This paper studies the Cauchy problem for three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic equations with vacuum as far field density. We prove the global existence and uniqueness of strong solutions provided that the quantity \(\Vert \rho _0\Vert _{L^\infty }+\Vert b_0\Vert _{L^3}\) is suitably small and the viscosity coefficients satisfy \(3\mu >\lambda \). Here, the

We consider a modified version of a suspension bridge model with a spatially variable stiffness parameter to reflect the discrete nature of the placement of the bridge hangers. We study the qualitative and quantitative properties of this model and compare the cases of constant and non-constant coefficients. In particular, we show that for certain values of the stiffness parameter, the bifurcation occurs

We establish the existence and multiplicity of solutions for Kirchhoff elliptic problems of type $$\begin{aligned} -m\left( \mathop \int \limits _{\mathbb {R}^3} |\nabla u|^2 \mathrm{{d}}x\right) \Delta u = f(x,u), \quad x \in \mathbb {R}^3, \end{aligned}$$ where \(m:\mathbb {R}_+\rightarrow \mathbb {R}\) is continuous, positive and satisfies appropriate growth and/or monotonicity conditions. We consider

This paper deals with the two-species chemotaxis-competition system with loop $$\begin{aligned} \left\{ \begin{array}{llll} \partial _{t} u_{1}=d_1\Delta u_{1}-\chi _{11}\nabla \cdot (u_{1}\nabla v_{1}) -\chi _{12}\nabla \cdot (u_{1}\nabla v_{2}) +\mu _{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ \partial _{t} u_{2}=d_2\Delta u_{2}-\chi _{21}\nabla \cdot (u_{2}\nabla v_{1}) -\chi _{22}\nabla \cdot (u_{2}\nabla

We consider the existence and uniqueness of a time periodic solution for the compressible magneto-micropolar fluids with time periodic forces in a periodic domain. More precisely, under some smallness and symmetry assumptions on the external forces, we prove the existence of the periodic solution by a regularized approximation scheme and the topological degree theory. The uniqueness of the periodic

In this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. We first establish some new qualitative properties

In this paper, it is proven an existence and uniqueness theorem for weak solutions of the equilibrium problem for linear isotropic dilatational strain gradient elasticity. Considered elastic bodies have as deformation energy the classical one due to Lamé but augmented with an additive term that depends on the norm of the gradient of dilatation: only one extra second gradient elastic coefficient is

The paper is devoted to a reaction-diffusion problem describing diffusion and consumption of nutrients in a biological tissue consisting of small cells periodically arranged in an extracellular matrix. Cells consume nutrients with a rate proportional to cell area and to nutrient concentration. The dependence on the nutrient concentration can be linear or nonlinear. The cells are modeled by a potential

A generalization of the Wiechert condition by introducing two independent dimensionless parameters instead of one parameter in the original Wiechert condition is proposed. Variation of Stoneley wave velocity at varying two parameters of the generalized Wiechert condition at different Poisson’s ratios is studied revealing a substantial discrepancy in Stoneley wave velocity profiles.

The main purpose of this manuscript is to study the well-posedness and decay estimates for strong solutions to the Cauchy problem of 3D smectic-A liquid crystals equations. First, applying Banach fixed point theorem, we prove the local existence and uniqueness of strong solutions. Then, by establishing some nontrivial estimates with energy method and a standard continuity argument, we prove that there

The work described in this paper is concerned with providing a rational asymptotic analysis of the wrinkling bifurcation experienced by a thin elastic hemispherical segment subjected to vertical tensile forces on its upper rim. This is achieved by considering the interplay between two boundary layers and matching the corresponding solutions associated with each separate region. Our key result is a

In this work, we consider the two-species chemotaxis system with logistic source in a two-dimensional bounded domain. We present the global existence of classical solutions under appropriate regularity assumptions on the initial data. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature.

In this paper, the non-isentropic Euler–Poisson system with the varying background charges in flat nozzles is investigated. Under physically acceptable boundary conditions, we establish the monotonicity between shock location and the density at the exit of the nozzle. Moreover, the unique existence of the transonic shock solution is obtained.

We derive an analytic solution to the plane problem associated with an edge dislocation interacting with a completely coated semi-infinite crack in the presence of remote in-plane loading. The semi-infinite crack and the parabolic coating–matrix interface are confocal, and the coating and the matrix have the same shear modulus but distinct Poisson’s ratios. Through the introduction of a conformal mapping

In this paper, we are concerned with the three-dimensional stationary equations of compressible magnetohydrodynamic (MHD) isentropic flow in a bounded cylinder domain \(\Omega =(0,1)\times \Omega _0\) with an inhomogeneous boundary condition. We obtain the existence and uniqueness of a strong solution provided that the data are around a given constant flow. From our result, it reveals that the magnetic

The paper deals with the homogenization of linear Boltzmann equations by the means of the sigma-convergence method. Replacing the classical periodicity hypothesis on the coefficients of the collision operator by an abstract assumption covering a great variety of physical behaviours, we prove that the density of the particles converges to the solution of a drift-diffusion equation. We then illustrate

In this paper, we consider large time behaviour of solutions to the 3D Navier–Stokes equations with damping term \(|u|^{\beta -1}u\)(\(\beta \ge 1\)). For \(\beta >\frac{7}{3}\), we derive decay rates of the \(L^2\)-norm of the solutions. By using Fourier splitting method, we also prove that the solutions are asymptotically equivalent to the solutions of the classic 3D Navier–Stokes equations.

This paper is devoted to the problem of wave propagation on a perfect conducting fluid under a vertical electric field, a kind of electrohydrodynamic waves. Three competing restoring forces resulting from gravity, surface tension, and normal electric field are taken into account. There are many numerical and experimental results on electrohydrodynamic waves, but the system’s well-posedness, which is

In the original publication of the article, Eq. (23) was incorrectly published.

The current paper is devoted to the study of the stability of space–time periodic traveling wave solutions and positive space–time periodic entire solutions of nonlocal dispersal cooperative systems in space–time periodic habitats. We first show the existence, uniqueness and stability of positive space–time periodic entire solution \({\mathbf {u}}^{*}(t,x)\) for such nonlocal dispersal cooperative

Models incorporating diffusion terms into tumor and immune cells interaction systems become more realistic to understand the scenarios with better compatibility. In this paper, we consider a tumor–immune model with diffusion and nonlinear functional response and investigate the effect of diffusion on the existence of non-constant positive steady states and the steady-state bifurcations. By using the

In a bounded smooth domain \(\Omega \subset {\mathbb {R}}^{2}\) and with a positive parameter \(\chi >0\), we consider the chemotaxis system $$\begin{aligned} \left\{ \begin{array}{ll} u_t=\Delta u -\chi \nabla \cdot (u \nabla v), \\ v_t=\Delta v +w-v-uv, \\ w_t=\Delta w-uw + w(1-w), \end{array} \right. \end{aligned}$$ for modeling the interactions between tumor and immune cells. It is shown that for

This work concerns the study of the effective balance equations governing linear elastic electrostrictive composites, where mechanical strains can be observed due to the application of a given electric field in the so-called small strain and moderate electric field regime. The formulation is developed in the framework of the active elastic composites. The latter are defined as composite materials constitutively

We address the stability of solutions corresponding to weakly damped two-layered Timoshenko beams with space variable coefficients. The mathematical model is composed of a system of coupled wave equations and describes the slip effect produced by a thin adhesive layer joining the beams. For the considered PDE system, combined with different boundary conditions, the main result is that the decay rates

In this paper, we establish some \(\varepsilon \)-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier–Stokes equations as follows: $$\begin{aligned}&\limsup \limits _{\varrho \rightarrow 0} \varrho ^{1-\frac{2}{p}-\sum \limits ^{3}_{j=1}\frac{1}{q_{j}}} \Vert u\Vert _{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(\varrho ))} \le \varepsilon , ~~\frac{2}{p}+\sum \limits

In this paper, we prove the global existence and uniqueness of the strong solution to the high-dimensional inhomogeneous incompressible Boussinesq equations with zero thermal diffusion.

This paper aims to prove that the three-dimensional periodic Burgers equation has a unique global in time solution, in the Lebesgue–Gevrey space. In particular, the initial data that belong to \(L_{a,\sigma }^{2}(\mathbb {T}^3)\) give rise to a solution in \(C(\mathbb {R}_+;L_{a,\sigma }^{2}(\mathbb {T}^3))\cap L^2(\mathbb {R}_+;H_{a,\sigma }^{1}(\mathbb {T}^3))\), where \(L_{a,\sigma }^2\) is identified

In this paper, we discuss an initial value problem for the semilinear time-fractional diffusion equation. The local well-posedness (existence and regularity) is presented when the source term satisfies a global Lipschitz condition. The unique continuation of solution and finite time blowup result are presented when the reaction terms are logarithmic functions (local Lipschitz types).

Plane and anti-plane dynamic problems for an elastic rectangle loaded along its sides are considered. Low-frequency perturbations to rigid body translations are calculated. The derivation involves the solutions of non-homogeneous boundary value problems for harmonic and bi-harmonic equations. The explicit solution for the harmonic problem for transverse anti-plane translation is expressed through Fourier

A nonlinear vorticity equation describing the behavior of a viscous incompressible fluid on a rotating sphere is considered. The viscosity term is modeled by a real degree of the Laplace operator. The smoothness of external forcing is established that guarantee the existence of a limited attractive set in the space of solutions. Theorems on the existence and uniqueness of non-stationary and stationary

We consider a fluid–structure interaction model for an incompressible fluid where the elastic response of the free boundary is given by a damped Kirchhoff plate model. Utilizing the Newton polygon approach, we first prove maximal regularity in \(L^p\)-Sobolev spaces for a linearized version. Based on this, we show existence and uniqueness of the strong solution of the nonlinear system for small data

Two triangular factorizations of the deformation gradient tensor are studied. The first, termed the Lagrangian formulation, consists of an upper-triangular stretch premultiplied by a rotation tensor. The second, termed the Eulerian formulation, consists of a lower-triangular stretch postmultiplied by a different rotation tensor. The corresponding stretch tensors are denoted as the Lagrangian and Eulerian

In hypoelastic constitutive models, an objective stress rate is related to the rate of deformation through an elasticity tensor. The Truesdell, Jaumann, and Green–Naghdi rates of the Cauchy and Kirchhoff stress tensors are examples of the objective stress rates. The finite element analysis software ABAQUS uses a co-rotational frame which is based on the Jaumann rate for solid elements and on the Green–Naghdi

We introduce a new model of the nonlocal wave equation with a logarithmic damping mechanism, which is rather weak as compared with frequently studied fractional damping cases. We consider the Cauchy problem for the new model in \(\mathbf{R}^{n}\) and study the asymptotic profile and optimal decay rates of solutions as \(t \rightarrow \infty \) in \(L^{2}\)-sense. The damping terms considered in this

This paper deals with the global stability of the following density-suppressed motility system $$\begin{aligned} \left\{ \begin{array}{ll} u_{t}=\Delta (\varphi (v)u), &{} x\in \Omega ,\quad t>0,\\ v_{t} =\Delta v+wz, &{} x\in \Omega ,\quad t>0,\\ w_{t}=-wz, &{} x\in \Omega ,\quad t>0,\\ z_{t}=\Delta z-z+u, &{} x\in \Omega ,\quad t>0 \end{array} \right. \end{aligned}$$ in a bounded domain \(\Omega

We consider a nonlinear elliptic Dirichlet problem driven by the anisotropic (p, q)-Laplacian and with a reaction which is nonparametric and has the combined effects of a singular and of a superlinear terms. Using variational tools together with truncation and comparison techniques, we show that the problem has at least two positive smooth solutions.

In this article, we focus on the inverse scattering transform for the Gerdjikov–Ivanov equation with nonzero boundary at infinity. An appropriate two-sheeted Riemann surface is introduced to map the original spectral parameter k into a single-valued parameter z. Based on the Lax pair of the Gerdjikov–Ivanov equation, we derive its Jost solutions with nonzero boundary. Further asymptotic behaviors,

We investigate a forager–exploiter model in a high-dimensional smooth bounded domain with zero-flux Neumann boundary condition: $$\begin{aligned} \begin{aligned} \left\{ \begin{array}{cc} {\displaystyle u_{t}=\Delta u-\chi _{1}\nabla \cdot (u\nabla w),}&{}{}\quad \,\,\,\,x\in \Omega ,\,t>0,\\ {\displaystyle v_{t}=\Delta v-\chi _{2}\nabla \cdot (v\nabla u),}&{}{}\quad \,\,\,\,x\in \Omega ,\,t>0,\\ w_{t}=\mathrm

In this paper, we consider a kinetic description of follow-the-leader traffic models, which we use to study the effect of vehicle-wise driver-assist control strategies at various scales, from that of the local traffic up to that of the macroscopic stream of vehicles. We provide theoretical evidence of the fact that some typical control strategies, such as the alignment of the speeds and the optimisation

In this paper, we study a Lotka–Volterra cooperative system with nonlocal dispersal under worsening habitats. By constructing appropriate vector super-/subsolutions combined with the monotone iteration scheme, we obtain the existence of bounded and positive forced waves connecting zero equilibrium to the coexistence state of the limiting system corresponding to the most favorable resource with the

The object of the present paper is to consider the Dirichlet boundary value problem of the coupled linear quasi-static theory of elasticity for porous isotropic elastic infinite strip. The general representations of a regular solutions of a system of considered equations for a homogeneous isotropic medium are constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions

In this paper, we are concerned with the existence and concentration of ground state solutions for the following nonlinear Schrödinger–Poisson-type system with doubly critical growth $$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^2\Delta u+V(x)u-\phi |u|^3u=|u|^4u+f(u),&{}\mathrm{in}\; \mathbb {R}^3,\\ -\varepsilon ^2\Delta \phi =|u|^5,&{}\mathrm{in}\; \mathbb {R}^3, \end{array}\right.

In this paper, we show that solutions of the cubic nonlinear Schrödinger equation are asymptotic limit of solutions to the Benney system. Due to the special characteristic of the one-dimensional transport equation, same result is obtained for solutions of the one-dimensional Zakharov and 1d-Zakharov–Rubenchik systems. Convergence is reached in the topology \(L^2({\mathbb R})\times L^2({\mathbb R})\)

We consider pure traction problems, and we show that incompressible linearized elasticity can be obtained as variational limit of incompressible finite elasticity under suitable conditions on external loads.

We study a reaction–diffusion–advection susceptible–infected–susceptible (SIS) epidemic model with saturated incidence rate and spontaneous infection. The existence of endemic equilibrium is established. We further analyze the effects of diffusion, advection, saturation and spontaneous infection on the asymptotic behavior of endemic equilibrium. In particular it is shown that advection may cause the

In this paper, we consider the following magnetic nonlinear Choquard equation $$\begin{aligned} -(\nabla +iA(x))^2u+ V(x)u = \left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2} u + \lambda f(u)\ \text { in }\ \mathbb {R}^N, \end{aligned}$$ where \(2_{\alpha }^{*}=\frac{2N-\alpha }{N-2}\) is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, \(\lambda

This paper reports the possibilities of predicting buckling of thin shells with nondestructive techniques during operation. It examines shallow shells fabricated from high-strength materials. Such structures are known to exhibit surface displacements exceeding the thickness of the elements. In the explored shells, relaxation oscillations of significant amplitude could be generated even under relatively

In this article, the authors investigate the non-autonomous magneto-micropolar fluids in a two-dimensional bounded domain. They first prove the existence of a pullback attractor for the associated process. Then, they construct a family of invariant Borel probability measures supported on the pullback attractor and prove that this family of probability measures is indeed a statistical solution for the

The Poisson–Boltzmann (P–B) equation is of fundamental importance in understanding solid–liquid electrolyte interfaces that are present in many fields. Due to the nonlinearity, it is usually challenging to find the explicit exact solutions of the P–B equation. The present work reports several perturbation solutions for the nonlinear P–B equation in the Cartesian and spherical coordinates. The new solutions

In this article, we study the stabilization problem for a hyperbolic-type Stokes system posed on a bounded domain. We show that when the damping effects are restricted to a subdomain satisfying the geometric control condition, the energy of the system decays exponentially. The result is a consequence of a new quasi-mode estimate for the Stokes system.

Stationary incompressible Newtonian fluid flow governed by external force and external pressure is considered in a thin rough pipe. The transversal size of the pipe is assumed to be of the order \(\varepsilon \), i.e., cross-sectional area is about \(\varepsilon ^{2}\), and the wavelength in longitudinal direction is modeled by a small parameter \(\mu \). Under general assumption \(\varepsilon ,\mu

This paper aims to clarify the homogenization results of the fluid–structure interaction in porous structures under the quasi-static and dynamic loading regimes. In the latter case, the acoustic fluctuations yield naturally a linear model which can be introduced in the configuration deformed as the consequence of the steady permanent flow. We consider a Newtonian slightly compressible fluid under the

We prove the existence and uniqueness of global strong solutions to the Maxwell–Navier–Stokes system in a two dimensional bounded domain with Navier-type boundary condition for the velocity.

Various non-local structural derivative diffusion models have been proposed based on different kernel functions to describe the anomalous time dependence of the mean-squared displacements. In the present study, the fundamental solutions for constant and variable-order structural derivative advection–dispersion models are achieved via scaling transformation and the generalized non-Euclidean Hausdorff

In this paper, we establish the global well-posedness of solutions for the magnetohydrodynamics with Hall and ion-slip effects in critical space by constructing the current function \(J=\nabla \times B\) as an additional unknown. Meanwhile, we consider the long-time behavior of the solutions and get some decay estimates.

This paper considers a one-dimensional generalized Allen–Cahn equation of the form $$\begin{aligned} u_t = \varepsilon ^2 (D(u)u_x)_x - f(u), \end{aligned}$$ where \(\varepsilon > 0\) is constant, \(D = D(u)\) is a positive, uniformly bounded below, diffusivity coefficient that depends on the phase field u, and f(u) is a reaction function that can be derived from a double-well potential with minima

This paper deals with the solution of two-species chemotaxis system $$\begin{aligned} \left\{ \begin{array}{llll} u_t=d_1\Delta u-\chi _1\nabla \cdot (u\nabla w)+\mu _{1}u(1-u-a_{1}v),&{}\quad x\in \Omega ,\quad t>0,\\ v_t=d_2\Delta v-\chi _2\nabla \cdot (v\nabla w)+\mu _{2}v(1-a_{2}u-v),&{}\quad x\in \Omega ,\quad t>0,\\ 0=d_3\Delta w-\gamma w+\alpha u+\beta v&{}\quad x\in \Omega ,\quad t>0 \end{array}

The evolution of a cloud of particles in a compressible fluid can be modeled with a Vlasov–Fokker–Planck equation for the distribution function of the particles coupled with Navier–Stokes or Euler equations for the density and velocity of the fluid. Formal calculations have established the convergence of solution to the mesoscopic model to solutions to the macroscopic Navier–Stokes or Euler model coupled

In this paper we consider the following quasilinear Schrödinger–Poisson system $$\begin{aligned} \left\{ \begin{array}[c]{ll} - \Delta u +u+\phi u = \lambda f(x,u)+|u|^{4}u &{}\ \text{ in } {\mathbb {R}}^{3} \\ -\Delta \phi -\varepsilon ^{4} \Delta _4 \phi = u^{2} &{} \ \text{ in } {\mathbb {R}}^{3}, \end{array} \right. \end{aligned}$$ depending on the two parameters \(\lambda ,\varepsilon >0\). We

This work studies the existence of positive solutions for fractional Schrödinger equations of the form $$\begin{aligned} (-\Delta )^s u + V(x) u = g(u)+\lambda |u|^{q-2}u\quad \text{ in }\quad {\mathbb {R}}^N, \end{aligned}$$ where \(s\in (0,1)\), \(N>2s\), \(V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is a potential function which can vanish at infinity, \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\)