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The paper treats a nonlinear eigenvalue problem that describes propagation of a transverse-magnetic wave in a plane dielectric waveguide having perfectly conducted walls at both sides. The dielectric's permittivity is characterised by an arbitrary monotonically increasing self-action nonlinearity. The full set of guided modes is described by eigenvalues of the corresponding (nonlinear) Maxwell operator with appropriate boundary conditions. We give a comprehensive analysis of this problem and develop an original approach to study its solvability and properties of solutions. Several results about existence of the eigenvalues are proved, their distribution and asymptotic are found; zeros of the eigenfunctions and their location are also determined; criterion of periodicity for the eigenfunctions is found, comparison theorem is derived, etc. It is shown that unbounded nonlinearities lead to appearance of nonlinearised solutions. This results in the existence of a novel guided regime that cannot be described within the framework of perturbation of linear guided wave regimes.

In this paper, we study the asymptotic behavior of the regular solution to a simplified Ericksen–Leslie model for the compressible nematic liquid crystal flow in a bounded smooth domain in R 2 as the Mach number tends to zero. The evolution system consists of the compressible Navier–Stokes equations coupled with the transported heat flow for the averaged molecular orientation. We suppose that the Navier–Stokes equations are characterized by a Navier's slip boundary condition, while the transported heat flow is subject to Neumann boundary condition. By deriving a differential inequality with certain decay property, the low Mach limit of the solutions is verified for all time, provided that the initial data are well-prepared.

The weak and strong ergodic properties of nonautonomous linear ordinary differential equations are considered. It is shown that if the coefficient matrix function is bounded, essentially nonnegative and uniformly irreducible, then the normalized positive solutions are asymptotically equivalent to the Perron vectors of the strongly positive transition matrix at infinity (weak ergodicity). If, in addition, the coefficient matrix function is uniformly continuous, then the convergence of the normalized positive solutions to the same strongly positive limiting vector (strong ergodicity) is equivalent to the convergence of the Perron vectors of the coefficient matrices.

In this article, we have developed a new method regarding asymptotic behavior and stability for the solutions of first order linear impulsive neutral delay differential equations with constant coefficients and constant delays. The asymptotic behavior of the solutions and the stability of the zero solution were investigated using a real root suitable for the characteristic equation. Furthermore, we obtained similar results for impulsive delay differential equations. Finally, three examples are given for the stability of the zero solution of the impulsive neutral delay differential equation.

This paper discusses how the oscillation of solutions changes by adding an impulsive term to a second-order linear differential equation having only a restoring force term but no damping term. Due to the effect of this impulsive term, the moving speed of a mass point of the equation of motion described by the second-order linear differential equation changes discontinuously. Oscillation theorems are given for this impulsive differential equation. As is well known, when the restoring force term is small, all nontrivial solutions of the equation with no impulsive effects do not oscillate. Even when the restoring force term is small, if the action of the impulsive term compensates, the oscillation criteria obtained are satisfied and all nontrivial solutions oscillate. Several examples are given to confirm this fact and some figures are shown to depict the solution curves of these examples. As shown by the obtained oscillation theorems, there are cases that the impulsive effect promotes the oscillation of solutions, but on the contrary, it may suppress the oscillation of solutions. It is shown by a simulation that there is a case where the impulsive effect suppress the oscillation of solutions. Finally, the obtained results are compared with the previous ones.

We consider the Cauchy problem associated to the recently derived higher order hamiltonian model for unidirectional water waves and prove global existence for given data in the Sobolev space $H^s$, $s\geq 1$. We also prove an ill-posedness result by showing that the flow-map is not continuous if the given data has Sobolev regularity $s< 1$. The results obtained in this work are sharp., 18 pages. To appear in JMAA - 2019

In this paper we study a free boundary problem modeling the growth of nonnecrotic tumors with angiogenesis. This model differs from the other tumor models studied in existing literatures at the point that it has a nonlinear boundary value condition for the nutrient concentration. We first study spherically symmetric version of this model. We prove that there exists a unique spherically symmetric stationary solution which is asymptotically stable under spherically symmetric perturbation. Next we make rigorous analysis to the spherically asymmetric version of this model. By using some abstract theory of parabolic differential equations in Banach manifold, we prove that this free boundary problem is locally well-posed in little Holder spaces and the radial stationary solution is asymptotically stable in case the surface tension coefficient γ is larger than a threshold value, whereas unstable in case γ is less than this threshold value.

This paper is concerned with the existence of pullback exponential attractors for the viscous Cahn–Hilliard–Navier–Stokes system with dynamic boundary conditions. Due to the lack of the Holder continuity in time of the semigroup and the smoothing property of the difference of two weak solutions, we construct a pullback exponential attractor in the original phase space by the l-trajectories methods.

In this paper, we mainly investigate global solutions and blow-up phenomena for the Cauchy problem of a generalized Degasperis–Procesi equation. We develop a new approach to consider the equation for ( 1 − ∂ x ) u instead of ( 1 − ∂ x 2 ) u . We first prove a new conserved quantity ‖ ( 1 − ∂ x ) u ‖ L 1 and give the local characterization of this conserved quantity. These properties then allow us to improve considerably a previous global existence result in [38] , and establish two new blow-up results for strong solutions to the equation. In the end, based on vanishing viscosity method and using the conserved quantity we deduce the existence of global weak solutions to the equation for any initial data belonging to L 1 ∩ B V .

A two-dimensional doubly-periodic, three-phase hexagonal structure is considered. The flow in the structure is generated by three sets of vortexes/sinks/sources, which are the same in each phase and are located in the centers of the hexagons. Complex analysis methods are utilized to reduce the doubly periodic R-linear conjugation problem to the simpler one, Riemann-Hilbert (RH) problem, on a three-sheeted Riemann surface. In turn, the latter problem is reduced to a RH problem involving three joined sectors on the plane, which was previously investigated in [3] . The limiting cases with one non-conducting phase and two phases of the same conductivities are investigated. All solutions derived are verified both numerically and analytically. Examples of relevant flow networks, streamlines and equipotentials, are plotted in the whole structure and separately in each phase.

For regime-switching diffusions processes with singular drifts, we introduce integrability conditions involving a nice reference probability measure and the $Q$-matrix of the jump part to study the existence of the invariant probability measures. Consequently, the generator of the regime-switching diffusions process has an extension in the $L^1$-space w.r.t the invariant probability measure to be a generator of $C_0$-semigroup. Moreover, we prove the uniqueness of the extension. Regularities and the uniqueness of the invariant probability density w.r.t the nice reference probability measure are also considered., 30 pages

In this paper we establish some results concerning existence and multiplicity of solution for a elliptic problem under integral Dirichlet boundary condition. We mainly use a lower and upper solution method and a result due to Rabinowitz [11] related to the existence of a continuum of solutions of an abstract nonlinear eigenvalue problem.

This paper is devoted to an investigation of blow-up phenomena occurring in high-contrast fiber-reinforced composites. When the distance between perfect conductors or between the conductors and the matrix boundary tends to zero, the electric field may appear blow-up. The major objective of this paper is to give a precise description for the singular behavior of such a high concentration in the presence of C 1 , α -inclusions with extreme conductivities. Our results contain the boundary and interior asymptotics of the concentrated field in all dimensions. In particular, the blow-up factor for each dimension is accurately captured.

We study the behavior of the longitudinal flow along a periodic array of cylinders upon perturbations of the shape of the cross section of the cylinders and the periodicity structure, when a Newtonian fluid is flowing at low Reynolds numbers around the cylinders. The periodicity cell is a rectangle of sides of length l and 1 / l , where l is a positive parameter, and the shape of the cross section of the cylinders is determined by the image of a fixed domain through a diffeomorphism ϕ. We also assume that the pressure gradient is parallel to the cylinders. Under such assumptions, for each pair ( l , ϕ ) , one defines the average of the longitudinal component of the flow velocity Σ [ l , ϕ ] . Here, we prove that the quantity Σ [ l , ϕ ] depends analytically on the pair ( l , ϕ ) , which we consider as a point in a suitable Banach space.

Whether or not weak solutions of classical Navier-Stokes equations decay to zero in L 2 as time tends to infinity was posed by Leray in his pioneering paper [12] , [13] and has been well solved (see [15] for instance). This paper examines the three dimensional Navier-Stokes equations with damping term | u | β − 1 u and proves that the weak solutions decay to zero in L 2 as time tends to infinity for any β ≥ 1 , by developing local-in-space estimate for weak solutions.

The algebraic decay for the total kinetic energy of weak solutions of the viscous Boussinesq system in an exterior domain is given by means of the spectral analysis method of the Stokes operator, which gives a positive answer to the L 2 -decay question in [11] .

In this work, we study the Cauchy problem for Fokker-Planck equation with the external potential. We show that the solution of the Cauchy problem enjoys Gelfand-Shilov S 1 1 ( R 3 ) regularizing property with respect to the velocity variable and Gevrey G 3 2 ( R 3 ) regularizing property with respect to the position variable.

In this paper, we investigate the critical points of solutions to the prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in a bounded smooth convex domain $\Omega$ of $\mathbb{R}^{n}(n\geq2)$. Firstly, we show the non-degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local Chen & Huang's comparison technique and the geometric properties of approximate surfaces at the non-degenerate critical points. Secondly, we deduce the uniqueness and non-degeneracy of the critical points of solutions in a rotationally symmetric domain of $\mathbb{R}^{n}(n\geq3)$ by the projection of higher dimensional space onto two dimensional plane., Comment: 15pages, 4figures

A novel infection-age structured eco-epidemiological model with saturation incidence is constructed. The model is regarded as an abstract non-densely defined Cauchy problem, and the condition of the existence of the interior equilibrium is derived. By employing the method of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain, we obtain that the model undergoes a Hopf bifurcation around the interior equilibrium which manifests that this model has a non-trivial periodic orbit that bifurcates from the interior equilibrium when bifurcation parameter τ crosses the bifurcation critical value τ 0 . In other words, the sustained periodic oscillation phenomenon appears. To support and extend our theoretically analytic results, numerical simulations are performed.

In this paper, we derive two Hamilton's gradient estimates for positive solutions of the nonlinear partial differential equation ∂ t u = Δ t u p + h u q under the Yamabe flow.

We use reflections involving analytic Dirichlet and Neumann data on a real-analytic curve in order to find a representation of solutions to Cauchy problems for harmonic functions in the plane. We apply this representation for finding solutions to Hele-Shaw problems. We also generalize the results by deriving the corresponding formulae for the Helmholtz equation and applying them to a uniform elliptic growth.

The aim of this paper is to study the regularity of the solution of some linear parabolic-elliptic problems in which parabolicity region depends on time. More specifically, this region is the position occupied by a body undergoing a motion (a deformation smoothly evolving in time). The main tool we introduce is a suitable extension of the motion to the entire spatial domain of the PDE. This enables us to reduce the original problem to a parabolic-elliptic problem with variable coefficients and with a parabolicity region independent of time. This problem can be seen as a Lagrangian formulation of our original problem. Next, we obtain regularity results for a class of parabolic-elliptic problems with variable coefficients and fixed parabolicity region. We apply these results to the Lagrangian formulation and, finally, we obtain a regularity result for our original problem.

We prove the Holder continuity of the unique solution to quaternionic Monge-Ampere equation with densities in L p , p > 2 , on a bounded strictly pseudoconvex domains.

In this paper, for the two-dimensional (2D) steady compressible magnetohydrodynamic equations, we construct a global piecewise smooth solution for a supersonic flow past a wedge with a large bended boundary. The uniform bounds are obtained by using the method of characteristic decompositions. Thus, we present a set of the characteristic decompositions for 2D steady magnetohydrodynamic equations. Then, under the assumption that the incoming flow has a limit speed, we could construct a degenerate solution and treat it as the background solution to obtain the piecewise smooth solution.

We address semigroup well-posedness for a linear, compressible viscous fluid interacting at its boundary with an elastic plate. We derive the model by linearizing the compressible Navier-Stokes equations about an arbitrary flow state, so the fluid PDE includes an ambient flow profile U. In contrast to model in [6] , we track the effect of this term at the flow-structure interface, yielding a velocity matching condition involving the material derivative of the structure; this destroys the dissipative nature of the coupling of the dynamics. We adopt here a Lumer-Phillips approach, with a view of associating fluid-structure solutions with a C 0 -semigroup { e A t } t ≥ 0 on a chosen finite energy space of data. Given this approach, the challenge becomes establishing the maximal dissipativity of an operator A , yielding the flow-structure dynamics.

The purpose of this paper is to study an inhomogeneous Dirichlet problem of a superquadratic nonlinear elliptic system with VMO coefficients in Reifenberg flat domains. We derive pointwise potential estimates for weak solution to the Dirichlet problem. As a consequence, we derive global integrability estimates for solutions with respect to Lorentz and Morrey norms.

In this paper, taking the Lax pairs of KdV equation as an illustrative example, with the help of a transformation of the spectral problem, another form of spectral function is used to give finite dimensional integrable systems. These integrable systems are generated by two different kinds of polynomial expansion of spectral function on spectral parameter. Further, the root variables of the spectral function are introduced to study the canonical equations of Hamilton. Finally, based on the Hamilton-Jacobi theory, the action-angle variables are built and the soliton solutions of KdV equation are obtained by inversion.

In this paper, we are concerned about the existence of solution to linear non-cutoff Boltzmann equation with boundary conditions. In [9] and [18] , the existence of the solutions to the kinetic equation and linear Boltzmann equation were obtained by characteristic line and iteration methods. However, their methods are not applicable in the context of non-cutoff linear operators. To overcome this difficulty, we cut off the non-cutoff linear operator L to L R , δ , the latter is compatible with the method in [9] . Via this process, we are able to use the sequence of solutions associated to L R , δ to approximate the solution associated to L, which is corresponding to the original equation, in turn, obtain the weak solutions of linear non-cutoff Boltzmann equation with specular and diffusion reflection boundary conditions.

This paper deals with the Eigenvalue estimates for Laplace operators with anti-Kirchhoff conditions on compact trees. The results include lower estimates for λ k ( Γ ) involving the edge lengths or diameter of Γ when Γ is a star, and upper estimates for λ k ( Γ ) involving the average edge length or the diameter of Γ when Γ is a general tree. We get the upper eigenvalue estimates by proving a monotonicity principle for the eigenvalues with respect to the domain (graph). Two geometric versions of the Ambartsumian theorem for − Δ Γ are also proven.

In this paper we deal with a singular Hamiltonian system of odd-order with several spectral parameters and we investigate the behavior of the solution of this system at singular point with the aid of the characteristic function theory. Moreover, some results have been introduced for the Weyl-Titchmarsh function for some special Hamiltonian systems of odd-order with several spectral parameters.

We study the linear stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere S 2 . The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution in time/spatial variable is derived.

It is well-known that the Rayleigh–Taylor (RT) problem of an inhomogeneous viscoelastic fluid defined on a bounded domain is unstable, if the elasticity coefficient κ is less than some threshold κ C . In this paper, we rigorously prove the existence of a unique unstable strong solution in the sense of L 1 -norm for the RT problem in Lagrangian coordinates based on a bootstrap instability method, when κ κ C . Applying an inverse transformation of Lagrangian coordinates to the obtained unstable solution, we can further get a unique unstable solution for the RT problem of inhomogeneous viscoelastic fluids in Eulerian coordinates.

In this paper, the magneto-heating coupling model is studied in details, with turbulent convection zone and the flow field involved. Our main work is to analyze the well-posed property of this model with the regularity techniques. We present the weak formulation of the coupled magneto-heating model and establish the regularity problem. Using Rothe's method, monotone theories of nonlinear operator, weak convergence theories, we prove that the limits of the solutions from Rothe's method converge to the solutions of the regularity problem with proper initial data. With the help of the spacial regularity technique, we derive the well-posedness of the original problems when the regular parameter ϵ ⟶ 0 . Moreover, with additional regularity assumption for both the magnetic field and temperature variable, we prove the uniqueness of the solutions.

This article deals with a semilinear hyperbolic equation with damping and conical singularity that was built by Alimohammady et al. (2017) [1] , where the weak solution with low initial energy ( I ( 0 ) d , d is the potential well depth) was considered. We extend the previous results in the following three aspects. Firstly, we consider the vacuum isolating behavior of the solution for initial energy I ( 0 ) ≤ 0 and 0 I ( 0 ) d , respectively. We find that there are two explicit vacuum regions: an annulus and a ball. Moreover, we obtain the asymptotic behavior of the energy functional as t tends to the maximal existence time, and then two necessary and sufficient conditions for the weak solution existing globally and blowing up in finite time are obtained. Secondly, we discuss the weak solution with critical initial energy and establish the global existence and nonexistence results. Finally, the weak solution with arbitrary positive initial energy is studied. In this case, the initial conditions such that the weak solution exists globally and blows up in finite time are given.

This paper is concerned with the fractional coupled Schrodinger system. By using the Nehari manifold and fibering map, we obtain the multiplicity and concentration of solutions for the given problem with steep potential wells, where some new estimates will be established. In particular, although there exist concave-convex nonlinearities in the coupled system, it is not necessary to assume that the corresponding Lebesgue norms of the weight functions of the convex terms need to be small enough.

This paper is concerned with the global stability of non-monotone traveling wave solutions to a nonlocal dispersion equation with time delay. It is proved that, all noncritical traveling wave solutions are globally stable with the exponential convergence rate t − 1 / α e − μ t for some constants μ > 0 and α ∈ ( 0 , 2 ] , and the critical traveling wave solutions are globally stable in the algebraic form t − 1 / α , where the initial perturbations around the monotone/non-monotone traveling wave solution in a weighted Sobolev space can be arbitrarily large. The adopted approach is the anti-weighted energy method combining with Fourier's transform.

In this paper, we are concerned with optimal temporal decay rates of solutions for the initial value problem of three-dimensional incompressible micropolar fluid system. By carefully examine the algebraic structure of micropolar fluid system, we show that some weighted negative Fourier–Besov norms of solutions are preserved along time evolution, which yields the optimal temporal decay rates of solutions in the framework of homogeneous Fourier–Besov spaces by solving an ordinary differential inequality through the interpolation techniques.

In this paper, quantitative upper estimates for the number of eigenvalues lying below the essential spectrum of Schrodinger operators with potentials generated by Ahlfors regular measures in a strip subject to two different types of boundary conditions (Robin and Dirichlet respectively) are presented. The estimates are presented in terms of weighted L 1 norms and Orlicz norms of the potential.

It was proved that the solution to a time-fractional partial differential equation lacks regularity at the initial time t = 0 , which is often inconsistent with the physical problem being modeled even though the nature of the physical problem away from t = 0 is well captured by a time-fractional partial differential equation. We argue that the fundamental reason why this phenomenon occurs lies in the incompatibility between the nonlocality of the time-fractional partial differential equations and the locality of the classical initial condition at t = 0 . We show that the variable-order time-fractional partial differential equations, in which the order varies in time as t goes to 0 to accommodate the impact of the local initial condition at time t = 0 , turn out to be a natural candidate to eliminate the nonphysical singularity of the solutions to the (constant-order) time-fractional partial differential equations and open up opportunities for modeling multiphysics phenomena from nonlocal to local dynamics and vice versa. We prove the wellposedness of a variable-order linear time-fractional diffusion partial differential equation in multiple space dimensions. We also prove that the regularity of its solution depends on the behavior of the variable order (and its derivatives) at time t = 0 , in addition to the usual smoothness assumptions. More precisely, we prove that its solution has full regularity like its integer-order analogue if the variable order has an integer limit at t = 0 or exhibits singular behaviors at t = 0 like in the case of the constant-order time-fractional partial differential equations if the variable order has a non-integer value at time t = 0 .

This paper is devoted to study the formation of stationary patterns for a chemotaxis model with nonlinear diffusion and volume-filling effect over a bounded rectangular domain. By using linear stability analysis around the homogeneous steady states we establish conditions for the existence of unstable mode bands that lead to the formation of spatial patterns. We derive the Stuart-Landau equations for the pattern amplitudes by means of weakly nonlinear multiple scales analysis and Fredholm theory. In particular, we find asymptotic expressions for a wide range of patterns sustained by the system. These patterns include mixed-mode, square, hexagonal, and roll stationary configurations. Our analytical results are corroborated by direct simulations of the underlying chemotaxis system.

The solutions of the stationary Navier-Stokes equations in R n for n ≥ 3 in the scaling invariant Besov spaces are investigated. It is proved that a sequence of bounded smooth external forces whose B ˙ ∞ , 1 − 3 norms converges to zero can produce a sequence of bounded smooth solutions whose B ˙ − 1 ∞ , ∞ norms never converges to zero. Such norm inflation phenomena are shown by constructing the sequence of external forces, as similar to those of initial data proposed by Bourgain-Pavlovic in the non-stationary problem.

In this paper, we establish the global regularity for the two-dimensional incompressible Leray-alpha-MHD equations with only magnetic diffusion. More precisely, when the diffusive term for the magnetic field is a half Laplacian, the solution initiated from data sufficiently smooth preserves its regularity.

This paper is concerned with the Cauchy problem of a generalized short pulse system in the periodic setting. Local well-posedness of this system is established in the Besov space with s > max ⁡ { 3 2 , 1 + 1 p } . It is then shown that the data-to-solution map is not uniformly continuous from B l , r s × B l , r s to C ( [ 0 , T ] ; B l , r s ) × C ( [ 0 , T ] ; B l , r s ) . Finally we provide a blow-up result of the solution.

We are concerned with the asymptotic behavior for a coupled system of two second order abstract evolution equations with one infinite memory and the damping effect given by only one equation, which describes the dynamics of many viscoelastic systems and covers the well-known Timoshenko system. We present an optimal energy decay estimate under very weak conditions. More specifically, we prove that the energy of the system decays to zero at least at the rate of t − 1 only under the condition that the related kernel functions are non-increasing and integrable. Moreover, we apply our abstract results to three classes of coupled systems of second-order partial differential equations arising in viscoelasticity.

In this paper, we consider a Cauchy problem of the 2D Leray-α regularized incompressible magneto-micropolar equations. The global smooth solution of the Cauchy problem for the equations with zero angular viscosity and zero magnetic diffusion or with only angular viscosity and magnetic diffusion are established.

In this paper, we study the diffusive mutualist model with advection and different free boundaries in one space dimension. These two free boundaries may intersect each other as time evolves and can be used to describe the spreading of invasive and native species directly. We first prove the existence and uniqueness, regularity and uniform estimates of global solution. Then we provide the criteria governing spreading and vanishing. At last we investigate long time behaviors and asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries.

In this short note, the author establishes a blow-up result for a semilinear heat equation with logarithmic nonlinearity, by using the logarithmic Sobolev inequality. This improves a recent blow-up result obtained in Chen et al. (2015) [1] .

In this work, we investigate the hepatitis B virus infection. We first derive a nonlinear PDE model for the studied biological phenomenon. The obtained initial boundary value problem is completely analyzed. To begin with the analysis of the model, we use Lou and Zhao Lemma concerning globally attractive steady states to prove boundedness of potential solutions. Then we prove global existence, uniqueness and positivity of the solution by a variational method combined with semigroups theory and some other useful tools from functional analysis. Moreover, the basic reproduction number R 0 determining the extinction or the persistence of the HBV infection is thoroughly computed via a new method based on spectral properties of differential operators, the residue Theorem and some arguments from numerical analysis. Also, the global asymptotical properties of the HBV-free equilibrium of the model are derived via a skillful construction of a suitable Lyapunov function. Local stability of the endemic equilibrium of the model is studied as well. Finally, numerical simulations are performed to support the theoretical results obtained.

This paper is concerned with the saddle-fold singularity that is a codimension two singularity for piecewise smooth planar differential systems. The approach consists in apply the Regularization's Method to the unfolding of the saddle-fold singularity in order to obtain a 2-parameter family of smooth vector fields. Our main result says that the regularized family undergoes thought a classical Bogdanov–Takens bifurcation.

This paper is concerned with the nonlinear stability of monostable traveling wavefronts for a discrete three species competition diffusion system. Applying weighted energy method combining with the comparison principle, we first show that the traveling wavefronts with large speed are exponentially stable when the initial perturbation around the traveling wave decays exponentially as x → − ∞ (but the initial perturbation can be arbitrarily large in other locations). Then, choosing different weight function, we improve the stability result to any traveling wavefronts with speed greater than the critical wave speed. We have to emphasize that the discrete dispersal operator in the system increases the difficulty of establishing energy estimates.