On the asymptotic behaviour of linear combinations of Mellin‐Picard type operators

Here we give a Voronovskaja formula for linear combination of Mellin‐Picard type convolution operators where is the Mellin‐Picard kernel. This approach provides a better order of pointwise approximation.     

Nuclear embeddings in function spaces

Let be the usual Besov spaces in bounded Lipschitz domains Ω in (bounded intervals if ). The paper clarifies under which conditions the continuous embedding between two such spaces with is nuclear.     

Hyponormality of bounded‐type Toeplitz operators

In this paper we deal with the hyponormality of Toeplitz operators with matrix‐valued symbols. The aim of this paper is to provide a tractable criterion for the hyponormality of bounded‐type Toeplitz operators (i.e., the symbol is a matrix‐valued function such that Φ and are of bounded type). In particular, we get a much simpler criterion for the hyponormality of when the co‐analytic part of the symbol Φ is a left divisor of the analytic part.     

Boundedness in a fully parabolic chemotaxis‐consumption system with nonlinear diffusion and sensitivity,and logistic source

In this paper we study the zero‐flux chemotaxis‐system Ω being a convex smooth and bounded domain of , , and where , and . For any the chemotactic sensitivity function is assumed to behave as the prototype , with and . We prove that for nonnegative and sufficiently regular initial data and , the corresponding initial‐boundary value problem admits a unique globally bounded classical solution provided μ is large enough.     

The estimates of Bessel integrals with oscillatory factors

In this paper, we establish some sharp estimates of Bessel integrals with oscillatory factors . As an application, we obtain the boundedness of the oscillatory singular integral operators with variable kernels.     

Fractional Sobolev inequalities associated with singular problems

In this paper we study Sobolev‐type inequalities associated with singular problems for the fractional p‐Laplacian operator in a bounded domain of , .     

On the global regularity of the 2D Boussinesq equations with fractional dissipation

In this paper, we consider the two‐dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Attention is focused on the subcritical case when the velocity dissipation dominates. More precisely, we establish the global regularity result of the 2D Boussinesq equations in a new range of fractional powers of the Laplacian, namely with . Therefore, this result significantly improves the previous work 31 which obtained the global regularity result for with , where is an explicit function.     

The invertibility of convolution type operators on a union of intervals and the corona theorem

The invertibility of convolution type operators on unions of intervals is studied. Sufficient conditions of invertibility for some classes of these operators are established. Solvability results forn-term corona problems are obtained using two different approaches: one involving reduction ton–1 Riemann-Hilbert problems in two variables and another involving reduction to two-term corona problems. The invertibility of the convolution operators on a union of intervals is also related to the invertibility of associated convolution operators on single intervals. Formulas for the inverse operators are given.Work partially supported by F. C. T. (Portugal) Project Praxis XXI/2/2.1/MAT/441/94 and by CONACYT (México) grant, Cátedra Patrimonial, No. 990017-EX.     

On nonlinear perturbations of a periodic fractional Schrödinger equation with critical exponential growth

In this paper we study the existence of solutions for fractional Schrödinger equations of the form where V is a potential bounded and the nonlinear term has the critical exponential growth. We prove the existence of at least one weak solution by combining the mountain‐pass theorem with the Trudinger–Moser inequality and a version of a result due to Lions for critical growth in .     

Global existence of solutions for a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source

This paper deals with a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source under homogeneous Neumann boundary conditions in a smooth bounded domain . The functions χ and f are assumed to generalize the chemotactic sensitivity function and logistic source respectively. Under some conditions, we obtain that the corresponding initial‐boundary value problem possesses a unique global classical solution that is uniformly bounded.     

On the discretisation in time of the stochastic Allen–Cahn equation

We consider the stochastic Allen–Cahn equation perturbed by smooth additive Gaussian noise in a bounded spatial domain with smooth boundary in dimension , and study the semidiscretisation in time of the equation by an Euler type split‐step method with step size . We show that the method converges strongly with a rate . By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.     

Approximation of Schrödinger operators with δ‐interactions supported on hypersurfaces

We show that a Schrödinger operator with a δ‐interaction of strength α supported on a bounded or unbounded C²‐hypersurface , can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator with a singular interaction is regarded as a self‐adjoint realization of the formal differential expression , where is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.     

Applications of ‐invariant distributions

In this sequel to the article 17 , a criterion for the regularity of fundamental solutions of differential operators with positive symbol is proved in analogy to Hörmander's criteria, see 9 , 12 . It implies that the temperate fundamental solution of the operator is not regular. Furthermore, a formula for the convolution of two ‐invariant distributions is presented, and, finally, L. Schwartz' question on the surjectivity of linear partial differential operators with constant coefficients on the space is completely answered in the case of ‐invariant differential operators.     

Well‐posedness,blow‐up phenomena and analyticity for a two‐component higher order Camassa–Holm system

In this paper, the local well‐posedness for the Cauchy problem of a two‐component higher‐order Camassa–Holm system (2HOCH) is established in Besov spaces with and (and also in Sobolev spaces with ), which improves the corresponding results for higher‐order Camassa–Holm in 7 , 24 , 25 , where the Sobolev index is required, respectively. Then the precise blow‐up mechanism and global existence for the strong solutions of 2HOCH are determined in the lowest Sobolev space with . Finally, the Gevrey regularity and analyticity of the 2HOCH are presented.     

On the bifurcation results for fractional Laplace equations

In this paper, we consider the bifurcation problem for the fractional Laplace equation where is an open bounded subset with smooth boundary,  stands for the fractional Laplacian. We show that a continuum of solutions bifurcates out from the principal eigenvalue λ₁ of the problem and, conversely.     

Scattering theory below energy for the cubic fourth‐order Schrödinger equation

We investigate the global existence and scattering for the cubic fourth‐order Schrödinger equation in a low regularity space with . We provide an alternative approach to obtain a new interaction Morawetz estimate which extends the range of the dimension of the interaction Morawetz estimate in Pausader 29 . We utilize the interaction Morawetz estimates and the I‐method to prove the global well‐posed and scattering result.     

Necessary and sufficient conditions for existence of blow‐up solutions for elliptic problems in Orlicz–Sobolev spaces

This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem where either with is a smooth bounded domain or . The function ϕ includes special cases appearing in mathematical models in nonlinear elasticity, plasticity, generalized Newtonian fluids, and in quantum physics. The proofs are based on comparison principle, variational methods and topological arguments on the Orlicz–Sobolev spaces.     

On a generalized nonlinear heat equation in Besov and Triebel–Lizorkin spaces

The paper deals with solutions of the Cauchy problem of a nonlinear generalized heat equation in the context of Besov and Triebel–Lizorkin spaces and where and with initial data belonging to some spaces where and .     

Spectral gaps for the Dirichlet‐Laplacian in a 3‐D waveguide periodically perturbed by a family of concentrated masses

We consider a spectral problem for the Laplace operator in a periodic waveguide perturbed by a family of “heavy concentrated masses”; namely, Π contains small regions of high density, which are periodically distributed along the z axis. Each domain has a diameter and the density takes the value in and 1 outside; m and ε are positive parameters, , . Considering a Dirichlet boundary condition, we study the band‐gap structure of the essential spectrum of the corresponding operator as . We provide information on the width of the first bands and find asymptotic formulas for the localization of the possible gaps.     

A Trudinger–Moser inequality in a weighted Sobolev space and applications

We establish a Trudinger–Moser type inequality in a weighted Sobolev space. The inequality is applied in the study of the elliptic equation where , f has exponential critical growth and h belongs to the dual of an appropriate function space. We prove that the problem has at least two weak solutions provided is small.