We study the influence of the factor of electron-ion collisions on the solution of the Cauchy problem in the one-dimensional relativistic model of cold plasma and show that, depending on their intensity and initial data, two scenarios are possible: either the solution remains smooth and stabilizes to a stationary state, or during a finite time the oscillations blowup. In contrast to the nonrelativistic

In this article, the existence and uniqueness of global solutions to the discrete Safronov–Dubovskiǐ aggregation equation is studied. The unbounded aggregation kernel exhibits at most linear growth at infinity. The solution exhibits mass conservation property without any further restriction over the kernels.

In this contribution, we propose a multiplicative decomposition of the deformation gradient corresponding to the imagined procedure that Eshelby (Proc R Soc Ser A 241:376–396, 1957) used to investigate the theory of inclusions in the case of infinitesimal deformations. The proposed multiplicative decomposition is inspired by classical multiplicative decompositions reported in the literature and encompasses

In this paper, we shall study the following tumor invasion model $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\nabla \cdot (\gamma (w)\nabla u)-\chi \nabla \cdot (u\nabla v)+au-bu^2, &{}\quad x\in \Omega , ~~t>0,\\ v_t=\Delta v+wz,&{}\quad x\in \Omega , ~~t>0,\\ w_t=-wz,&{}\quad x\in \Omega , ~~t>0,\\ z_t=\Delta z-z+u,&{}\quad x\in \Omega , ~~t>0,\\ \frac{\partial u}{\partial \nu }=\frac{\partial

We are concerned with a two-phase fluid model in \({\mathbb {R}}^3\). This model was first derived by Choi (SIAM J. Math. Anal. 48: 3090–3122, 2016) by taking the hydrodynamic limit from the Vlasov–Fokker–Planck/isentropic Navier–Stokes equations with strong local alignment forces. Under the assumption that the \(H^3\) norm of the initial data is small but its higher-order Sobolev norm can be arbitrarily

In this paper, we study the Cauchy problem for the three-dimensional incompressible Keller–Segel–Navier–Stokes equations. By taking advantage of the geometry of axisymmetric flow without swirl, we obtain the global well-posedness for the system.

We establish blow-up results for systems of NLS equations with quadratic interaction in anisotropic spaces. We precisely show finite time blow-up or grow-up for cylindrical symmetric solutions. With our construction, we moreover prove some polynomial lower bounds on the kinetic energy of global solutions in the mass-critical case, which in turn implies grow-up along any diverging time sequence. Our

We consider the nonlinear biharmonic Schrödinger equation $$\begin{aligned} i\partial _tu+(\Delta ^2+\mu \Delta )u+f(u)=0,\qquad (\text {BNLS}) \end{aligned}$$ in the critical Sobolev space \(H^s({{\mathbb {R}}}^N)\), where \(N\ge 1\), \(\mu =0\) or \(-1\), \(0

Foundations of the modified Poisson kernel method were laid out by Finkelstein and Scheinberg in 1975 in the context of explicit solvability for the adaptive Dirichlet problem in a half plane. Over the past decade, this method has been further developed, and new applications have appeared both in the field of harmonic analysis and operator theory and in practical Schrödinger problems. In this paper

In this paper, we consider the interaction of elementary waves for the Aw–Rascle (AR) traffic flow model with variable lane width. The model is established based on the conservation laws where the lane width is considered, and the variable spaces ahead of the vehicles can be viewed as a source term which is incorporated into the model. Stationary waves are involved as the ingredient of the elementary

This paper is devoted to the study of the difference gap (for brevity, D-gap) function and global error bounds for a class of elliptic variational inequalities (for brevity, EVIs). Firstly, we establish the regularized gap function introduced by Yamashita and Fukushima for EVIs under some suitable conditions. Then the D-gap function for EVIs is proposed by employing these regularized gap functions

We propose a model for rate-independent evolution in elastoplastic materials under external loading, which allows large strains. In the setting of strain-gradient plasticity with multiplicative decomposition of the deformation gradient, we prove the existence of the so-called energetic solution. The stored energy density function is assumed to depend on gradients of minors of the deformation gradient

The problem of flow past fluid spheres was first solved by Rybczynski and Hadmard where they had assumed both the flow field to be Newtonian. However, there are many practical cases where one or both of these immiscible fluids are non-Newtonian. One such case of importance is emulsions, where the non-Newtonian droplets are in the Newtonian liquid. To facilitate studies in this area, we present a systematic

In this paper, we study the existence and behavior of positive solutions of the following quasilinear elliptic problems with discontinuous nonlinearities: $$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+V(x)u-\kappa u\Delta (u^2)=H(u-\delta )f(x,u),~~\text {in}~{\mathbb {R}}^{N}, \qquad \qquad (P_{\delta })\\&u\in D^{1,2}({\mathbb {R}}^{N})\cap W^{2,2}_{\mathrm{loc}}({\mathbb {R}}^{N}), \end{aligned}

With aid of the Stroh octet formalism, we obtain the elastic field in an infinite Kirchhoff laminated anisotropic plate containing a parabolic Eshelby inclusion undergoing uniform mid-plane in-plane eigenstrains and eigencurvatures. We prove the uniformity of the internal elastic field of membrane stress resultants, bending moments, total mid-plane in-plane strains, total mid-plane curvatures and in-plane

This paper deals with the chemotaxis consumption model with signal-dependent motility $$\begin{aligned} \left\{ \begin{array}{llll} u_t=\Delta (r(v)u)+\mu u(1-u),\quad &{}x\in \Omega ,\quad t>0,\\ v_t=\Delta v-uv,\quad &{}x\in \Omega ,\quad t>0 \end{array} \right. \end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset {\mathbb {R}}^n\) (\(n\ge 2\))

The onset of thermal convection in anisotropic rotating bidisperse porous media is investigated. The optimal result concerning the coincidence between linear instability and nonlinear stability thresholds with respect the \(L^2\)-norm is obtained.

This paper deals with the following p-Laplacian equation $$\begin{aligned} -\varepsilon ^{p}\Delta _{p}u+V(x)|u|^{p-2}u=|u|^{p^{*}-2}u,\quad u\in D^{1,p}({\mathbb {R}}^N), \end{aligned}$$ where \(p\in (1,N)\), p-Laplacian operator \(\Delta _{p}{:}{=}\)div\((|\nabla u|^{p-2}\nabla u) \), \(p^{*}=Np/(N-p)\), \(\varepsilon \) is a positive parameter, \(V(x)\in L^{{N}/{p}}({\mathbb {R}}^N)\cap L^{\infty

This paper considers a one-dimensional piezoelectric beams with magnetic effect damped with a weakly nonlinear feedback in the presence of a nonlinear delay term. Under appropriate assumptions on the weight of the delay, we establish an energy decay rate, using a perturbed energy method and some properties of a convex functions. Our result generalizes the recent result obtained in Ramos et al. (Z Angew

In this article, numerical schemes are proposed for approximating the solutions, possibly measure-valued with concentration (delta shocks), for a class of nonstrictly hyperbolic systems. These systems are known to model physical phenomena such as the collision of clouds and dynamics of sticky particles, for example. The scheme is constructed by extending the theory of discontinuous flux for scalar

This work is concerned with the time-harmonic eddy current problem in a bi-dimensional setting with a high contrast of magnetic permeabilities between a conducting medium and a dielectric medium. We describe a magnetic skin effect by deriving rigorously a multiscale expansion for the magnetic potential in power series of a small parameter \(\varepsilon \) which represents the inverse of the square

A general reaction–diffusion equation with nonlinear advection term having neither monotonicity nor variational structure is considered. This work is concerned with traveling waves for this type of equations. By constructing an invariant region with a lower curve smoothly connecting two fixed points where a heteroclinic orbit exists, we establish several theorems on the existence of wave fronts. The

We are concerned with 3-D compressible micropolar fluid system in the critical Besov space. We will focus on the global well-posedness, which is based on the results for the incompressible case given by Chen and Miao (J Differ Equ 252:2698–2724, 2012). To deal with the linear system which is a couple system with \((a,u,\omega )\), inspired by Wu and Wang (J Differ Equ 265:2544–2576, 2018), we find

According to the formulation of the nonlinear problem of stationary heat conduction in a homogeneous ellipsoid, when the intensity of volume energy release increases with temperature, a variational form of a mathematical model of a thermal explosion has been developed. This form contains a functional defined on a set of continuous and piecewise differentiable functions. These functions approximate

We consider a damped plate model with rotational forces in a bounded domain. The plate is either clamped or hinged. The rotational forces and damping involve the spectral fractional Laplacian with powers \(\theta \) in [0, 1] and \(\delta \) in [0, 2], respectively. Using the frequency domain approach, and appropriate interpolation inequalities, we show that the underlying semigroup is: (a) analytic

Describing the emerging macro-scale behavior by accounting for the micro-scale phenomena calls for microstructure-informed continuum models accounting properly for the deformation mechanisms identifiable at the micro-scale. Classical continuum theory, in contrast to the micromorphic continuum theory, is unable to take into account the effects of complex kinematics and distribution of elastic energy

We present a complete classification of the central configurations of the 5-body problem in a plane having the following properties: three bodies, denoted by 1, 2, 3, are at the vertices of an isosceles triangle, and the other two bodies are symmetrically located with respect to the mediatrix of the segment joining the bodies 1 and 2.

We are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph

In this paper, a variable-coefficient symbolic computation approach is proposed to solve the multiple rogue wave solutions of nonlinear equation with variable coefficients. As an application, a (\(2+1\))-dimensional variable-coefficient Kadomtsev–Petviashvili equation is investigated. The multiple rogue wave solutions are obtained and their dynamic features are shown in some 3D and contour plots.

In this article, the inverse scattering transform is considered for the Gerdjikov–Ivanov equation with zero and non-zero boundary conditions by a matrix Riemann–Hilbert (RH) method. The formula of the soliton solutions is established by Laurent expansion to the RH problem. The method we used is different from computing solution with simple poles since the residue conditions here are hard to be obtained

This paper is concerned with the asymptotic profiles of positive solutions for diffusive logistic equations. The aim is to study the sharp effect of nonlinear diffusion functions. Both the classical reaction–diffusion equation and nonlocal dispersal equation are investigated. We prove the sharp change occurs in reaction–diffusion equation by regularity estimates and compact principle. Due to lack of

We consider the transmission problem of couple stress elasticity in a fixed domain \(\Omega _{-}\) juxtaposed with a thin layer \(\Omega _{+}^{\delta }\). Our aim is to model the effect of the thin layer \(\Omega _{+}^{\delta }\) on the fixed domain \(\Omega _{-}\) by an impedance boundary condition. For that we use the techniques of asymptotic expansion to approximate the transmission problem by an

In this paper, we consider a new Timoshenko beam model with thermal and mass diffusion effects where heat and mass diffusion flux are governed by Cattaneo’s law. Necessary and sufficient conditions for exponential stability are provided in terms of the physical parameters of the model. Firstly, by the \(C_0\)-semigroup theory, we prove the well-posedness of the considered problem. Then, we prove the

The problem of axially symmetric TM wave diffraction from the finite, perfectly conducting bicone formed by a pair of finite cones of an arbitrary lengths and opening angles is considered. The problem is formulated in the spherical coordinate system as a mixed boundary value problem for the Helmholtz equation with respect to the unknown diffracted field. The solution of the problem is reduced to the

The dynamics of biofilm lifecycle are deeply influenced by the surrounding environment and the interactions between sessile and planktonic phenotypes. Bacterial biofilms typically develop in three distinct stages: attachment of cells to a surface, growth of cells into colonies, and detachment of cells from the colony into the surrounding medium. The attachment of planktonic cells from the surrounding

The stabilization properties of dissipative Timoshenko systems have been attracted the attention and efforts of researchers over the years. In the past 20 years, the studies in this scenario distinguished primarily by the nature of the coupling and the type or strength of damping. Particularly, under the premise that the Timoshenko beam model is a two-by-two system of hyperbolic equations, a large

Viscous flow through a reservoir with porous boundary is studied via asymptotic analysis and homogenization. Under the assumption of periodicity of the pores, the effective boundary condition is derived and rigorously justified. The velocity on the boundary satisfies a version of the Darcy law. The Darcy law for tangential component can also be seen as the Beavers–Joseph law.

In this paper, we establish transfer matrix for an incompressible orthotropic elastic layer. It is explicit and expressed compactly in terms of square brackets. This transfer matrix is a very convenient tool for solving various problems of wave propagation in layered elastic media including incompressible orthotropic layers. To prove this point, we apply it to investigate the reflection of SV-waves

Based on the equations of the f-plane shallow water model, a new set of nonlinear models for the interaction of the vortices in the f-plane shallow water system are established by using perturbation expansion and multi-scale method. These models have different nonlinear characteristics. The asymptotic analytical solution(vortex solution) is obtained by using Newton iteration method and quasi-spectral

We consider the inhomogeneous nonlinear Schrödinger equation with inverse-square potential in \({\mathbb {R}}^N\)$$\begin{aligned} i u_t -{\mathcal {L}}_a u+\lambda |x|^{-b}|u|^\alpha u = 0,\;\;{\mathcal {L}}_a=-\Delta +\frac{a}{|x|^2}, \end{aligned}$$ where \(\lambda =\pm 1\), \(\alpha ,b>0\) and \(a>-\frac{(N-2)^2}{4}\). We first establish sufficient conditions for global existence and blow-up in

The paper is devoted to the study of the asymptotic speed of spread and traveling wave solutions for a time-periodic reaction–diffusion SI epidemic model which lacks the comparison principle. By using the basic reproduction number \(R_0\) of the corresponding periodic ordinary differential system and the minimal wave speed \(c^*\), the spreading properties of the corresponding solution of the model

We derive Papkovich–Neuber type representations for the solutions of Navier–Lamé equations in linear elastostatics and of the stationary Stokes equations using exterior calculus on the Euclidean space. We generalize the result for two-dimensional Riemannian manifolds.

We prove the existence and uniqueness of the solutions to a Caputo–Hadamard fractional stochastic differential equation driven by a multiplicative white noise, which may describe the random phenomena in the ultraslow diffusion processes. The moment estimates are given in terms of the logarithmic Mittag–Leffler function. We also prove the regularity of the solutions via the logarithmic Hölder continuity

In a private communication with D. M. Barnett, W. D. Nix presented a series of numerical computations in which he calculated the net interaction force between two skew dislocations separated by a distance h that are parallel to the traction-free surface of an isotropic elastic half-space. Nix drew a series of conclusions from his numerical computations including the independence of the net interaction

In this paper, we study a Timoshenko system only with thermodiffusion effects. We establish exponential energy decay of the system with two kinds of boundary conditions under the assumption of the equal wave speeds. Our result extends the recent result by Aouadi et al. in Z. Angew. Math. Phys., 70, 117 (2019).

In this paper, we deal with issues related to global multiplicity of \(W^{1,p}_{\mathrm {loc}}(\Omega )\)-solutions for the very-singular and non-local \(\mu \)-problem $$\begin{aligned} -{g\left( \int \limits _\Omega u^q\right) }\Delta _pu={\mu } u^{-\delta } + u^{\beta } \ \ \text{ in } \ \ \Omega , \ \ \ \ u > 0 \ \ \ \text{ in } \ \Omega \ \ \ \text{ and } \ \ u=0 \ \ \text{ on } \ \partial \Omega

Consider weak solutions u of the 3D Navier–Stokes equations in the critical space $$\begin{aligned} u\in L^{p}\left( 0,\infty ; {\dot{B}}^{\frac{2}{p}+\frac{3}{q}-1}_{q,\infty }({\mathbb {R}}^3)\right) , \quad 2

In this paper, we concerned with a haptotaxis system proposed as a model for fusogenic oncolytic virotherapy and syncytia, accounting for interaction between uninfected cancer cells, infected cancer cells, syncytia cancer cells, extracellular matrix and oncolytic virus. We present the global classical solution in two-dimensional bounded domains under appropriate regularity assumptions on the initial

The main purpose of this research is to find new and more exact closed form solutions of an important integrable model in theoretical physics namely coupled Gerdjikov–Ivanov equation by utilizing the Lie symmetry approach. Lie group analysis has been implemented to find the infinitesimal generators and symmetry reductions. The coupled Gerdjikov–Ivanov equation has been reduced into a system of nonlinear

This focuses on the large time behavior of the solution to the IVP of the 2D fully dissipative magnetohydrodynamics system. We first compute the decay rates of the solutions, then, we show that these rates are sharp. Moreover, the explicit asymptotic profiles are identified. It turns out that these profiles resemble the Oseen vortices.

Experimental tests of rotation sensitivity in elastic materials and in empty space are surveyed. Pioneers in the theory for each field of study were aware of the other field. Sensitivity to rotation has been demonstrated experimentally in elastic solids but tests for rotation gradient sensitivity in gravity have found no such effects. Insight from comparison of experimental approaches may lead to new

Taking into account the action of inhomogeneous zonal wind (shear flow), nonlinear dynamic equations describing the propagation of planetary ULF magnetized Rossby waves in the ionospheric D-, E-, and F-layers are obtained and investigated. The influence of existence of charged particles through Hall and Pedersen conductivities on such dynamic equations is studied in detail. It is shown that the existence

In Grebenev, Wacławczyk, Oberlack (2019 Phys A: Math. Theor. 52, 33), the conformal invariance (CI) of the characteristic \({\varvec{X}}_{1}(t)\) (the zero-vorticity Lagrangian path) of the first equation (i.e. for the evolution of the 1-point PDF \(f_1({\varvec{x}}_{1},\omega _{1},t)\), \({\varvec{x}}_{1} \in D_1 \subset {\mathbb {R}}^2)\) of the inviscid Lundgren–Monin–Novikov (LMN) equations for

We investigate the stability of a specific stationary solution to Boussinesq equations without thermal conduction in the flat strip \(\Omega = \mathbb {T}\times (0,1)\). Explicit decay rates of the vorticity/velocity are given as well as the limit state of the temperature. Our method is based on time-weighted energy estimates.

In this paper, we investigate the stabilization of a linear Bresse system with one discontinuous local internal viscoelastic damping of Kelvin–Voigt type acting on the axial force, under fully Dirichlet boundary conditions. First, using a general criteria of Arendt–Batty, we prove the strong stability of our system. Finally, using a frequency domain approach combined with the multiplier method, we

We study the behavior in time of the solutions to several systems of equations for porous-thermo-elastic problems when one of the variables is considered to be quasi-static or, in other words, whose second time derivative can be neglected. We analyze three different situations using the classical Fourier law and also the type II or type III Green–Naghdi heat conduction models.

The balance of pseudomomentum is discussed and applied to simple elasticity, ideal fluids, and the mechanics of inextensible rods and sheets. A general framework is presented in which the simultaneous variation of an action with respect to position, time, and material labels yields bulk balance laws and jump conditions for momentum, energy, and pseudomomentum. The example of simple elasticity of space-filling

We propose a modified Laurent series for investigating the elastic field around holes/inclusions under plane deformation by dividing the classical Laurent series over the first derivative of the conformal mapping, which is justified from a mathematical point of view. Through a group of numerical examples concerning the stress concentration around holes of common shapes, the accuracy of the modified