In this paper, we investigate the λ-nuclearity in the system of completely 1-summing mapping spaces (Π1(·, ·), π1). In Section 2, we obtain that ℂ is the unique operator space that is nuclear in the system (Π1(·, ·), π1). We generalize some results in Section 2 to λ-nuclearity in Section 3.

In this paper, the Cauchy problem for the two layer viscous shallow water equations is investigated with third-order surface-tension terms and a low regularity assumption on the initial data. The global existence and uniqueness of the strong solution in a hybrid Besov space are proved by using the Littlewood-Paley decomposition and Friedrichs’ regularization method.

In this article, we study multivariate approximation defined over weighted anisotropic Sobolev spaces which depend on two sequences a = {aj}j≥1 and b = {bj}j≥1 of positive numbers. We obtain strong equivalences of the approximation numbers, and necessary and sufficient conditions on a, b to achieve various notions of tractability of the weighted anisotropic Sobolev embeddings.

Although QP-free algorithms have good theoretical convergence and are effective in practice, their applications to minimax optimization have not yet been investigated. In this article, on the basis of the stationary conditions, without the exponential smooth function or constrained smooth transformation, we propose a QP-free algorithm for the nonlinear minimax optimization with inequality constraints

The theory of increasing and convex-along-rays (ICAR) functions defined on a convex cone in a real locally convex topological vector space X was already well developed. In this paper, we first examine abstract convexity of increasing plus-convex-along-rays (IPCAR) functions defined on a real normed linear space X. We also study, for this class of functions, some concepts of abstract convexity, such

This article is devoted to studying the initial-boundary value problem for an ideal polytropic model of non-viscous and compressible gas. We focus our attention on the outflow problem when the flow velocity on the boundary is negative and give a rigorous proof of the asymptotic stability of both the degenerate boundary layer and its superposition with the 3-rarefaction wave under some smallness conditions

This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation. We obtain a critical value λ D1 (Ω0), and demonstrate that the existence of the predator in \({\overline \Omega _0}\) only depends on the relationship of the growth rate μ of the predator and λ D1 (Ω0), not on the prey. Furthermore, when μ < λ D1 (Ω0), we obtain the existence and uniqueness of

In this article we study the two-dimensional complete λ-translators immersed in the Euclidean space ℝ3 and Minkovski space ℝ31 . We obtain two classification theorems: one for two-dimensional complete λ-translators x: M2 → ℝ3 and one for two-dimensional complete space-like λ-translators x: M2 → ℝ31 , with a second fundamental form of constant length.

We show that the spatial Lq-norm (q > 5/3) of the vorticity of an incompressible viscous fluid in ℝ3 remains bounded uniformly in time, provided that the direction of vorticity is Hölder continuous in space, and that the space-time Lq-norm of vorticity is finite. The Hölder index depends only on q. This serves as a variant of the classical result by Constantin-Fefferman (Direction of vorticity and

In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term: $${( - \Delta )^s}u - \gamma {u \over {{{\left| x \right|}^{2s}}}} = {{{{\left| u \right|}^{2_s^ * (\beta ) - 2}}u} \over {{{\left| x \right|}^\beta }}} + \left[ {{I_\mu } * {F_\alpha }( \cdot ,u)} \right](x){f_\alpha }(x,u),\;\;\;\;u \in {\dot H^{^s}}({\mathbb{R}^n})

We prove that the group of weighted composition operators induced by continuous automorphism groups of the upper half plane \(\mathbb{U}\) is strongly continuous on the weighted Dirichlet space of \(\mathbb{U}\), \({{\cal D}_\alpha }(\mathbb{U})\). Further, we investigate when they are isometries on \({{\cal D}_\alpha }(\mathbb{U})\). In each case, we determine the semigroup properties while in the

In this article we consider the compressible viscous magnetohydrodynamic equations with Coulomb force. By spectral analysis and energy methods, we obtain the optimal time decay estimate of the solution. We show that the global classical solution converges to its equilibrium state at the same decay rate as the solution of the linearized equations.

In this article, we first give two simple examples to illustrate that two types of parametric representation of the family of Σ0K have some gaps. Then we also find that the area derivative formula (1.6), which is used to estimate the area distortion of Σ0K , cannot be derived from [6], but that formula still holds for Σ0K through our amendatory parametric representation for the one obtained by Eremenko

In this article, we study the following fractional (p, q)-Laplacian equations involving the critical Sobolev exponent: $$({P_{\mu ,\lambda }})\left\{ {\begin{array}{*{20}{l}} {( - \Delta )_p^{{s_1}}u + ( - \Delta )_q^{{s_2}}u = \mu |u{|^{q - 2}}u + \lambda |u{|^{p - 2}}u + |u{|^{p_{{s_1}}^* - 2}}u,}&{\text{in}\;\Omega ,} \\ {u = 0,}&{\text{in}\;{\mathbb{R}^N}\backslash \Omega ,} \end{array}} \right

Let K be the familiar class of normalized convex functions in the unit disk. In [14], Keogh and Merkes proved that for a function \(f(z) = z + \sum\limits_{k = 2}^\infty {{a_k}} {z^k}\) in the class K, $$\left| {{a_3} - \lambda a_2^2} \right| \le \max \left\{ {{1 \over 3},\left| {\lambda - 1} \right|} \right\},\;\;\;\;\lambda \in \mathbb{C}.$$ The above estimate is sharp for each λ. In this article

In this article, we study optimal reinsurance design. By employing the increasing convex functions as the admissible ceded loss functions and the distortion premium principle, we study and obtain the optimal reinsurance treaty by minimizing the VaR (value at risk) of the reinsurer’s total risk exposure. When the distortion premium principle is specified to be the expectation premium principle, we also

This article is devoted to a deep study of the Roper-Suffridge extension operator with special geometric properties. First, we prove that the Roper-Suffridge extension operator preserves ϵ starlikeness on the open unit ball of a complex Banach space ℂ × X, where X is a complex Banach space. This result includes many known results. Secondly, by introducing a new class of almost boundary starlike mappings

Let \({D_{{p_1},{p_2}, \cdots ,{p_n}}} = {\rm{\{ }}z \in {\mathbb{C}^n}{\rm{:}}\sum\limits_{l = 1}^n {{{\left| {{z_l}} \right|}^{{p_l}}} < 1{\rm{\} }},{p_l} > 1,l = 1,2, \cdots \;,n}.\). In this article, we first establish the sharp estimates of the main coefficients for a subclass of quasi-convex mappings (including quasi-convex mappings of type \(\mathbb{A}\) and quasi-convex mappings of type \(\mathbb{B}\))

In this paper, we study the coupled system of Kirchhoff type equations $$\left\{ {\matrix{ { - \left( {a + b\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}{\rm{d}}x} } \right)\Delta u + u = {{2\alpha } \over {\alpha + \beta }}{{\left| u \right|}^{\alpha - 2}}u{{\left| v \right|}^\beta },\;x \in {\mathbb{R}^3},} \hfill \cr { - \left( {a + b\int_{{\mathbb{R}^3}} {{{\left| {\nabla v} \right|}^2}{\rm{d}}x}

We show that the Weyl symbol of a Born-Jordan operator is in the same class as the Born-Jordan symbol, when Hörmander symbols and certain types of modulation spaces are used as symbol classes. We use these properties to carry over continuity, nuclearity and Schatten-von Neumann properties to the Born-Jordan calculus.

In this article, we derive the Lp-boundedness of the variation operators associated with the heat semigroup which is generated by the high order Schrödinger type operator (−Δ)2 + V2 in ℝn(n ≥ 5) with V being a nonnegative potential satisfying the reverse Hölder inequality. Furthermore, we prove the boundedness of the variation operators on associated Morrey spaces. In the proof of the main results

The Editor-in-Chief has retracted this article [1], because one of the Editorial Board Mem- bers pointed out that Lemma 3.5 in [1] has an uncorrectable gap which can only be true when n = 1, 2, 3, indicating that the major theorem 1.1 in [1] is only a special case of Theorem 1 in [2], and Theorem 1.2 in [1] is only a trivial result in [3]. The author refused to retract the article [1] in his first

In this article, by the mean-integral of the conserved quantity, we prove that the one-dimensional non-isentropic gas dynamic equations in an ideal gas state do not possess a bounded invariant region. Moreover, we obtain a necessary condition on the state equations for the existence of an invariant region for a non-isentropic process. Finally, we provide a mathematical example showing that with a special

The numerical simulation of a three-dimensional semiconductor device is a fundamental problem in information science. The mathematical model is defined by an initial-boundary nonlinear system of four partial differential equations: an elliptic equation for electric potential, two convection-diffusion equations for electron concentration and hole concentration, and a heat conduction equation for temperature

In this paper we prove a central limit theorem and a moderate deviation principle for a class of semilinear stochastic partial differential equations, which contain the stochastic Burgers’ equation and the stochastic reaction-diffusion equation. The weak convergence method plays an important role.

This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator. Naturally, the resulting linear system is symmetric and positive definite, and thus the algorithm is easy to implement and analyze. Convergence analysis in the H2 equivalent norm is established on an arbitrary shape regular

In 2018, Duan [1] studied the case of zero heat conductivity for a one-dimensional compressible micropolar fluid model. Due to the absence of heat conductivity, it is quite difficult to close the energy estimates. He considered the far-field states of the initial data to be constants; that is, \(\mathop {\lim }\limits_{x \to \pm \infty } ({v_0},{u_0},{\omega _0},{\theta _0})(x) = (1,0,0,1)\). He proved

We consider the large time behavior of solutions of the Cauchy problem for the one-dimensional compressible Navier-Stokes equations for a reacting mixture. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave which corresponds to the contact discontinuity is asymptotically stable, provided the strength of contact

We study the existence and the regularity of solutions for a class of nonlocal equations involving the fractional Laplacian operator with singular nonlinearity and Radon measure data.

In this article, we study the initial boundary value problem of the two-dimensional nonhomogeneous incompressible primitive equations and obtain the local existence and uniqueness of strong solutions. The initial vacuum is allowed.

We give characterizations for Bergman-Orlicz spaces with standard weights via a Lipschitz type condition in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As an application, we obtain the boundeness of the symmetric lifting operator from Bergman-Orlicz spaces on the unit disk into Bergman-Orlicz spaces on the bidisk.

We study a Schrödinger system with the sum of linear and nonlinear couplings. Applying index theory, we obtain infinitely many solutions for the system with periodic potentials. Moreover, by using the concentration compactness method, we prove the existence and nonexistence of ground state solutions for the system with close-to-periodic potentials.

This article is devoted to the identification, from observations or field measurements, of the hydraulic conductivity K for the saltwater intrusion problem in confined aquifers. The involved PDE model is a coupled system of nonlinear parabolic-elliptic equations completed by boundary and initial conditions. The main unknowns are the saltwater/freshwater interface depth and the freshwater hydraulic

In this article, we study the generalized quasilinear Schrödinger equation$$ - \text{div}({\varepsilon ^2}{g^2}(u)\nabla u) + {\varepsilon ^2}g(u)g'(u){\left| {\nabla u} \right|^2} + V(x)u = K(x){\left| u \right|^{p - 2}}u,\;\;\;\;x \in {\mathbb{R}^N},$$where N ≥ 3, ε> 0, 4

In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator \({\cal L} = \Delta + {\bf{V}}\) in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Hölder’s inequality. Our results are new and

Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [26] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces 〈Lcosh−1, L log(L + 1)〉, since this framework gives a better description of regular observables, and also allows for a

We apply the Davies method to give a quick proof for the upper estimate of the heat kernel for the non-local Dirichlet form on the ultra-metric space. The key observation is that the heat kernel of the truncated Dirichlet form vanishes when two spatial points are separated by any ball of a radius larger than the truncated range. This new phenomenon arises from the ultra-metric property of the space

In this paper, we study the following perturbation problem with Sobolev critical exponent:$$\left\{ {\begin{array}{*{20}{c}} { - \Delta u = (1 + \varepsilon K(x)){u^{2* - 1}} + \frac{\alpha }{{{2^*}}}{u^{\alpha - 1}}{v^\beta } + \varepsilon h(x){u^p},}&{x \in {\mathbb{R}^N},} \\ { - \Delta v = (1 + \varepsilon Q(x)){v^{2* - 1}} + \frac{\beta }{{{2^*}}}{u^\alpha }{v^{\beta - 1}} + \varepsilon l(x){v^q}

In this article, we investigate a fractional-order singular Leslie-Gower prey-predator bioeconomic model, which describes the interaction between populations of prey and predator, and takes into account the economic interest. We firstly obtain the solvability condition and the stability of the model system, and discuss the singularity induced bifurcation phenomenon. Next, we introduce a state feedback

This article extends the results of Arkeryd and Cercignani [6]. It is shown that the Cauchy problem for the relativistic Enskog equation in a periodic box has a global mild solution if the mass, energy and entropy of the initial data are finite. It is also found that the solutions of the relativistic Enskog equation weakly converge to the solutions of the relativistic Boltzmann equation in L1 if the

This article is concerned with the existence of traveling wave solutions for a discrete diffusive ratio-dependent predator-prey model. By applying Schauder’s fixed point theorem with the help of suitable upper and lower solutions, we prove that there exists a positive constant c* such that when c > c*, the discrete diffusive predator-prey system admits an invasion traveling wave. The existence of an

In this paper, we study the global existence of weak solutions for the Cauchy problem of the nonlinear hyperbolic system of three equations (1.1) with bounded initial data (1.2). When we fix the third variable s, the system about the variables ρ and u is the classical isentropic gas dynamics in Eulerian coordinates with the pressure function \(P(\rho ,s) = {{\rm{e}}^s}{{\rm{e}}^{ - {1 \over \rho }}}\)

In this article, we extend the well-known Roper-Suffridge operator on Bn+1 and bounded complete Reinhardt domains in ℂn+1, then we investigate the properties of the generalized operators. Applying the Loewner theory, we obtain the mappings constructed by the generalized operators that have parametric representation on Bn+1. In addition, by using the geometric characteristics and the parametric representation

In this article, two results concerning the periodic points and normality of meromorphic functions are obtained: (i) the exact lower bound for the numbers of periodic points of rational functions with multiple fixed points and zeros is proven by letting R(z) be a non-polynomial rational function, and if all zeros and poles of R(z) − z are multiple, then Rk(z) has at least k + 1 fixed points in the

In this article, we discuss, by Nevanlinna theory, the influence of multiple values and deficiencies on the uniqueness problem of algebroid functions. We get several uniqueness theorems of algebroid functions which include an at most 3v-valued theorem. These results extend the existing achievements of some scholars.

In this paper, we study the generalized lower order of entire functions defined by Dirichlet series. By constructing the Newton polygon based on Knopp-Kojima’s formula, we obtain a relation between the coefficients of the Dirichlet series and its generalized lower order.

We consider the Schrödinger-Poisson system with nonlinear term Q(x)|u|p−1u, where the value of \(\mathop {\lim}\limits_{\left| x \right| \to \infty} \,\,Q\left(x \right)\)Q(x) may not exist and Q may change sign. This means that the problem may have no limit problem. The existence of nonnegative ground state solutions is established. Our method relies upon the variational method and some analysis tricks

In this article, we study the pointwise estimates of solutions to the nonlinear viscous wave equation in even dimensions (n ≥ 4). We use the Green’s function method. Our approach is on the basis of the detailed analysis of the Green’s function of the linearized system. We show that the decay rates of the solution for the same problem are different in even dimensions and odd dimensions. It is shown

This article investigates the algebraic differential independence concerning the Euler Γ-function and the function F in a certain class \({\mathbb F}\) which contains Dirichlet ℒ-functions, ℒ-functions in the extended Selberg class, or some periodic functions. We prove that the Euler Γ-function and the function F cannot satisfy any nontrivial algebraic differential equations whose coefficients are

We discuss the set-valued dynamics related to the theory of functional equations. We look for selections of convex set-valued functions satisfying set-valued Euler-Lagrange inclusions. We improve and extend upon some of the results in [13, 20], but under weaker assumptions. Some applications of our results are also provided.

In this paper, we give a simpler proof for Ohta’s theorems [1995, Ann. Inst. Henri Poincare, 63, 111; 1995, Diff. Integral Eq., 8, 1775] on the strong instability of the ground states for a generalized Davey-Stewartson system. In addition, a sufficient condition is given to ensure the nonexistence of a minimizer for a variational problem, which is related to the stability of the standing waves of the

In this article, we study the blow-up solutions for a case of b-family equations. Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution

Multi-parameter mixed Hardy space \(H_{{\rm{mix}}}^p\) is introduced by a new discrete Calderón’s identity. As an application, we obtain the \(H_{{\rm{mix}}}^p \to {L^p}\left({{ℝ^{{n_1} + {n_2}}}} \right)\) boundedness of operators in the mixed Journé’s class.

We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline

In this paper we consider one dimensional mean-field backward stochastic differential equations (BSDEs) under weak assumptions on the coefficient. Unlike [3], the generator of our mean-field BSDEs depends not only on the solution (Y, Z) but also on the law PY of Y. The first part of the paper is devoted to the existence and uniqueness of solutions in Lp, 1 < p ≤ 2, where the monotonicity conditions

In this paper, we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit. As an application, we obtain the convergence of random attractors for non-autonomous stochastic reaction-diffusion equations on unbounded domains, when the density of stochastic noises approaches zero. The weak convergence of solutions is proved by

We consider a Neumann problem driven by the (p, q)-Laplacian under the Landesman-Lazer type condition. Using the classical saddle point theorem and other classical results of the calculus of variations, we show that the problem has at least one nontrivial weak solution.

This paper is concerned with the large-time behavior of solutions to the Cauchy problem of a one-dimensional viscous radiative and reactive gas. Based on the elaborate energy estimates, we develop a new approach to derive the upper bound of the absolute temperature by avoiding the use of auxiliary functions Z(t) and W(t) introduced by Liao and Zhao [J. Differential Equations, 2018, 265(5): 2076–2120]

In this article, we investigate a stochastic Galerkin method for the Maxwell equations with random inputs. The generalized Polynomial Chaos (gPC) expansion technique is used to obtain a deterministic system of the gPC expansion coefficients. The regularity of the solution with respect to the random is analyzed. On the basis of the regularity results, the optimal convergence rate of the stochastic Galerkin

We establish uniqueness theorems of L-functions in the extended Selberg class, which show how an L-function and a meromorphic function are uniquely determined by their shared values in two finite sets. This can be seen as a new solution of a problem proposed by Gross.