The effects of cross-diffusion on linear and weak nonlinear stability of double diffusive convection in an electrically conducting horizontal fluid layer with an imposed vertical magnetic field are investigated. The criterion for the onset of stationary and oscillatory convection is obtained analytically by performing the linear instability analysis. Several noteworthy departures from those of doubly

We show some regularity criteria (Prodi–Serrin type regularity) to weak solutions of the 3D generalized Navier–Stokes equations in viewpoint of the velocity vector u or the vorticity vector \(\omega :=\nabla \times u\) in Lorentz space. Moreover, we briefly mention some results for coupled equations with Navier–Stokes equation (see Remark 1.5 and 1.8).

This paper is a development of the results and techniques of the two papers (Carvalho-Neto and Planas in J Differ Equ 259:2948–2980, 2015; Oka in J Math Anal Appl 473:382–407, 2019) for the aim of addressing the existence and uniqueness of local and global mild solutions, on the Heisenberg group \(\mathbb {H}^d\), of the time-fractional Navier–Stokes system with time derivative of order \(\alpha \in

In this paper, we consider the standard linear solid model in \(\mathbb {R}^N\) coupled with the Fourier law of heat conduction. First, we give the appropriate functional setting to prove the well-posedness of this model under certain assumptions on the parameters (that is, \(0<\tau \le \beta \)). Second, using the energy method in the Fourier space, we obtain the optimal decay rate of a norm related

This paper devotes to establish an improved regularity criterion for weak solution of 3D incompressible MHD equations. We show that the weak solution is regular, provided that $$\begin{aligned} \int _{0}^{T}{\frac{\Vert \pi (\cdot ,t)\Vert _{\dot{B}_{\infty ,\infty }^{-1}}^{2}}{\left( e+\ln \left( e+\Vert \pi (\cdot ,t)\Vert _{\dot{B} _{\infty ,\infty }^{-1}}\right) \right) \ln \left( e+\ln \left(

In this paper, we consider the Fornberg–Whitham equation and a family of solitary wave solutions is found by using minimization principle, where a related penalization function and the concentration-compactness lemma play a key role in the proof. Besides, we also prove that the family of solitary solutions is orbitally stable and decays exponentially when wave speed c is bigger than 1.

This paper addresses some results about mild solution to time-fractional Navier–Stokes equations with bounded delay existed in the convective term and the external force. Based on the Schauder’s fixed point theorem, we establish the sufficient conditions for the existence and uniqueness of the global mild solution in Banach spaces. Moreover, the polynomial decay estimate of the mild solution is given

This paper considers the problem of the local existence for the generalized MHD equations with fractional dissipative terms \(\Lambda ^{2\alpha } u\) for the velocity field and \(\Lambda ^{2\beta } b\) for the magnetic field, respectively. Based on some new commutator estimates, local existence for the generalized MHD equations is established, which recovers and improves previous results.

This paper aims to study the existence and concentration of solutions for the stationary Dirac equation in \(\mathbb {R}^3\) with critical nonlinearities: $$\begin{aligned} i\varepsilon \sum \limits _{k=1}^3\alpha _k\partial _k u-a\beta u+V(x)u=P(x)f(|u|)u+Q(x)g(|u|)u, \end{aligned}$$ where \(\varepsilon >0\) is a small parameter and \(a>0\) is a constant. We also show the semiclassical solutions \(\omega

The problem of the small oscillations of a spherical layer of an inviscid fluid about a rigid spherical body in the gravitational field has been studied by Laplace in the case of a fluid layer of small depth. His results have been rediscovered by R. Wavre by using his method of the uniform process. The second author and his collaborators have studied the case of a layer of viscous fluid by means of

This paper deals with the propagation dynamics for lattice differential equations in a time-periodic shifting habitat. We prove the existence, uniqueness and global exponential stability of the periodic forced waves. We also establish the spreading properties of the solutions. Our results indicate that the long-time behaviors of solutions depend on the speed of the shifting habitat and a number that

A one-dimensional Stefan problem with spherical symmetry corresponding to the evaporation process of a droplet is considered. An equivalent integral formulation is obtained, and through a fixed point theorem, the existence and uniqueness of the solution are proved.

In this paper, we consider a three-dimensional chemotaxis-Stokes system $$\begin{aligned} \left\{ \begin{aligned}&n_t+u\cdot \nabla n=\Delta n^m-\nabla \cdot (nS(x,n,c)\cdot \nabla c),&\quad&x\in \Omega , t>0,\\&c_t+u\cdot \nabla c=\Delta c-nf(c),&\quad&x\in \Omega ,t>0,\\&u_t+\nabla P=\Delta u+n\nabla \phi ,&\quad&x\in \Omega ,t>0,\\&\nabla \cdot u=0,&\quad&x\in \Omega ,t>0,\\&(\nabla n^m- n S(x,n

This paper deals with a fully parabolic singular chemotaxis-(growth) system with indirect signal production or consumption $$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\chi \nabla \cdot \left( \frac{u}{v}\nabla v\right) +f(u),&(x,t)\in \Omega \times (0,\infty ), \\&v_t=\Delta v+h(v,w),&(x,t)\in \Omega \times (0,\infty ), \\&w_t=\Delta w- w+u,&(x,t)\in \Omega \times (0,\infty ), \end{aligned}

In this paper, we are concerned with the initial-boundary value problem to the 2D magneto-micropolar system with zero angular viscosity in a smooth bounded domain. We prove that there exists a unique global strong solution of such a system by imposing natural boundary conditions and regularity assumptions on the initial data, without any compatibility condition.

In this paper, we obtain the existence of positive solution for a system of quasilinear Schrödinger equations with concave nonlinearities which is related to several applications in Hydrodynamics, Heidelberg Ferromagnetism and Magnus Theory, Condensed Matter Theory, Dissipative Quantum Mechanics and nanotubes and fullerene-related structures. The quasilinear Schrödinger problem is studied by considering

We study the dynamics of a two-level crossing model with a polynomial modification of the linear Landau–Zener tunneling. For a cubic modification, we express the non-adiabatic transition amplitudes analytically via the bi-confluent Heun functions. We find a closed-form series expression of the transition probability at the long time limit, and derive tractable approximate formulas to the state populations

We analyze the well-posedness and smoothing properties of a variable-order time-fractional wave partial differential equation in multiple space dimensions. Accordingly, we prove the unique determination of the variable order in this model with the observations of the unknown solutions on an arbitrarily small spatial domain over a sufficiently small time interval. The proved theorem provides a guidance

An investigation is made on the Gurtin–Murdoch (GM) model for soft spherical elastic solids with surface tension-induced bulk residual stress. Unlike the original GM model which treats the surface tension as a finite value but the surface tension-induced bulk residual stress as infinitesimal, the present model treats both consistently as finite values. Detailed comparisons between the present model

In this paper, we study a PDE–ODE system as a simplification of a glioblastoma model. Mainly, we prove the existence and uniqueness of global in time classical solution using a fixed point argument. Moreover, we show some stability results of the solution depending on some conditions on the parameters.

We consider a stationary boundary value problem describing a radiative–conductive heat transfer in a semitransparent body with absolutely black inclusions. To describe the radiative transfer, the integro-differential radiative transfer equation is used. We do not take into account the dependence of the radiation intensity and the properties of semitransparent materials on the radiation frequency. We

We consider Robin problems driven by the anisotropic p-Laplace operator and with a logistic reaction. Our analysis covers superdiffusive, subdiffusive and equidiffusive equations. We examine all three cases, and we prove multiplicity properties of positive solutions (superdiffusive case) and uniqueness (subdiffusive and equidiffusive cases). The equidiffusive equation is studied only in the context

In this paper we study the approximation of random diffusion by nonlocal diffusion with properly rescaled kernels in age-structured models. First we show that solutions of age-structured models with nonlocal diffusion under Dirichlet and Neumann boundary conditions converge to solutions of the corresponding age-structured models with random diffusion under Dirichlet and Neumann boundary conditions

In this paper, we study the nonlinear viscoelastic plate equation modeling the deformation of a suspension bridge, given by $$\begin{aligned} u_{tt}(x,y,t)+\Delta ^2 u(x,y,t) -\displaystyle \int \limits _0^{t} g(t-\tau )\Delta ^2 u(\tau )\;\hbox {d}\tau +a(x, y) \vert u_t\vert ^m u_t+\vert u\vert ^{\rho }u=0, \end{aligned}$$ in the bounded domain \(\Omega = (0, \pi )\times (-d, d)\), which is assumed

The equations governing turbulent flow of micropolar incompressible media are studied using the Reynolds decomposition. The average force-stress is augmented by the Reynolds stress of the same form as in classical continuum mechanics, while the average couple-stress is augmented by a new turbulent couple-stress. Additionally, the average heat flux and internal energy density are modified from the expressions

This paper is concerned with the low Mach number limit of the compressible Hall-magnetohydrodynamic model for quantum plasmas in \( {\mathbb {T}}^{3}\). It is justified rigorously that, for the well-prepared initial data, as the Mach number tends to zero, the classical solution of the compressible quantum Hall-magnetohydrodynamic model converges to that of the incompressible Hall-magnetohydrodynamic

We study the Cauchy problem for the higher-order nonlinear 3D Hartree-type equation $$\begin{aligned} \left\{ \begin{array} [c]{ll} i\partial _{t}u+\frac{1}{2}\Delta u-\frac{1}{4}\Delta ^{2}u=\left( \left| x\right| ^{-1}*\left| u\right| ^{2}\right) u,&{}{}\quad t>0, x\in {\mathbb {R}}^{3},\\ u\left( 0,x\right) =u_{0}\left( x\right) ,&{}{}\quad x\in {\mathbb {R}}^{3}, \end{array} \right. \end{aligned}$$

In this paper, we are concerned with the large-time behavior of the solutions in the full quantum hydrodynamic model, which can be used to analyze the thermal and quantum influences on the transport of carriers (electrons or holes) in semiconductor device. For the Cauchy problem in \({\mathbb {R}}^3\), the global existence and uniqueness of smooth solutions, when the initial data are small perturbations

The implication of a uniform horizontal throughflow on the linear stability of buoyancy-driven convection in a vertical fluid layer is studied. The vertical boundaries of the fluid layer are rigid-permeable and holding at uniform but different temperatures. The instability to small-amplitude perturbations is tested by parameterizing the basic stationary flow through the Péclet number and the Prandtl

In this work, we study a thermoelastic Bresse system from both mathematical and numerical points of view. The dual-phase-lag heat conduction theory is used to model the heat transfer. An existence and uniqueness result is obtained by using the theory of linear semigroups. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A priori error

A Correction to this paper published: https://doi.org/10.1007/s00033-021-01480-3

For the non-isentropic compressible micropolar fluids in three dimensions, we show that the space–time behaviors of the fluid density, momentum and energy contain both generalized Huygens’ wave and diffusion wave as the non-isentropic compressible Navier–Stokes equation, while it only contains the diffusion wave for the micro-rational velocity. All of the previous estimates containing Huygens’ waves

The novelty of the approach contains several aspects: the method of derivation of the governing equations, the form and number of equations, the optimal choice of elastic constants. While majority of published articles derives the governing equations from the Green’s function for the compound space and the use of the reciprocal theorem, we use the combination of Green’s functions for 2 different half-spaces

In this short note, our aim is to correct a mistake in Zhao et al. (Z Angew Math Phys 69:146, 2018, Lemma 3.2) and consider the case of \(c=c^*\) in the right way.

Within the framework of linearised theory of water waves, a model of oblique wave scattering by obstacles in the form of thin multiple surface-piercing porous barriers having non-uniform porosity is analysed. Herein, we consider a flexible base in an ocean of uniform finite depth. The flexible base surface is modelled as a thin elastic plate under the acceptance of Euler–Bernoulli beam equation. With

We present a rigorous analysis of the plane problem of an elastic half-plane with a fixed cycloid wavy boundary under biaxial tension using Muskhelishvili’s complex variable method. A closed-form expression for the pair of analytic functions characterizing the stresses and displacements in the elastic body is derived by means of analytic continuation. Furthermore, elementary expressions for the interfacial

In this paper, we study the well-posedness and stability of a wave equation with infinitely structural memory, herein the memory kernel function does not satisfy the monotonicity. For the model, the history function space setting is a main difficulty because the usual space setting will lead the shift semigroup to be a unbounded semigroup. In the present paper, we modify the history function space

Previously, the incompressible limit of the equations of Hookean elastodynamics has been established in the whole space. The present paper is devoted to the study of the incompressible limit in a bounded domain. Here, the main difficulty lies in the interactions between the large operator and the boundary. By designing some delicate semi-norms and energy estimates, we obtain the uniform estimates of

This paper is concerned with the scattering resonances of open cavities. It is a follow-up of Ammari et al. (ZAMP 71:102, 2020), where the transverse magnetic polarization was assumed. In that case, using the method of matched asymptotic expansions, the leading-order term in the shifts of scattering resonances due to the presence of small particles of arbitrary shapes was derived and the effect of

The initial boundary value problems for compressible Navier–Stokes–Poisson are considered on a bounded domain in \({\mathbb {R}}^3\) in this paper. The global existence of smooth solutions near a given steady state for compressible Navier–Stokes–Poisson with physical boundary conditions is established with the exponential stability. An important feature is that the steady state (except velocity) and

This paper is concerned with the long-time behavior of Moore-Gibson-Thompson equation with degenerate memory effect, given by $$\begin{aligned} \tau u_{ttt}+\alpha (x)u_{tt}-b\Delta u_t-c^2\Delta u+\int \limits _0^tg(t-s)div\big (\beta (x)\nabla u(s)\big )ds=0, \quad ~in ~\Omega \times (0,\infty ), \end{aligned}$$ where structural damping and molecular relaxation are both locally distributed, satisfying

This paper is devoted to classifying the dynamical behavior of solutions to a free boundary problem of a predator–prey model with a nonlocal reaction term. We investigate the spreading-vanishing dichotomy, that is, the asymptotic behavior of predator and prey species. Moreover, some criteria are also established for spreading and vanishing. At last, when spreading occurs, we obtain an upper and a lower

Fluids, as a phase of matter including liquids, gases and plasmas, are seen to be common in nature, the study of which helps the design in the related industries. In this paper, we optimize the Pfaffian technique and investigated the Boiti–Leon–Manna–Pempinelli equation for an irrotational incompressible fluid. Higher-order hybrid solutions consisting of the L lumps, M breathers and N solitons are

A closed-form dispersion equation for Lamb waves propagating in functionally graded (FG) plates with arbitrary varying anisotropic physical properties in transverse direction is constructed by applying Cauchy sextic formalism. Comparative analyses of the dispersion curves related to the fundamental modes of homogeneous and FG plates reveal their substantial and visually detected differences even at

This paper aims to provide the complete analysis on the threshold dynamics of an age-space structured malaria epidemic model. We formulate the model in a spatially bounded domain by assuming that: (i) the density of susceptible humans at space x stabilizes at H(x); (ii) the force of infection between human population and mosquitoes is given by the mass action incidence. By appealing to the theory of

The flux-limited Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{lcl} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot \Big ( n f(|\nabla c|^2) \nabla c\Big ), \\ c_t + u\cdot \nabla c &{}=&{} \Delta c - c + n, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ is

In this paper, we consider the following Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases: $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\lambda Q(x)u+\omega _{1}|u|^{2}u+\omega _{2}(K\star |u|^{2})u=0, &{} x\in {\mathbb {R}}^{3},\\ \displaystyle u>0,\quad u\in H^{1}({\mathbb {R}}^{3}),\\ \end{array}\right. } \end{aligned}$$(0.1)

In this paper, we are concerned with the existence and multiplicity of multi-bump type nodal solutions for the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{ll} -\Delta u+ \lambda V(x)u=u \log u^2, &{}\quad \text{ in } \quad {\mathbb {R}}^{N}, \\ u \in H^1({\mathbb {R}}^{N}), \\ \end{array} \right. $$ where \(N \ge 1\), \(\lambda >0\) is a real parameter and the nonnegative continuous

The Kuramto–Sivashinsky equation with anisotropy effects models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. Written in terms of the step slope, it can be represented in a form similar to

In this article, the propagation of nonlinear shear horizontal waves for some comparisons between the heterogeneous and homogeneous plates is considered. It is assumed that one plate is made of up hyper-elastic, heterogeneous, isotropic, and generalized neo-Hookean materials, and the other consists of hyper-elastic, homogeneous, isotropic, and generalized neo-Hookean materials. Using the known solitary

In this paper, we consider the nonlinear abstract equation $$\begin{aligned} u_{tt}+Au-\int _{0}^{t}g(t-s)Au(s)\mathrm{d}s+h(u_{t})=j(u) \end{aligned}$$ subject to a competing effect of viscoelastic and frictional dampings. With very general assumptions on the behavior of g at infinity and the behavior of h near 0, we establish explicit and optimal energy decay result. To the best of our knowledge

We consider the problem of a semi-infinite mode III crack partially penetrating a three-phase elliptical inhomogeneity. The two elliptical interfaces are confocal, and the semi-infinite crack passes through the vertices of the two interfaces with its crack tip located at one of the two common foci of the two elliptical interfaces. Full-field closed-form solutions are derived by means of conformal mapping

Via the Hirota bilinear method combined with the Kadomtsev–Petviashvili (KP) hierarchy reduction method, two types of solitons, namely general solitons on a background of periodic waves and rational soliton solutions, to the parity-time-symmetric nonlocal nonlinear Schrödinger equation with the defocusing-type nonlinearity for nonzero boundary condition are investigated. These two types of soliton

A class of perturbed discrete nonlinear Schrödinger equations with sublinear terms is studied. The new features are: (1) negative and positive energy homoclinic solutions are obtained; (2) the continuity of solutions in the perturbed terms at zero is studied; (3) the positive-definite condition can be removed.

We prove by using an iteration argument some blow-up results for a semilinear damped wave equation in generalized Einstein–de Sitter spacetime with a time-dependent coefficient for the damping term and power nonlinearity. Then, we conjecture an expression for the critical exponent due to the main blow-up results, which is consistent with many special cases of the considered model and provides a natural

We obtain the global large solutions to the compressible Navier–Stokes equations in \({\mathbb {R}}^2\). The solution is large in the sense that there is no smallness assumption applied to one component of the initial incompressible velocity.

We consider self-similar solutions to the 1-dimensional isothermal Euler system for compressible gas dynamics. For each\(\beta \in {\mathbb {R}}\), the system admits solutions of the form $$\begin{aligned} \rho (t,x)=t^\beta \Omega (\xi )\qquad u(t,x)=U(\xi )\qquad \qquad \textstyle \xi =\frac{x}{t}, \end{aligned}$$ where \(\rho \) and u denote the density and velocity fields. The ODEs for \(\Omega

This paper focuses on the stability problem for perturbations near a hydrostatic equilibrium associated with the three-dimensional Boussinesq equations with partial dissipation. We mainly study the global stability and large-time behavior of solutions to this system with only horizontal dissipation and obtain three results. The first establishes the global \(H^2\)-stability, the second obtains the

This article is devoted to studying one-dimensional piston problem for the compressible Magnetohydrodynamics when the piston is pushed forward relatively. If the initial data and the piston speed are both given some small \(\mathrm {BV}\) perturbations, then global stability of large shock waves to this problem has been established, and the long time behavior has been also considered.

This paper shows how the classical representation techniques for the solution of elasticity problems, based on the Green’s functions, can be generalized to second-gradient continua focusing on the specific case of pantographic lattices. As these last are strongly anisotropic, the fundamental solutions of isotropic second-gradient continua involving bi-Helmholtz-type operators are not applicable. More