In this paper, we study a semilinear Timoshenko system with heat conduction having two damping effects. The observation that two damping effects might lead to smaller decay rates for solutions in comparison to one damping effect is rigorously proved here in providing optimality results. Moreover, the global well-posedness for small data is presented.

In this work, we study the Cauchy problem for the radially symmetric spatially homogeneous Boltzmann equation with Debye-Yukawa potential. We prove that this Cauchy problem enjoys the same smoothing effect as the Cauchy problem defined by the evolution equation associated to a fractional logarithmic harmonic oscillator. To be specific, we can prove the solution of the Cauchy problem belongs to Shubin spaces.

In this paper, by using the method of moving planes, we are concerned with the symmetry and monotonicity of positive solutions for the fractional Hartree equation.

In this paper we consider a one-dimensional laminated beam system. The only dissipation in the system is through heat conduction in the interfacial slip equation. We prove that this unique dissipation is strong enough to exponentially stabilize the system provided the wave speeds of the system are equal. This result extends previous works where additional internal or boundary controls were used together with a frictional damping in the interfacial slip.

We prove the global existence of classical solutions to a fluid-particle interaction model in R³, namely, compressible Navier-Stokes-Smoluchowski equations, when the initial data are close to the stationary state (ρ_⋆, 0, η_⋆) and the external potential satisfies the smallness assumption. Furthermore, optimal decay rates of classical solutions in H³-framework are obtained.

The purpose of this work is to study the global existence and asymptotic behavior of solutions to a coupled reaction-diffusion system describing epidemiological or chemical situations. Our analytical proofs are based on the Lyapunov functional methods.

This paper concerns the Cauchy problem of the 3D generalized incompressible magneto-hydrodynamic (GMHD) equations. By using the Fourier localization argument and the Littlewood-Paley theory, we get local well-posedness results of the GMHD equations with large initial data (u₀, b₀) belonging to the critical Fourier-Besov-Morrey spaces FṄ_p,λ,q^{1-2α+ 3/p'+λ/p} (R³). Moreover, stability of global solutions is also discussed.

In this paper, we discuss the existence, uniqueness and stability of boundary value problem for differential equation with Hilfer-Katugampola fractional derivative. The arguments are based upon Schaefer's fixed point theorem, Banach contraction principle and Ulam type stability.

Let A denote the family of all analytic functions f(z) in the unit disk D={z ∈ C:|z|<1}, normalized by the conditions f(0)=0 and f'(0)=1. Let U denote the set of all functions f ∈ A satisfying the condition
|(z/f(z))² f'(z) -1|<1 for z ∈ D.
Let Ω be the class of all f ∈ A for which
|zf'(z) -f(z)|<1/2, z ∈ D.
In this paper, the relations between the two classes are discussed. Furthermore, some new results on the class Ω are obtained.

By using the weighted-energy method combining continuation method, the exponential stability of monostable traveling waves for a delayed equation without quasi-monotonicity is established, including even the slower waves whose speed are close to the critical speed. Particularly, the nonlinearity is nonlocal in the equation and the initial perturbation is uniformly bounded only at x=+∞ but may not be vanishing.

The (integer order) Halanay inequality with distributed delays is extended to the fractional order case. It is proved that solutions decay to zero as a Mittag-Leffler function as time goes to infinity provided that the delay feedback are bounded by similar functions. An application to a problem arising in neural network theory is provided showing that the equilibrium is Mittag-Leffler stable.

Let λ_G(z)|dz|be the hyperbolic metric on a simply connected proper domain G ⊂ C containing the origin, and let||·||_j be the Banach norms of C^n_j for j=1, 2, …, k.This note is to prove that if f is a normalized biholomorphic convex function on G, then
Φ_{N,1/p1,…,1/pk}(f)(z)=F_{1/p1,…,1/pk}(z)=f(z₁), (f'(z₁))^1/p1z, …, (f'(z₁))^1/pkw)
is a normalized biholomorphic convex mapping on
Ω_N={(z₁, z, …, w) ∈ C×C^n₁×…×C^n_k:||z||₁^p1 + … +||w||_k^pk<1/λ_G (z₁)},
where N=1 + n₁ + … + n_k and the branch is chosen such that (f'(z₁))^1/p_j|_z1=0=1, j=1, …, k. Applying to the Roper-Suffridge extension operator, we obtain a new convex mappings construction of an unbounded domain and a refinement of convex mappings construction on a Reinhardt domain, respectively.

In this article, we prove the existence and uniqueness of solutions of the Navier-Stokes equations with Navier slip boundary condition for incompressible fluid in a bounded domain of R³. The results are established by the Galerkin approximation method and improved the existing results.

The purpose of this work is to investigate the initial value problem for a general isothermal model of capillary fluids derived by Dunn and Serrin[12], which can be used as a phase transition model. Motivated by[9], we aim at extending the work by Danchin-Desjardins[11] to a critical framework which is not related to the energy space. For small perturbations of a stable equilibrium state in the sense of suitable L^p-type Besov norms, we establish the global existence. As a consequence, like for incompressible flows, one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true.

In the paper we introduce an idea of harmonic functions with correlated coefficients which generalize the ideas of harmonic functions with negative coefficients introduced by Silverman and harmonic functions with varying coefficients defined by Jahangiri and Silverman. Next we define classes of harmonic functions with correlated coefficients in terms of generalized Dziok-Srivastava operator. By using extreme points theory, we obtain estimations of classical convex functionals on the defined classes of functions. Some applications of the main results are also considered.

We use two simple methods to derive four important explicit graphical solutions of the curve shortening flow in the plane. They are well-known as the circle, hairclip,paperclip, and grim reaper solutions of the curve shortening flow. By the methods, one can also see that the hairclip and the paperclip solutions both converge to the grim reaper solutions as t→-∞.

In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in C². Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in C² than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant Kähler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant Kähler angle is generally non-vanishing. Secondly, we obtain two pinching results for the Kähler angle which imply rigidity theorems of self-shrinkers with Kähler angle under the condition that ∫_M|h|²e^-|x|2/2 dV_M<∞, where h and x denote, respectively, the second fundamental form and the position vector of the surface.

Real epidemic spreading networks are often composed of several kinds of complex networks interconnected with each other, such as Lyme disease, and the interrelated networks may have different topologies and epidemic dynamics. Moreover, most human infectious diseases are derived from animals, and zoonotic infections always spread on directed interconnected networks. So, in this article, we consider the epidemic dynamics of zoonotic infections on a unidirectional circular-coupled network. Here, we construct two unidirectional three-layer circular interactive networks, one model has direct contact between interactive networks, the other model describes diseases transmitted through vectors between interactive networks, which are established by introducing the heterogeneous mean-field approach method. Then we obtain the basic reproduction numbers and stability of equilibria of the two models. Through mathematical analysis and numerical simulations, it is found that basic reproduction numbers of the models depend on the infection rates, infection periods, average degrees, and degree ratios. Numerical simulations illustrate and expand these theoretical results very well.

In this article, we discuss the stability of ε-isometries for L_∞,λ-spaces. Indeed, we first study the relationship among separably injectivity, injectivity, cardinality injectivity and universally left stability of L_∞,λ-spaces as well as we show that the second duals of universally left-stable spaces are injective, which gives a partial answer to a question of Bao-Cheng-Cheng-Dai, and then we prove a sharpen quantitative and generalized Sobczyk theorem which gives examples of nonseparable L_∞-spaces X (but not injective) such that the couple (X, Y) is stable for every separable space Y. This gives a new positive answer to Qian's problem.