper, we study the inhomogeneous boundary value problem for the steady MHD system of a viscous incompressible fluid in an arbitrary bounded multiply connected domain. We prove the existence of generalized solutions of the steady MHD system with some smallness conditions on the boundary values. PubDate: 2021-08-27

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Abstract: Abstract If we let \(n(k, d)\) denote the order of the largest undirected graphs of maximum degree \(k\) and diameter \(d\) , and let \(M(k,d)\) denote the corresponding Moore bound, then \(n(k,d) \leq M(k,d)\) , for all \(k \geq 3\) , \(d \geq 2 \) . While the inequality has been proved strict for all but very few pairs \(k\) and \(d\) , the exact relation between the values \(n(k,d)\) and \(M(k,d)\) is unknown, and the uncertainty of the situation is captured by an open question of Bermond and Bollobás who asked whether it is true that for any positive integer \(c>0\) there exist a pair \(k\) and \(d\) , such that \(n(k, d)\leq M(k,d)-c\) . We present a connection of this question to the value \(2\sqrt{k-1}\) , which is also essential in the definition of the Ramanujan graphs defined as \(k\) -regular graphs whose second largest eigenvalue (in modulus) does not exceed \(2 \sqrt{k-1}\) . We further reinforce this surprising connection by showing that if the answer to the question of Bermond and Bollobás were negative and there existed a \(c > 0\) such that \(n(k,d) \geq M(k,d) - c \) , for all \(k \geq 3\) , \(d \geq 2 \) , then, for any fixed \(k\) and all sufficiently large even \(d\) ’s, the largest undirected graphs of degree \(k\) and diameter \(d\) would have to be Ramanujan graphs. This would imply a positive answer to the open question whether infinitely many non-bipartite \(k\) -regular Ramanujan graphs exist for any degree \(k\) . PubDate: 2021-08-24

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Abstract: Abstract In this paper, we consider the following system $$ \left \{ \textstyle\begin{array}{ll} n_{t}+u\cdot \nabla n&=\Delta n-\nabla \cdot (n\mathcal{S}( \nabla c ^{2}) \nabla c)-nm, \\ c_{t}+u\cdot \nabla c&=\Delta c-c+m, \\ m_{t}+u\cdot \nabla m&=\Delta m-mn, \\ u_{t}&=\Delta u+\nabla P+(n+m)\nabla \Phi ,\qquad \nabla \cdot u=0 \end{array}\displaystyle \right . $$ which models the process of coral fertilization, in a smoothly three-dimensional bounded domain, where \(\mathcal{S}\) is a given function fulfilling $$ \mathcal{S}(\sigma ) \leq K_{\mathcal{S}}(1+\sigma )^{- \frac{\theta }{2}},\qquad \sigma \geq 0 $$ with some \(K_{\mathcal{S}}>0\) . Based on conditional estimates of the quantity \(c\) and the gradients thereof, a relatively compressed argument as compared to that proceeding in related precedents shows that if $$ \theta >0, $$ then for any initial data with proper regularity an associated initial-boundary problem under no-flux/no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded, and which also enjoys the stabilization features in the sense that $$\begin{aligned} &\ n(\cdot ,t)-n_{\infty }\ _{L^{\infty }(\Omega )}+\ c(\cdot ,t)-m_{ \infty }\ _{W^{1,\infty }(\Omega )} +\ m(\cdot ,t)-m_{\infty }\ _{W^{1, \infty }(\Omega )}\\ &\quad{}+\ u(\cdot ,t)\ _{L^{\infty }(\Omega )}\rightarrow 0 \quad \text{as}~t\rightarrow \infty \end{aligned}$$ with \(n_{\infty }:=\frac{1}{ \Omega }\left \{ \int _{\Omega }n_{0}-\int _{ \Omega }m_{0}\right \} _{+}\) and \(m_{\infty }:=\frac{1}{ \Omega }\left \{ \int _{\Omega }m_{0}-\int _{ \Omega }n_{0}\right \} _{+}\) . PubDate: 2021-08-06

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Abstract: Abstract A novel approach to the exponential stability in mean square of stochastic functional differential equations and neutral stochastic functional differential equations with infinite delay is presented. Consequently, some new criteria for the exponential stability in mean square of the considered equations are obtained. Lastly, some examples are investigated to illustrate the theory. PubDate: 2021-07-29

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Abstract: Abstract We investigate here the following weighted degenerate elliptic system $$\begin{aligned} &-\Delta _{s} u =\Big(1+\ \mathbf{x}\ ^{2(s+1)}\Big)^{ \frac{\alpha }{2(s+1)}} v^{p}, \quad -\Delta _{s} v= \Big(1+\ \mathbf{x}\ ^{2(s+1)}\Big)^{\frac{\alpha }{2(s+1)}}u^{\theta }, \\ &\quad u,v>0 \quad \mbox{in }\; \mathbb{R}^{N}:=\mathbb{R}^{N_{1}}\times \mathbb{R}^{N_{2}}, \end{aligned}$$ where \(\Delta _{s}=\Delta _{x}+ x ^{2s}\Delta _{y}\) , is the Grushin operator, \(s\) , \(\alpha \geq 0\) and \(1< p\leq \theta \) . Here $$ \ \mathbf{x}\ =\Big( x ^{2(s+1)}+ y ^{2}\Big)^{\frac{1}{2(s+1)}} \; \mbox{and}\quad \mathbf{x}:=(x, y)\in \mathbb{R}^{N}:=\mathbb{R}^{N_{1}} \times \mathbb{R}^{N_{2}}. $$ In particular, we establish some new Liouville-type theorems for stable solutions of the system, which recover and considerably improve upon the known results (Duong and Phan in J. Math. Anal. Appl. 454(2):785–801, 2017; Hajlaoui et al. in Discrete Contin. Dyn. Syst. 37:265–279, 2017). PubDate: 2021-07-29

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Abstract: Abstract We study two closely related problems in the online selection of increasing subsequence. In the first problem, introduced by Samuels and Steele (Ann. Probab. 9(6):937–947, 1981), the objective is to maximise the length of a subsequence selected by a nonanticipating strategy from a random sample of given size \(n\) . In the dual problem, recently studied by Arlotto et al. (Random Struct. Algorithms 49:235–252, 2016), the objective is to minimise the expected time needed to choose an increasing subsequence of given length \(k\) from a sequence of infinite length. Developing a method based on the monotonicity of the dynamic programming equation, we derive the two-term asymptotic expansions for the optimal values, with \(O(1)\) remainder in the first problem and \(O(k)\) in the second. Settling a conjecture in Arlotto et al. (Random Struct. Algorithms 52:41–53, 2018), we also design selection strategies to achieve optimality within these bounds, that are, in a sense, best possible. PubDate: 2021-07-26

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Abstract: Abstract We consider the dynamics of point particles which are confined to a bounded, possibly nonconvex domain \(\Omega \) . Collisions with the boundary are described as purely elastic collisions. This turns the description of the particle dynamics into a coupled system of second order ODEs with discontinuous right-hand side. The main contribution of this paper is to develop a precise solution concept for this particle system, and to prove existence of solutions. In this proof we construct a solution by passing to the limit in an auxiliary problem based on the Yosida approximation. In addition to existence of solutions, we establish a partial uniqueness theorem, and show by means of a counterexample that uniqueness of solutions cannot hold in general. PubDate: 2021-07-26

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Abstract: Abstract For this paper, we first obtain the Riemann solutions for the one-dimensional non-isentropic Euler equations of extended Chaplygin gas. The solutions are composed of shock waves, contact discontinuity and rarefaction waves. Furthermore, the behavior of these solutions under the limiting conditions are considered. As a certain parameter tends to zero, we find that the Riemann solutions of the one-dimensional non-isentropic Euler equations for extended Chaplygin gas can be transformed into the solutions of the generalized Chaplygin gas unidirectional. Finally, we are done to verify that the theoretical analysis is accurate by using numerical simulation. PubDate: 2021-07-22