Communications on Pure and Applied Analysis
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Published By American Institute Of Mathematical Sciences

1534-0392

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Juhong Kuang ◽  
Weiyi Chen ◽  
Zhiming Guo

<p style='text-indent:20px;'>In this paper, we develop a new method to study Rabinowitz's conjecture on the existence of periodic solutions with prescribed minimal period for second order even Hamiltonian system without any convexity assumptions. Specifically, we first study the associated homogenous Dirichlet boundary value problems for the discretization of the Hamiltonian system with given step length and obtain a sequence of nonnegative solutions corresponding to different step lengths by using discrete variational methods. Then, using the sequence of nonnegative solutions, we construct a sequence of continuous functions which can be shown to be precompact. Finally, by utilizing the limit function of convergent subsequence and the symmetry of the potential, we will obtain the desired periodic solution. In particular, we prove Rabinowitz's conjecture in the case when the potential satisfies a certain symmetric assumption. Moreover, our main result greatly improves the related results in the literature in the case where <inline-formula><tex-math id="M1">\begin{document}$ N = 1 $\end{document}</tex-math></inline-formula>.</p>


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xujie Yang

<p style='text-indent:20px;'>Motivated by the studies of the hydrodynamics of the tethered bacteria <i>Thiovulum majus</i> in a liquid environment, we consider the following chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{split} &amp; n_t = \Delta n-\nabla\cdot\left(n\chi(c)\nabla{c}\right)+nc, &amp;x\in \Omega, t&gt;0, \ &amp; c_t = \Delta c-{\bf u}\cdot\nabla c-nc, &amp;x\in \Omega, t&gt;0\ \end{split} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded convex domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^d(d\in\{2, 3\}) $\end{document}</tex-math></inline-formula> with smooth boundary. For any given fluid <inline-formula><tex-math id="M2">\begin{document}$ {\bf u} $\end{document}</tex-math></inline-formula>, it is proved that if <inline-formula><tex-math id="M3">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula>, the corresponding initial-boundary value problem admits a unique global classical solution which is uniformly bounded, while if <inline-formula><tex-math id="M4">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>, such solution still exists under the additional condition that <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\chi\leq \frac{1}{16\|c(\cdot, 0)\|_{L^\infty(\Omega)}} $\end{document}</tex-math></inline-formula>.</p>


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Thierry Horsin ◽  
Mohamed Ali Jendoubi

<p style='text-indent:20px;'>In the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.</p>


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Le Thi Phuong Ngoc ◽  
Khong Thi Thao Uyen ◽  
Nguyen Huu Nhan ◽  
Nguyen Thanh Long

<p style='text-indent:20px;'>In this paper, we investigate a system of pseudoparabolic equations with Robin-Dirichlet conditions. First, the local existence and uniqueness of a weak solution are established by applying the Faedo-Galerkin method. Next, for suitable initial datum, we obtain the global existence and decay of weak solutions. Finally, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative, then we establish here the lifespan for the equations via finding the upper bound and the lower bound for the blow-up times.</p>


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhiwen Zhao

<p style='text-indent:20px;'>In the perfect conductivity problem arising from composites, the electric field may become arbitrarily large as <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula>, the distance between the inclusions and the matrix boundary, tends to zero. In this paper, by making clear the singular role of the blow-up factor <inline-formula><tex-math id="M2">\begin{document}$ Q[\varphi] $\end{document}</tex-math></inline-formula> introduced in [<xref ref-type="bibr" rid="b27">27</xref>] for some special boundary data of even function type with <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math></inline-formula>-order growth, we prove the optimality of the blow-up rate in the presence of <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula>-convex inclusions close to touching the matrix boundary in all dimensions. Finally, we give closer analysis in terms of the singular behavior of the concentrated field for eccentric and concentric core-shell geometries with circular and spherical boundaries from the practical application angle.</p>


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Marcello Lucia ◽  
Guido Sweers

<p style='text-indent:20px;'>We consider fully coupled cooperative systems on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> with coefficients that decay exponentially at infinity. Expanding some results obtained previously on bounded domain, we prove that the existence of a strictly positive supersolution ensures the first eigenvalue to exist and to be nonzero. This result is applied to show that the topological solutions for a Chern-Simons model, described by a semilinear system on <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> with exponential nonlinearity, are nondegenerate.</p>


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fanfan Chen ◽  
Dingbian Qian ◽  
Xiying Sun ◽  
Yinyin Wu

<p style='text-indent:20px;'>We prove the existence and multiplicity of subharmonic solutions for bounded coupled Hamiltonian systems. The nonlinearities are assumed to satisfy Landesman-Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is based on phase plane analysis and a higher dimensional version of the Poincaré-Birkhoff twist theorem by Fonda and Ureña. The results obtained generalize the previous works for scalar second-order differential equations or relativistic equations to higher dimensional systems.</p>


2021 ◽  
Vol 20 (11) ◽  
pp. 3921
Author(s):  
Wei Yang ◽  
Jinliang Wang

<p style='text-indent:20px;'>In this paper, we are concerned with the threshold dynamics of a diffusive cholera model incorporating latency and bacterial hyperinfectivity. Our model takes the form of spatially nonlocal reaction-diffusion system associated with zero-flux boundary condition and time delay. By studying the associated eigenvalue problem, we establish the threshold dynamics that determines whether or not cholera will spread. We also confirm that the threshold dynamics can be determined by the basic reproduction number. By constructing Lyapunov functional, we address the global attractivity of the unique positive equilibrium whenever it exists. The theoretical results are still hold for the case when the constant parameters are replaced by strictly positive and spatial dependent functions.</p>


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Boya Li ◽  
Hongjie Ju ◽  
Yannan Liu

<p style='text-indent:20px;'>In this paper, a generalitzation of the <inline-formula><tex-math id="M2">\begin{document}$ L_{p} $\end{document}</tex-math></inline-formula>-Christoffel-Minkowski problem is studied. We consider an anisotropic curvature flow and derive the long-time existence of the flow. Then under some initial data, we obtain the existence of smooth solutions to this problem for <inline-formula><tex-math id="M3">\begin{document}$ c = 1 $\end{document}</tex-math></inline-formula>.</p>


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xuan Wu ◽  
Huafeng Xiao

<p style='text-indent:20px;'>In this paper, we study the existence of periodic solutions of the following differential delay equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} z^{\prime\prime}(t) = \sum\limits_{k = 1}^{M-1}(-1)^kf(z(t-k)), \notag \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ f\in C(\mathbf{R}^N, \mathbf{R}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ M,N\in \mathbf{N} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> is odd. By making use of <inline-formula><tex-math id="M4">\begin{document}$ S^1 $\end{document}</tex-math></inline-formula>-geometrical index theory, we obtain an estimation about the number of periodic solutions in term of the difference between eigenvalues of asymptotically linear matrices at the origin and at infinity.</p>


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