Cauchy problem for doubly singular parabolic equation with gradient source

This paper examines the Cauchy problem for doubly singular parabolic equation with a source term depending solely on the gradient. We establish the local and global existence of solutions when initial data is merely a function in (). Moreover, the uniform ‐estimates and gradient estimates of solutions are obtained.     

Positive solutions of Kirchhoff type elliptic equations in with critical growth

In this paper, we study the following Kirchhoff type elliptic problem with critical growth: where , and , and the nonlinear growth term reaches the Sobolev critical exponent since for four spatial dimensions. In a non‐radial symmetric function space, we establish a local compactness splitting lemma of critical version to investigate the existence of positive ground state solutions.     

The Cauchy problem for the generalized hyperelastic rod wave equation

In this paper we consider the Cauchy problem for the generalized hyperelasticrod wave equation which includes the Camassa‐Holm equation and the hyperelastic rod wave equation. Firstly, by using the Kato's theory, we prove that the Cauchy problem for the generalized hyperelastic rod wave is locally well‐posed in Sobolev spaces with . Secondly, we give some conservation laws, some useful conclusions and the precise blow‐up scenario and show that the Cauchy problem for the generalized hyperelastic rod wave equation has smooth solutions which blows up in finite time. Thirdly, we give the blow‐up rate of the strong solutions to the Cauchy problem for the generalized hyperelastic rod wave equation. Finally, we give the lower bound of the maximal existence time of the solution and the lower semicontinuity of existence time of solutions to the generalized hyperelastic rod wave equation.     

Variational and Perron–Wiener solutions in stratified Lie groups

We consider sub‐Laplacians in stratified Lie groups and we compare Perron–Wiener and weak‐variational solutions of the Dirichlet problem , where Ω is a bounded open set in and φ is the restriction to the boundary of a function such that . The result we obtained extends a previous theorem by Arendt and Daners, related to the classial Laplacian in .     

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.     

Multi‐peak standing waves for nonlinear Schrödinger equations involving critical growth

For the following singularly perturbed problem we construct a solution which concentrates at several given isolated positive local minimum components of V as . Here, the nonlinearity f is of critical growth. Moreover, the monotonicity of and the so‐called Ambrosetti–Rabinowitz condition are not required.     

Bifurcation and multiplicity results for critical fractional p‐Laplacian problems

We prove a bifurcation and multiplicity result for a critical fractional p‐Laplacian problem that is the analog of the Brézis‐Nirenberg problem for the nonlocal quasilinear case. This extends a result in the literature for the semilinear case to all , in particular, it gives a new existence result. When , the nonlinear operator , has no linear eigenspaces, so our extension is nontrivial and requires a new abstract critical point theorem that is not based on linear subspaces. We prove a new abstract result based on a pseudo‐index related to the ‐cohomological index that is applicable here.     

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.     

Solutions to the Navier–Stokes equations with mixed boundary conditions in two‐dimensional bounded domains

In this paper we consider the system of the non‐steady Navier–Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y, respectively, to be the space of “possible” solutions of this problem and the space of its data. We define the operator and formulate our problem in terms of operator equations. Let and be the Fréchet derivative of at . We prove that is one‐to‐one and onto Y. Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. The next result proved in the Appendix of this paper is W^{2, 2}‐regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.     

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 analyticity of the ‐Stokes semigroup for some non‐Helmholtz domains

Consider the Stokes equations in a sector‐like C ³ domain . It is shown that the Stokes operator generates an analytic semigroup in for . This includes domains where the ‐Helmholtz decomposition fails to hold. To show our result we interpolate results of the Stokes semigroup in and L ² by constructing a suitable non‐Helmholtz projection to solenoidal spaces.     

Riemann boundary value problem for harmonic functions in Clifford analysis

In this article, we mainly deal with the boundary value problem for harmonic function with values in Clifford algebra: where is a Liapunov surface in , the Dirac operator , are unknown functions with values in a universal Clifford algebra Under some assumptions, we show that the boundary value problem is solvable.     

The asymptotic behavior of the linear transmission problem in viscoelasticity

We consider a transmission problem with localized Kelvin‐Voigt viscoelastic damping. Our main result is to show that the corresponding semigroup is not exponentially stable, but the solution of the system decays polynomially to zero as when the initial data are taken over the domain Moreover, we prove that this rate of decay is optimal. Finally, using a second order scheme that ensures the decay of energy (Newmark‐β method), we give some numerical examples which demonstrate this polynomial asymptotic behavior.     

Global existence and nonexistence of solutions for a viscoelastic wave equation with nonlinear boundary source term

In this paper, we consider the initial boundary value problem for a viscoelastic wave equation with nonlinear boundary source term. First of all, we introduce a family of potential wells and prove the invariance of some sets. Then we establish the existence and nonexistence of global weak solution with small initial energy under suitable assumptions on the relaxation function , nonlinear function , the initial data and the parameters in the equation. Furthermore, we obtain the global existence of weak solution for the problem with critical initial conditions and .     

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.     

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.     

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 .     

Holomorphic Campanato spaces on the unit ball

As outlined below, this paper is devoted to a Carleson‐type‐measure‐based study of the holomorphic Campanato 2‐space on the open unit ball of , comprising all Hardy 2‐functions whose oscillations in non‐isotropic metric balls on the compact unit sphere are proportional to some power of the radius other than the dimension .     

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.