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We study an inhomogeneous minimization problems associated to the $p(x)$-Laplacian. We make a thorough analysis of the essential properties of their minimizers and we establish a relationship with a suitable free boundary problem. On the one hand, we study the problem of minimizing the functional $J(v)=\int_\Omega\Big(\frac{|\nabla v|^{p(x)}}{p(x)}+\lambda(x)\chi_{\{v>0\}}+fv\Big)\,dx$. We show that nonnegative local minimizers $u$ are solutions to the free boundary problem: $u\ge 0$ and \begin{equation} \label{fbp-px}\tag{$P(f,p,{\lambda}^*)$} \begin{cases} \Delta_{p(x)}u:=\mbox{div}(|\nabla u(x)|^{p(x)-2}\nabla u)= f & \mbox{in }\{u>0\}\\ u=0,\ |\nabla u| = \lambda^*(x) & \mbox{on }\partial\{u>0\} \end{cases} \end{equation} with $\lambda^*(x)=\Big(\frac{p(x)}{p(x)-1}\,\lambda(x)\Big)^{1/p(x)}$ and that the free boundary is a $C^{1,\alpha}$ surface. On the other hand, we study the problem of minimizing the functional $J_{\varepsilon}(v)= \int_\Omega \Big(\frac{|\nabla v|^{p_\varepsilon(x)}}{p_\varepsilon(x)}+B_{\varepsilon}(v)+f_\varepsilon v\Big)\, dx$, where $B_\varepsilon(s)=\int _0^s\beta_\varepsilon(\tau) \, d\tau$, $\varepsilon>0$, ${\beta}_{\varepsilon}(s)={1 \over \varepsilon} \beta({s \over \varepsilon})$, with $\beta$ a Lipschitz function satisfying $\beta>0$ in $(0,1)$, $\beta\equiv 0$ outside $(0,1)$. We prove that if $u_\varepsilon$ are nonnegative local minimizers, then any limit function $u$ ($\varepsilon\to 0$) is a solution to the free boundary problem $P(f,p,{\lambda}^*)$ with $\lambda^*(x)=\Big(\frac{p(x)}{p(x)-1}\,M\Big)^{1/p(x)}$, $M=\int \beta(s)\, ds$, $p=\lim p_\varepsilon$, $f=\lim f_\varepsilon$, and that the free boundary is a $C^{1,\alpha}$ surface. In order to obtain our results we need to overcome deep technical difficulties and develop new strategies, not present in the previous literature for this type of problems.

We study the question of when two weighted variable exponent Bergman spaces or Hardy spaces are equivalent. As an application, we show that variable exponent Hardy spaces have a close relation to classical Hardy spaces when the exponent is log-Holder continuous and has bounded harmonic conjugate (when extended from its boundary values to be harmonic in the disc). We use this to characterize Carleson measures for these variable exponent Hardy spaces. We also prove under certain conditions an analogue of Littlewood subordination and a result on the boundedness of composition operators.

In this paper we establish sufficient conditions for an asymptotically linear elliptic boundary value problem to have at least five solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues. We make extensive use of variational methods and characterizations of the local degree of critical points. Furthermore, uniqueness and qualitative properties of the solutions are investigated.

We consider the space H ( c e s p ) of all Dirichlet series whose coefficients belong to the Cesaro sequence space c e s p , consisting of all complex sequences whose absolute Cesaro means are in l p , for 1 p ∞ . It is a Banach space of analytic functions, for which we study the maximal domain of analyticity and the boundedness of point evaluations. We identify the algebra of analytic multipliers on H ( c e s p ) as the Wiener algebra of Dirichlet series shifted to the vertical half-plane C 1 / q : = { s ∈ C : ℜ s > 1 / q } , where 1 / p + 1 / q = 1 .

In this paper, we study a reaction-diffusion system with prey taxis which models the dynamics of a two-predator-one-prey ecosystem in which the predators collaboratively take advantage of the prey's strategy. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and small prey-taxis sensitivity coefficients are proved. It is also shown that such prey-taxis qualitatively affect the stability of coexistence steady state solutions in some cases, and the emergence of forceful prey-taxis will enhance the spatial patterns.

We study the Schrodinger systems with linear and nonlinear coupling terms (doubly coupled nonlinear Schrodinger system for short) which arise naturally in nonlinear optics, and in the Hartree–Fock theory for Bose–Einstein condensates, among other physical problems. First, for small linear coupling constant, we get existence of a nontrivial bound state solution to the system via perturbation method, furthermore, we prove each component of the bound state solution is nonnegative by energy estimate. Second, we establish a version of Pohozaev–Nehari identity and prove a nonexistence result for the more general system when the spatial dimension N ≥ 4 .

Contiguous relations are a powerful tool for dealing with hypergeometric functions. In this paper, four different types of contiguous relations for the Fox–Wright function are given. We demonstrate the usefulness of our formulas by showing how they can be used to recover the famous Hagen–Rothe convolution.

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.

We deduce a product formula for the Whittaker $W$ function whose kernel does not depend on the second parameter. Making use of this formula, we define the positivity-preserving convolution operator associated with the index Whittaker transform, which is seen to be a direct generalization of the Kontorovich-Lebedev convolution. The mapping properties of this convolution operator are investigated; in particular, a Banach algebra property is established and then applied to yield an analogue of the Wiener-L\'evy theorem for the index Whittaker transform. We show how our results can be used to prove the existence of a unique solution for a class of convolution-type integral equations., Comment: 24 pages. Accepted for publication in Journal of Mathematical Analysis and Applications

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.

Let φ : R n × [ 0 , ∞ ) → [ 0 , ∞ ) be an anisotropic growth function and A a general expansive matrix on R n . Let H A φ ( R n ) be the anisotropic Musielak–Orlicz Hardy space associated with A defined via the non-tangential grand maximal function. In this article, the authors establish a new molecular characterization of H A φ ( R n ) via its known atomic characterizations, which positively answer a question mentioned by Li et al. (2016) [34] . As an application, the authors obtain a criterion on the boundedness of linear operators on H A φ ( R n ) , which further implies the boundedness of integral anisotropic Calderon–Zygmund operators on H A φ ( R n ) . In addition, the boundedness of these operators from H A φ ( R n ) to the Musielak–Orlicz space L φ ( R n ) is also presented.

Let H be a Hilbert space. Suppose A is a positive self-adjoint operator on H and φ ∈ H is a cyclic unit vector. For each λ ∈ R , we can define the rank one perturbation of A by A λ = A + λ 〈 φ , ⋅ 〉 φ . To each A λ we can consider the spectral measure of φ, which we denote by μ λ . This generates a family of measures, { μ λ } , and we analyze the packing dimension of this family. Past results have determined that the Hausdorff dimension of this family can be determined if the limit inferior of a ratio involving μ is constant on a Lebesgue typical set. This ratio is sometimes called the pointwise dimension of μ and is related to the upper derivative of μ. Work has been done to make a similar argument for the packing dimension, but with little success. Using the theory of rank one perturbations and Borel transforms, we introduce the concept of Lebesgue exact dimension for μ, which allows us to determine the packing dimension of spectral measures of almost every rank one perturbation μ λ . If the Lebesgue exact dimension for μ is 1 α 2 then the packing dimension of Lebesgue almost every μ λ is 2 − α . As a corollary, we find that this limit condition implies a stronger result: the Hausdorff and packing dimensions are equal for almost every μ λ .

In this paper, we investigate the relation between Sobolev-type embeddings of Hajlasz-Besov spaces (and also Hajlasz-Triebel-Lizorkin spaces) defined on a metric measure space ( X , d , μ ) and lower bound for the measure μ. We prove that if the measure μ satisfies μ ( B ( x , r ) ) ≥ c r Q for some Q > 0 and for any ball B ( x , r ) ⊂ X , then the Sobolev-type embeddings hold on balls for both these spaces. On the other hand, if the Sobolev-type embeddings hold in a domain Ω ⊂ X , then we prove that the domain Ω satisfies the so-called measure density condition, i.e., μ ( B ( x , r ) ∩ Ω ) ≥ c r Q holds for any ball B ( x , r ) ⊂ X , where X = ( X , d , μ ) is an Ahlfors Q-regular and geodesic metric measure space.

We present a new sufficient condition for existence of semiattractors for set-valued semiflows of state multifunctions associated with general cocycle mappings. We assume that a given cocycle has at least one weakly pullback contracting fiber. In particular, one can apply the result to iterated function systems having no contracting mappings. We also introduce and analyze a fiber structure of semiattractors.

In this paper, we continue the study of the properties of the generalised projection on nonconvex sets in reflexive smooth Banach spaces with smooth dual norm started by the first author in [9] . We define a new class of nonconvex sets in Banach spaces containing the convex closed sets and having various important properties whenever the space is assumed to be uniformly convex with smooth norm.

In this paper, the topological entropy dimension for nonautonomous dynamical systems is studied. This entropy dimension measures the complexity of the systems with zero entropy via the subexponential growth rate of orbits. Some basic properties of this topological entropy dimension are developed. Moreover, we construct concrete nonautonomous dynamical systems with zero entropy, such that their topological entropy dimension can attain any real number between 0 and 1.

This paper concerns the nonlinear stability properties of discontinuous Galerkin (DG) method for stiff Volterra functional differential equations (VFDEs). We derive that the DG method leads to global and analogously asymptotical stability for VFDEs, and it is shown that the perturbations of the numerical solution are controlled by the initial perturbations. This general results provide unified theoretical treatment for numerical stability analysis of Volterra integro-differential equations (VIDEs) with constant or variable delay, delay differential equations (DDEs), ordinary differential equations (ODEs) and so on. Numerical examples are given to confirm our theoretical results.

In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the p-Laplacian, in general, the regularity of solutions appears in terms of functional spaces with nonlinear order of smoothness. Moreover, despite its own interest, fractional regularity methods may be used as a tool for the investigation of some Partial Differential Equations which are not usually addressed in this manner. Thus, the purpose of the present paper is to exploit such methods in order to provide some results regarding existence and regularity of solutions to a class nonlocal elliptic equations which are linked to the p-Laplacian. This is done by means of explicit a priori estimates regarding Lebesgue and Nikolskii spaces, which are part of the present contribution. As a consequence, this approach allows a relaxation on some of the standard conditions employed in this class of problems.

We consider the Cauchy problem for the relativistic Vlasov-Poisson-Fokker-Planck system in the whole space. For perturbative initial data with suitable regularity, we obtain the global classical solutions and prove the exponential time decay rate to the equilibrium around a global relativistic Maxwellian.

For a function space $X(\OO)$ satisfying weak assumptions we prove that the generic function in $X(\OO)$ is totally unbounded, hence non-extendable. We provide several examples of such spaces; they are mainly localized versions of classical function spaces and intersections of them., 17 pages

Liu, Lu and Wang (1991) introduced a method for determining the LaSalle's invariant sets of Lotka-Volterra prey-predator chain systems. The current paper modifies their method and combines it with resultant and Grobner basis techniques. For a class of 8-dimensional Lotka-Volterra prey-predator chain systems, the necessary and sufficient conditions for their global stability are obtained.

© 2019 Elsevier Inc. We consider the construction of insurance premiums that are monotonically increasing with respect to a loading parameter. By introducing weight functions that are totally positive of higher order, we derive higher monotonicity properties of generalized weighted premiums; in particular, we deduce for weight functions that are totally positive of order three a monotonicity property of the variance-to-mean ratio, or index of dispersion, of the loss variable. We derive the higher order total positivity properties of some ratios that arise in actuarial and insurance analysis of combined risks. Further, we examine seven classes of weight functions that have appeared in the literature and we ascertain the higher order total positivity properties of those functions.

In the paper, it is proved that a pair of analytic functions is simultaneously approximated by shifts ( ζ ( s + i γ k h ) , ζ ( s + i γ k h , α ) ) of the Riemann and Hurwitz zeta-functions with transcendental parameter α, where 0 γ 1 γ 2 ⋯ ⩽ γ n ⩽ ⋯ is a sequence of imaginary parts of non-trivial zeros of ζ ( s ) . For this, the weak Montgomery conjecture on pair correlation of zeros of ζ ( s ) is used.

In this paper, we study the following quasilinear Schrodinger equation of Choquard type − △ u + V ( x ) u − △ ( u 2 ) u = ( I α ⁎ | u | p ) | u | p − 2 u , x ∈ R N , where N ≥ 3 , 0 α N , 2 ( N + α ) N ≤ p 2 ( N + α ) N − 2 and I α is a Riesz potential. Under appropriate assumptions on V ( x ) , we establish the existence of positive solutions.

This note contains a short account of various mixing properties of coarse bimodules and its relations with associated completely positive maps.

In this paper, we establish Buser type inequalities, i.e., upper bounds for eigenvalues in terms of Cheeger constants. We prove the Buser's inequality for an infinite but locally finite connected graph with Ricci curvature lower bounds. Furthermore, we derive that the graph with positive curvature is finite, especially for unbounded Laplacians. By proving Poincare inequality, we obtain a lower bound for Cheeger constant in terms of positive curvature.

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.

In this paper, we study the one-dimensional nematicon equations which model the propagation of coherent and polarized light in nonlocal nematic liquid crystals. The existences of local and global solutions are derived first upon applying the Strichartz's estimates. Then the existence of ground state solitary wave solutions is proved by using the concentration-compactness technique and the critical point theory.

The fundamental role of slowly varying functions in the Zygmund sense is examined. We derive a necessary and sufficient condition for a positive measurable function to be slowly varying in the Zygmund sense. Analytical equivalences involving Zygmund SVF's in probabilistic setting are related to necessary and sufficient analytical conditions for classical limit theorems. In this context an analytical conjecture is presented. Finally, the role of Zygmund SVF's in recent generalizations of regular variation is presented as a unifying principle.

In this paper we are interested in the following nonlocal nonhomogeneous elliptic equation in R 2 , − Δ u + V ( x ) u = ( 1 | x | μ ⁎ F ( u ) | x | β ) f ( u ) | x | β + e h ( x ) in R 2 , where V is a positive continuous potential, 0 μ 2 , β ≥ 0 , 2 β + μ ≤ 2 , e is a small parameter and F ( s ) is the primitive function of f ( s ) . Suppose that the nonlinearity f ( s ) is of critical exponential growth in the sense of Trudinger-Moser inequality, we prove the existence and multiplicity of solutions by variational methods.

In this paper, we apply a logarithmic approximation introduced by Boussetila and Rebbani in order to study an ill-posed Cauchy problem in Banach space. The proposed method induces a well-posed problem whose approximation parameter yields continuous dependence on modeling. Throughout, properties of holomorphic semigroups are utilized culminating in the motivating application which is the backward heat equation in L p ( R ) , 1 p ∞ .

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.

This paper is devoted to study a general diffusive predator-prey system with strong Allee effect in the prey. A priori estimate and nonexistence of nonconstant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation. The Hopf bifurcation and steady-state bifurcation from the positive constant solution branch are investigated. These results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulations.

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.

The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. We extend the Laguerre calculus for nilpotent groups of step two, and test it in the determining of the fundamental solution of the sub-Laplace operator. We also apply it to find the Szego kernels of the projection operators to a kind of regular functions on the quaternion Heisenberg group.

We consider the question of exponential decay to equilibrium of solutions of an abstract class of degenerate evolution equations on a Hilbert space modeling the steady Boltzmann and other kinetic equations. Specifically, we provide conditions suitable for construction of a stable manifold in a particular “reverse L ∞ ” norm and examine when these do and do not hold.

We prove the existence and uniqueness of the viscosity solutions to the obstacle problems for second order differential partial equations under some conditions. As applications, we show some existence and uniqueness theorems of the viscosity solutions to the obstacle problems for k-Hessian equations, which include Monge-Ampere type equations when k = n .

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.

The main purpose of this article is to determine the existence of a unique selection of convex set-valued functions satisfying some generalized set-valued inclusions. From Radstrom's cancelation lemma and the main result of Piszczek [20] , we extend the same conclusions as in [19] , but under weaker assumptions. Finally, we present a few applications of set-valued dynamics.

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.

The Ragnisco–Tu equation with the Heaviside initial condition is studied in this paper through the inverse spectral transform method. The procedure based on the regularization is applied to construct the relation between potentials and dressing matrices. The explicit solutions are obtained and their dynamic behaviors are discussed.

We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self similar or homogeneous. The calculation is based on the local behavior of the natural probability measure supported on the sets.

In this paper, we study some differential inequalities on Finsler manifolds. Firstly, we prove a Liouville type theorem for the solution of a differential inequality. Secondly, we obtain a priori boundedness of solutions of the differential inequality and give some applications to problems in geometry and differential equations.

The authors present sharp Landen transformation inequalities for the hypergeometric functions F 1 2 ( a , b ; a + b ; x ) and F 1 2 ( a , b ; ( a + b + 1 ) / 2 ; x ) , by showing the monotonicity properties of certain combinations defined in terms of one of these two hypergeometric functions and linear (or rational) functions, thus giving complete solutions of the problem on extending the well-known Landen transformation identities for the complete elliptic integrals of the first kind to these two hypergeometric functions, and substantially improving the related known results. As applications of these results, sharp Landen transformation inequalities are obtained for the generalized Grotzsch ring functions and the modular functions, which appear in Ramanujan's modular equations. Some other properties of hypergeometric functions and several properties of the Ramanujan constant are obtained, too.

In this paper we characterize those exponents p ( ⋅ ) for which corresponding variable exponent Lebesgue space L p ( ⋅ ) ( [ 0 ; 1 ] ) has in common with L ∞ the property that the space of continuous functions is a closed linear subspace in it. In particular, we obtain necessary and sufficient condition on decreasing rearrangement of exponent p ( ⋅ ) for which exists equimeasurable exponent of p ( ⋅ ) which corresponding variable exponent Lebesgue space have the above mentioned property.

In a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element x we associate the eigenvalue vector λ ( x ) whose components are the eigenvalues of x written in the decreasing order. For any p ∈ [ 1 , ∞ ] , we define the spectral p-norm of x to be the p-norm of λ ( x ) in R n . In this paper, we show that ‖ x ∘ y ‖ 1 ≤ ‖ x ‖ p ‖ y ‖ q , where x ∘ y denotes the Jordan product of two elements x and y in V and q is the conjugate of p. For a linear transformation on V , we state and prove an interpolation theorem relative to these spectral norms. In addition, we compute/estimate the norms of Lyapunov transformations, quadratic representations, and positive transformations on V .

Let Γ n denote the n-th level Sierpinski graph of the Sierpinski gasket K. We consider, for any given conductance ( a 0 , b 0 , c 0 ) on Γ 0 , the Dirichlet form E on K obtained from a recursive construction of compatible sequence of conductances ( a n , b n , c n ) on Γ n , n ≥ 0 . We prove that there is a dichotomy situation: either a 0 = b 0 = c 0 and E is the standard Dirichlet form, or a 0 > b 0 = c 0 (or the two symmetric alternatives), and E is a non-self-similar Dirichlet form independent of a 0 , b 0 . The second situation has been studied in [13] , [9] as a one-dimensional asymptotic diffusion. The analytic approach here is more direct and yields sharper results; in particular, for the spectral property, we give a precise estimate of the eigenvalue distribution of the associated Laplacian, which improves a similar result in [9] .

In this paper, we introduce two perturbations in the classical deterministic susceptible–infected–susceptible epidemic model. Greenhalgh and Gray [4] in 2011 use a perturbation on β in SIS model. Based on their previous work, we consider another perturbation on the parameter μ + γ and formulate the original model as a stochastic differential equation (SDE) with two independent Brownian Motions for the number of infected population. We then prove that our Model has a unique and bounded global solution I ( t ) . Also we establish conditions for extinction and persistence of the infected population I ( t ) . Under the conditions of persistence, we show that there is a unique stationary distribution and derive its mean and variance. Computer simulations illustrate our results and provide evidence to back up our theory.

A Banach algebra A is said to be zero Lie product determined if every continuous bilinear functional φ : A × A → C satisfying φ ( a , b ) = 0 whenever a b = b a is of the form φ ( a , b ) = ω ( a b − b a ) for some ω ∈ A ⁎ . We prove that A has this property provided that any of the following three conditions holds: (i) A is a weakly amenable Banach algebra with property B and having a bounded approximate identity, (ii) every continuous cyclic Jordan derivation from A into A ⁎ is an inner derivation, (iii) A is the algebra of all n × n matrices, where n ≥ 2 , over a cyclically amenable Banach algebra with a bounded approximate identity.

In this paper, we aim to study quasi-sure exponential stabilization of stochastic systems induced by G-Brownian motion with discrete time feedback control. As for an unstable system, we use discrete stochastic feedback control to stabilize it. If stochastic differential equation driven G-Brownian motion (G-SDE, in short) is quasi-sure exponential stability, it reveals that there exist the discrete step size h > 0 and a positive constant h ˜ with h h ˜ , and the responding stochastically controlled system is also quasi-sure exponential stability. Finally, a simple example is also provided to show the validity of the control strategy.