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We consider the 2-d isentropic compressible Euler equations. It was shown in [E. Chiodaroli, C. De Lellis and O. Kreml, Global ill-posedness of the isentropic system of gas dynamics, Comm. Pure Appl. Math. 68(7) (2015) 1157–1190] that there exist Riemann initial data as well as Lipschitz initial data for which there exist infinitely many weak solutions that fulfill an energy inequality. In this paper, we will prove that there is Riemann initial data for which there exist infinitely many weak solutions that conserve energy, i.e. they fulfill an energy equality. As in the aforementioned paper, we will also show that there even exist Lipschitz initial data with the same property.

We demonstrate the asymptotic analysis of a semi-linear optimal control problem posed on a smooth oscillating boundary domain in the present paper. We have considered a more general oscillating domain than the usual “pillar-type” domains. Consideration of such general domains will be useful in more realistic applications like circular domain with rugose boundary. We study the asymptotic behavior of the problem under consideration using a new generalized periodic unfolding operator. Further, we are studying the homogenization of a non-linear optimal control problem and such non-linear problems are limited in the literature despite the fact that they have enormous real-life applications. Among several other technical difficulties, the absence of a sufficient criteria for the optimal control is one of the most attention-grabbing issues in the current setting. We also obtain corrector results in this paper.

In this paper, we present all types of 3-point quadrature rules on continuous function on interval [Formula: see text] with constant weight function and compare there with the composite type also. All 3-point (open and closed) rules are taken from available papers and some new results like non-polynomial fitting and derivative type are introduced. Also, differences, comparisons, and errors between method procedures have been shown.

In this paper, we prove some common fixed point theorems by using the generalized distance in a cone metric space. The results of this paper extend well-known results in the literature.

We consider the initial value problem for the semilinear wave equation with time-dependent effective damping. The interest is the behavior of lifespan of solutions in view of the asymptotic profile of the damping as $t\to \infty$. The result of this paper is the sharp lifespan estimates of blowup solutions for general time-dependent damping including threshold cases between effective and overdamping., 20 pages, typos are corrected, references are updated, to appear in J. Hyperbolic Differ. Equ

In this paper, We establish new homogenization results for stochastic nonlinear hyperbolic equations with periodically oscillating coefficients. We use a delicate blending of Tartar’s method of oscillating test functions and deep probabilistic compactness results due to Prokhorov and Skorokhod. We prove that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized stochastic hyperbolic problem with constant coefficients. We also prove the convergence of the associated energies.

We study a general system of hyperbolic-parabolic balance laws in [Formula: see text] space dimensions ([Formula: see text]). The system has rank deficient viscosity matrices and a lower order term whose Jacobian matrix is rank deficient as well. We consider the Cauchy problem when initial data are small perturbations of a constant equilibrium state. Under a set of reasonable assumptions including Kawashima–Shizuta condition, we establish the existence of solution global in time via energy method. The proposed assumptions are sufficiently general for applications to physical models such as electro-magneto flows and physical gas flows. In particular, we study the gas flow with an internal non-equilibrium mode besides the translational non-equilibrium. The general result in this paper recovers the existing results in literature on hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively, as two special cases.

In this paper, we propose a non-autonomous and diffusive SIR epidemic model based on the fact that the infection rate, the removal rate and the death rate often vary in time. The explicit formulas of the basic reproduction number [Formula: see text] and the minimum wave speed [Formula: see text] are derived. Applying upper-lower solution method and Schauder’s fixed point theorem, we show that when [Formula: see text], [Formula: see text] and the diffusion rates satisfy a certain condition, a time periodic traveling wave solution exists in the model. By the method of contradiction analysis and the comparison arguments together with the properties of the spreading speed of an associated subsystem, we prove that when [Formula: see text] and [Formula: see text] or [Formula: see text] and [Formula: see text], the model possesses no time periodic traveling wave solutions.

This paper presents a new non-local expanding flow for convex closed curves in the Euclidean plane which increases both the perimeter of the evolving curves and the enclosed area. But the flow expands the evolving curves to a finite circle smoothly if they do not develop singularity during the evolving process. In addition, it is shown that an additional assumption about the initial curve will ensure that the flow exists on the time interval [Formula: see text]. Meanwhile, a numerical experiment reveals that this flow may blow up for some initial convex curves.

In this paper, regularity properties, Strichartz type estimates for solution of integral problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator [Formula: see text] defined in a Banach space [Formula: see text], in which by choosing [Formula: see text] and [Formula: see text] we can obtain numerous classis of initial value problems for Schrödinger equations which occur in a wide variety of physical systems.

In an attempt to resolve the von Neumann triple point paradox in shock reflection phenomenon, a new type of reflection configuration, called Guderley Mach reflection, was observed both in numerical simulations and physical experiments recently. In this type of reflection configuration, there exists a sequence of triple points, with a centered expansion fan and a supersonic patch at each triple point. In this paper, we present a mathematical analysis of the centered wave flow patches of Guderley Mach reflection. In order to construct such a flow patch, a centered wave problem is introduced. The existence of a global classical solution to the centered wave problem for the two-dimensional isentropic irrotational steady Euler equations is established.

In this paper, we study the Ricci–Bourguignon flow of all locally homogenous geometries on closed three-dimensional manifolds. We also consider the evolution of the Yamabe constant under the Ricci–Bourguignon flow. Finally, we prove some results for the Bach-flat shrinking gradient soliton to the Ricci–Bourguignon flow.

In this paper, we mainly investigate the time fractional nonlinear wave equation which can be usually used to express nonlinear phenomena appearing in shallow water waves by using group analysis scheme. First, the symmetry can be obtained make uses of the group analysis to the time fractional nonlinear wave equation. Based on the above-found symmetry, this equation was able to reduce into an ordinary differential equation of fractional order. As a result, some new invariant solutions were also constructed for this considered equation. Second, the scaling transformation was also obtained by introducing new independent and dependent variables. Finally, the conservation laws were also found to satisfy the time fractional nonlinear wave equation with the help of the Ibragimov theorem. These novel results show unique nonlinear phenomena.

In this paper, we study the local well-posedness for the Cauchy problem of a semilinear fractional diffusion equation where the perturbations behave like [Formula: see text] and [Formula: see text], and [Formula: see text] is the characteristic function of a ball [Formula: see text]. Here, we are interested in the solvability of the problem when singular initial data [Formula: see text] are taken in [Formula: see text]. Eventually, we give sufficient conditions to the nonexistence of positive global solutions.

Asymptotic analysis of an incompressible Stokes fluid in a dynamic regime in a three-dimensional thin domain [Formula: see text] with mixed boundary conditions and Tresca friction law is studied in this paper. The problem statement and variational formulation of the problem are reformulated in a fixed domain. In which case, the estimates on velocity and pressure are proved. These estimates will be useful in order to give a specific Reynolds equation associated with variational inequalities and prove the uniqueness.

This paper studies the global bifurcation phenomenon for the following homogeneous nonlocal eigenvalue problem [Formula: see text] Under some natural hypotheses on [Formula: see text] and [Formula: see text], we show that [Formula: see text] is a bifurcation point of the nontrivial solution set of the above problem. As application of the above result, we determine the interval of [Formula: see text], in which there exist positive solutions for the following Kirchhoff type problem [Formula: see text] where [Formula: see text] is asymptotically 3-linear at zero and infinity. Our results provide a positive answer to an open problem. Moreover, we also study the spectral structure for a homogeneous nonlocal eigenvalue problem.

To tackle a linear Fredholm integral equation on great interval, two numerical processes are involved: discretization and iterative scheme. The conventional numerical process is discretize first then use an iterative scheme as Jacobi’s method to approach the solutions of the huge algebraic system. In this paper, we propose an alternative numerical process, we apply an iterative scheme based on construction of a generalization of the iterative scheme for Jacobi method which is adapted to the system of linear bounded operators, then we use Nyström method to discretize only the diagonal part of the system. The convergence analysis of this new method is proved and numerical tests developed show its effectiveness.

In this paper, we present a monotone domain decomposition iterative technique for boundary value problems of a nonlinear fractional differential equation. We construct two monotone domain decomposition sequences of upper and lower solutions which converge uniformly to the actual solution of the problem. The accuracy and efficiency of our new approach are demonstrated through two examples.

The aim of this paper is to study the existence and uniqueness of solutions for a boundary value problem associated with a fractional nonlinear differential equation with higher order posed on the half-line. An appropriate continuous embedding for suitable Banach spaces are proved and the Minty–Browder theorem for monotone operators is used in the proof of existence of solutions for a boundary value problem of fractional order posed on the half-line.

In this paper, we study some type of equations which may model the behavior of species inhabiting in some habitat. For our purpose, using a priori bounded techniques, we obtain a positive solution to a family of non-local partial differential equations with non-homogeneous boundary conditions.

This paper not only presents a criterion for the Cordes–Nirenberg space [Formula: see text] imbedding between the associate Morrey space [Formula: see text] and the Morrey space [Formula: see text], but also characterizes a Radon measure [Formula: see text] such that the Cordes–Nirenberg potential space [Formula: see text] is restricted to the [Formula: see text]-based Campanato space [Formula: see text] — thereby giving an application of the discovered characterization to the regularity for a class of the elliptic equations with symmetric [Formula: see text]-coefficients.

Dispersive SQG equation have been studied by many works (see, e.g., [M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Londen. Math. Soc. 106 (2013) 650–674; T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684; A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity 23 (2010) 549–554; R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys. 67 (2016) 104]), which is very similar to the 3D rotating Euler or Navier–Stokes equations. Long time stability for the dispersive SQG equation without dissipation was obtained by Elgindi–Widmayer [Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684], where the initial condition [Formula: see text] [Formula: see text] plays a important role in their proof. In this paper, by using the Strichartz estimate, we can remove this initial condition. Namely, we only assume the initial data is in the Sobolev space like [Formula: see text]. As an application, we can also obtain similar result for the 2D Boussinesq equations with the initial data near a nontrivial equilibrium.

In this paper, existence and multiplicity of positive solutions of the elliptic system − Δu + V1(x)u = μ1(x)u3 + β(x)uv2in Ω 𝜀, − Δv + V2(x)v = β(x)u2v + μ 2(x)v3 in Ω 𝜀,u = v = 0 on ∂Ω𝜀 is proved, where Ω𝜀 is an exterior domain in ℝN such that ℝN∖Ω 𝜀 is far away from the origin and contains a sufficiently large ball, N = 1, 2, 3, and the coefficients Vj,μj,β are continuous functions on ℝN which tend to positive constants at infinity. We do not assume μ1,μ2,β to be positive functions.

This paper extracts the analytical soliton solutions of the perturbed NLSE given in (1). We use successfully two integration methods namely the extended simple equation method and generalized Kudryashov method. In view of the results obtained, some new additional ones have been obtained. The results are dark, bright and exact solutions that propagate in the fiber optic and left-handed metamaterials (LHMs).

This study aimed to give definitions and relations between strip theory and harmonic curvatures of the strip in Minkowski space. Previously, the same was done in Euclidean Space (see [F. Ertem Kaya, Y. Yayli and H. H. Hacısalihoglu, A characterization of cylindrical helix strip, Commun. Fac. Sci. Univ. Ank. Ser. A1 59(2) (2010) 37–51]). The present paper gives for the first time a generic characterization of the harmonic curvatures of the strip, helix strip and inclined strip in Minkowski space.

In this paper, Pachpatte’s inequality is employed to discuss the Ulam–Hyers stabilities for Volterra integrodifferential equations and Volterra delay integrodifferential equations in Banach spaces on both finite and infinite intervals. Examples are given to show the applicability of our obtained results.

This paper deals with the curvature bound for a nonlocal curve flow with a prescribed rate of change in enclosed area via Andrews–Bryan’s distance comparison. As a by-product, a partial answer to a conjecture given by Dallaston and McCue is obtained and the [Formula: see text] convergence of the curvature for the nonlocal flow is achieved.

In this paper, we study an evolution equation involving the normalized [Formula: see text]-Laplacian and a bounded continuous source term. The normalized [Formula: see text]-Laplacian is in non-divergence form and arises for example from stochastic tug-of-war games with noise. We prove local [Formula: see text] regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration.

In this paper, we characterize a family of solitary waves for nonlinear Schrödinger equation (NLS) with derivative (DNLS) by the structure analysis and the variational argument. Since DNLS does not enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters [Formula: see text] and the critical parameters [Formula: see text], we show the existence and uniqueness of the solitary waves for DNLS, up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters [Formula: see text], [Formula: see text] and the supercritical parameters [Formula: see text], there is no nontrivial solitary wave for DNLS. At last, we make use of the invariant sets, which is related to the variational characterization of the solitary wave, to obtain the global existence of solution for DNLS with initial data in the invariant set [Formula: see text], with [Formula: see text], [Formula: see text] or [Formula: see text]. On the one hand, different with the scattering result for the [Formula: see text]-critical NLS in [B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math. 285(5) (2015) 1589–1618], the scattering result of DNLS does not hold for initial data in [Formula: see text] because of the existence of infinity many small solitary/traveling waves in [Formula: see text] with [Formula: see text], [Formula: see text] or [Formula: see text]. On the other hand, our global result improves the global result in [Y. Wu, Global well-posedness of the derivative nonlinear Schrödinger equations in energy space, Anal. Partial Differential Equations 6(8) (2013) 1989–2002; Global well-posedness on the derivative nonlinear Schrödinger equation, Anal. Partial Differential Equations 8(5) (2015) 1101–1112] (see Corollary 1.6).

In this paper, we consider two types of set-valued Volterra–Hammerstein integral equations and prove the existence and uniqueness theorem.

In this paper, we study diagonal hyperbolic systems in one space dimension. Based on a new gradient entropy estimate, we prove the global existence of a continuous solution, for large and non-decreasing initial data. We remark that these results cover the case of systems which are hyperbolic but not strictly hyperbolic. Physically, this kind of diagonal hyperbolic system appears naturally in the modeling of the dynamics of dislocation densities.

In this paper, we give new conditions for existence and uniqueness of best proximity point. Also, we introduce the concept of cyclic contraction and nonexpansive for multivalued map and we give existence and convergence theorems for best proximity point in the complete metric space.

In this paper, we define a new integral operator in the open unit disk. This operator is considered as a complex Volterra operator. Moreover, we define a new subspace of Hardy space involving the normalized analytic functions. We shall show that the new integral operator is closed in the subspace of normalized functions. Geometric characterizations are established in the sequel. Our display is maintained by the Jack Lemma.

In this paper, we consider the following singular elliptic problem involving an exponential nonlinearity in two dimensions: [Formula: see text] [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We show the existence and multiplicity of positive solutions globally with respect to the bifurcation parameter [Formula: see text].

In this paper, we study a [Formula: see text]th-order impulsive eigenvalue problem with an advanced argument. We shall establish several criteria for the optimal intervals of the parameter [Formula: see text] so as to ensure existence of single or many positive solutions. Our methods are based on transformation technique, Hölder’s inequality and the eigenvalue theory.

In this paper, an approximate solution for solving weakly singular kernel partial integro-differential equations with time fractional order is proposed. The method is based on using a second-order time difference approximation followed by applying the fractional integral operator and piecewise linear interpolation to compute the singularity of the kernel that appear in the discretization process. The stability of the method is also considered in the sense of von Neumann analysis. Numerical examples are solved to demonstrate the validity and applicability of the presented technique.

We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of Bartolucci-Chen-Lin-Tarantello it is proved that the profile of the solutions differs from global solutions of a Liouville type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate., Comment: 21 pages. Communications on contemporary mathematics, in press

In this article we study a class of mixed monotone operators with convexity on ordered Banach spaces and present some new tripled fixed point theorems by means of partial order theory, we get the existence and uniqueness of tripled fixed points without assuming the operator to be compact or continuous, which extend the existing corresponding results. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for a fractional differential equation boundary value problem.

In this paper, we prove the global gradient estimates on the generalized Lebesgue spaces for weak solutions to elliptic quasilinear obstacle problems. It is worth noticing that the coefficients related to the obstacle problems are merely measurable with small BMO norms and the underlying domain does not satisfy any smoothness conditions.

In this paper, we prove the local existence and uniqueness of the solution with discontinuous density for the inhomogeneous non-resistive magnetohydrodynamics (MHD) equations on a [Formula: see text] bounded domain [Formula: see text] or [Formula: see text], if the initial data [Formula: see text] with [Formula: see text] satisfies [Formula: see text].

In this paper, the existence and multiplicity of solutions for a quasilinear elliptic problem driven by the [Formula: see text]-Laplacian operator is established. These solutions are also built as ground state solutions using the Nehari method. The main difficulty arises from the fact that the [Formula: see text]-Laplacian operator is not homogeneous and the nonlinear term is indefinite.

This paper is devoted to investigate the symmetry and monotonicity properties for positive solutions of fractional Laplacian equations. Especially, we consider the following fractional Laplacian equation with homogeneous Dirichlet condition: [Formula: see text] Here [Formula: see text] is a domain (bounded or unbounded) in [Formula: see text] which is convex in [Formula: see text]-direction. [Formula: see text] is the nonlocal fractional Laplacian operator which is defined as [Formula: see text] Under various conditions on [Formula: see text] and on a solution [Formula: see text] it is shown that [Formula: see text] is strictly increasing in [Formula: see text] in the left half of [Formula: see text], or in [Formula: see text]. Symmetry (in [Formula: see text]) of some solutions is proved.

In this paper, by Morse theory, we study the existence and multiplicity of solutions for the [Formula: see text]-Laplacian equation with a “concave” nonlinearity and a parameter. In our results, we do not need any additional global condition on the nonlinearities, except for a subcritical growth condition.

In this paper, we show some relations between traveling fronts of curvature flow and constant [Formula: see text]-curvature curves. In the plane with density [Formula: see text] satisfying [Formula: see text], we prove a Fenchel’s type theorem for the class of simple, closed, convex curves. We also prove that among all curves joining two given points, a curve minimizes weighted arc-length if and only if its [Formula: see text]-curvature is 0. Another proof is also presented for the fact “the plane with density [Formula: see text] contains no isoperimetric region”.

In this paper, initial and final value Abelian theorems for fractional wavelet transform of function and tempered distributions are obtained. Using Mexican hat wavelet function, an application for Abelian theorems is investigated.

In this paper, we define a sub-class of analytic bi-univalent functions associated with a certain differential operator [Formula: see text]. Bounds for the general Taylor–Maclaurin coefficients [Formula: see text] for the functions in this class are obtained. Estimates for the coefficient [Formula: see text] and the estimate for the functional [Formula: see text] for any real [Formula: see text], are also found. Results for the specific values of the parameters [Formula: see text], are also given mentioning some of the results obtained earlier.

The paper develops some crucial steps in extending the first-order cone or edge calculus to higher singularity orders. We focus here on order 2, but the ideas are motivated by an iterative approach for higher singularities.

In this paper, we are concerned with the zero Dirichlet boundary value problem associated to the quasilinear elliptic equation [Formula: see text] where [Formula: see text] is an open and bounded set in [Formula: see text] ([Formula: see text]), [Formula: see text] is a continuously differentiable real function in [Formula: see text], [Formula: see text] is an elliptic, bounded and symmetric matrix, [Formula: see text] is non-negative and may be singular at zero and [Formula: see text]. We give sufficient conditions on [Formula: see text], [Formula: see text] and [Formula: see text] in order to have a comparison principle and, as a consequence, uniqueness of positive solutions being continuous up to the boundary.

In this paper, we consider translating solitons in [Formula: see text] which is foliated by spheres. In three-dimensional Euclidean space, we show that such a translating soliton is a surface of revolution and the axis of revolution is parallel to the translating direction of the translating soliton. We also show that the same result holds for a higher dimension case with a hypersurface foliated by spheres in parallel hyperplanes that are perpendicular to the translating direction.

Let [Formula: see text] be a symplectic diffeomorphism on a closed [Formula: see text][Formula: see text]-dimensional Riemannian manifold [Formula: see text]. In this paper, we show that [Formula: see text] is Anosov if any of the following statements holds: [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the limit shadowing property or [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the limit weak shadowing property or [Formula: see text] belongs to the [Formula: see text]-interior of the set of symplectic diffeomorphisms satisfying the s-limit shadowing property.