In this paper, we prove the exponential stability of the solution of the nonlinear dissipative Schrödinger equation on a star-shaped network \(\mathcal {R}\), where the damping is localized on one branch at the infinity and the initial data are assumed to be in \(L^{2}(\mathcal {R})\). We use the fixed point argument and Strichartz estimates on a star-shaped network to obtain results of local and global

In this note, we show that in some cases, via the use of Hardy-type inequality, there is a non-trivial nonnegative \(W^{1,p}(R^n)\) weak solution to quasi-linear elliptic problem with the p-Laplacian on \(R^n\) and with Hardy-type singularity term. We also study the behavior of solutions to the Poisson equation of p-Laplacian on the whole space and this Poisson equation has a close relationship with

In this paper, we study the multiplicity and concentration of positive solutions for the following (p, q)-Laplacian problem: $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p} u -\Delta _{q} u +V(\varepsilon x) \left( |u|^{p-2}u + |u|^{q-2}u\right) = f(u) &{} \text{ in } {\mathbb {R}}^{N}, \\ u\in W^{1, p}({\mathbb {R}}^{N})\cap W^{1, q}({\mathbb {R}}^{N}), \quad u>0 \text{ in } {\mathbb {R}}^{N}

The generalized Navier–Stokes–Landau–Lifshitz equations are considered in this paper. The well-posedness for the multi-dimensional hyperviscous Navier–Stokes–Landau–Lifshitz equations is proved for \(n\ge 3\). The existence and uniqueness of the strong solutions for the generalized Navier–Stokes–Landau–Lifshitz equations are proved for \(\frac{5}{4}\le \alpha <\frac{5}{2}\) and \(\frac{5}{4}\le \beta

In this paper, we study an elliptic obstacle problem with a generalized fractional Laplacian and a multivalued operator which is described by a generalized gradient. Under quite general assumptions on the data, we employ a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping to prove that the set

This paper analyzes prey–predator models with indirect predator-taxis in such a way that chemical secreted by the predator triggers the repellent behavior of prey against the predator. Under the assumption of quadratic decay of predator, we prove the global existence and uniform boundedness of classical solutions up to two spatial dimensions. Moreover, via the linear stability analysis, we show that

In the present article, we investigate the thermal and elastic behaviour of an infinite thermoelastic material with a cylindrical cavity caused by a time-dependent moving heat source. The cavity surface is assumed to be subjected to a thermal shock. The formulation of the problem is applied for the recently proposed generalized thermoelastic model [Modified Green–Lindsay (MGL)] that takes into account

In this paper, we investigate the initial value problem for the 3D magneto-micropolar fluid equations with mixed partial viscosity. The main purpose of this paper is to establish global well-posedness of classical small solutions. More precisely, we prove that the global stability of perturbations near the steady solution is given by a background magnetic field. The proof is mainly based on the energy

Due to a variety of applications of nanoscaled materials, several researchers further investigate a joining between two nanostructures as a candidate for new potential applications. Here, the vertically joining between the nanotube with the nano-torus is investigated. Variational calculus is used to predict the joining curve between two nanostructures based on minimizing the elastic energy. Moreover

Let \(\Omega \subset {\mathbb {R}}^3\) be a sheared waveguide, i.e., \(\Omega \) is built by translating a cross section in a constant direction along an unbounded spatial curve. Consider \(-\Delta _{\Omega }^D\) the Dirichlet Laplacian operator in \(\Omega \). Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of \(-\Delta

In this paper, we consider the signal-dependent diffusion and sensitivity in a chemotaxis–competition population system with two different signals in a two-dimensional bounded domain. We consider more general signal production functions and assume that the signal-dependent diffusion is a decreasing function which may be degenerate with respect to the density of the corresponding signal. We first obtain

It is proved that if the solution of the Navier–Stokes system satisfies $$\begin{aligned} \partial _3\varvec{u}\in L^p(0,T;L^q(\mathbb {R}^3)),\quad \frac{2}{p}+\frac{3}{q} =\frac{22}{13}+\frac{3}{13q},\quad 3

In this paper, we consider the classical solutions to a model of two-dimensional incompressible inviscid Boussinesq–Bénard equations. Notice that, in the case when the source term of temperature equation in this model is the second component of velocity \(u_2\) or no source term, there is no global-in-time existence result for the general initial data. Here, if the source term is only chosen as \(\Delta

In this paper, we study the following class of fractional Choquard-type equations $$\begin{aligned} (-\Delta )^{1/2}u + u=\Big ( I_\mu *F(u)\Big )f(u), \quad x\in \mathbb {R}, \end{aligned}$$ where \((-\Delta )^{1/2}\) denotes the 1/2-Laplacian operator, \(I_{\mu }\) is the Riesz potential with \(0<\mu <1\), and F is the primitive function of f. We use variational methods and minimax estimates to study

The article is devoted to the study of Cauchy problem to the Rayleigh–Bénard convection model for the micropolar fluid in two dimensions. We first prove the unique local solvability of smooth solution to the system when the system has only velocity dissipation, and then establish a criterion for the breakdown of smooth solutions imposed only the maximum norm of the gradient of scalar temperature field

This paper is concerned with the periodic problem to the two-fluid non-isentropic Euler–Maxwell (N-E-M) equations. The equations arises in the modeling of magnetic plasma, in which appear two physical parameters, the mass of an electron \(m_\mathrm{e}\) and the mass of an ion \(m_{\mathrm{i}}\). With the help of methods of asymptotic expansions, we prove the local-in-time convergence of smooth solutions

This paper proposes a nonlocal formulation regarding the modeling of Antarctic Circumpolar Current by introducing flow functions to encode horizontal flow components without considering vertical motion. Using topological degree, zero exponent theory and fixed point technique, we show the existence of positive solutions to nonlocal boundary value problems with nonlinear vorticity.

Fully dynamic system of equations for a single piezoelectric beam strongly couples the mechanical (longitudinal) vibrations with the total charge distribution across the beam. Unlike the electrostatic (or quasi-static) assumption of Maxwell’s equations, the hyperbolic-type charge equations have been recently shown to affect the stabilizability of the high-frequency vibrational modes if one considers

In this paper, we study the dynamics of the FitzHugh–Nagumo system \(\dot{x}=z,\;\dot{y}=b\left( x-dy\right) ,\;\dot{z}=x\left( x-1\right) \left( x-a\right) +y+cz\) having invariant algebraic surfaces. This system has four different types of invariant algebraic surfaces. The dynamics of the FitzHugh–Nagumo system having two of these classes of invariant algebraic surfaces have been characterized in

This paper is concerned with the doubly tactic model $$\begin{aligned} \left\{ \begin{array}{ll} u_t=\Delta u-\chi _u\nabla \cdot (u\nabla w)+uw,&{} x\in \Omega , t>0, \\ v_t=\Delta v-\chi _v\nabla \cdot (v\nabla u)+vw, &{}x\in \Omega , t>0, \\ w_t=\Delta w-\lambda (u+v)w-\mu w,&{} x\in \Omega , t>0, \end{array}\right. \end{aligned}$$ in a smoothly bounded domain \(\Omega \subset {\mathbb {R}}^N\)

This paper is concerned with the following memory-type Timoshenko system $$\begin{aligned} {\left\{ \begin{array}{ll} \rho _1 \varphi _{tt}-K(\varphi _x+\psi )_x =0,\\ \rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+\displaystyle \int \limits _0^{+\infty } g(s)\psi _{xx}(t-s){\mathrm{d}}s=0,\\ \end{array}\right. } \end{aligned}$$ with Dirichlet boundary conditions, where g is a positive nonincreasing

In this paper, we propose a new viscosity for a coupled compressible Navier–Stokes/Allen–Cahn system because it describes the motion of a gas in a flowing liquid. The viscosity depends on two different variables (the density and the unknown function in Allen–Cahn equations). We establish blow-up criterions for strong solutions to initial-boundary value problem. We also show that the strong solutions

The Cauchy problem for a quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations of electrons in a cold plasma is considered. For some simplified formulation of the problem, a criterion for the existence of a global in time solutions is obtained. For the original problem, a sufficient condition for blow-up is found, as well as a sufficient condition for

We theoretically investigate the deformation of a perfect dielectric drop suspended in a second dielectric liquid subject to a uniform electric field. Axisymmetric equilibrium shapes are found by solving simultaneously the Young–Laplace equation at the interface and Laplace equation for the electric field. Analytical solutions are constructed for the governing nonlinear boundary-value problem using

In the context of linear theories of generalized elasticity including those for homogeneous micropolar media, quasicrystals, and piezoelectric and piezomagnetic media, we explore the concept of null Lagrangians. For obtaining the family of null Lagrangians, we employ the sufficient conditions of H. Rund. In some cases, a nonzero null Lagrangian is found and the stored energy admits a split into a null

In this paper, we are concerned with existence, uniqueness, regularity, and continuous dependence upon the initial data for mild solutions of an equation of viscoelasticity in power type materials in the \(L^q\)-setting. We also analyze the continuation of this solution up a maximal time of existence and a blow-up alternative. Finally, we obtain the global well-posedness for the problem.

This paper shows time-asymptotic nonlinear stability and existence of the viscous shock profiles to the Cauchy problem of the one-dimensional compressible planar magnetohydrodynamics system, which describes the motions of a conducting fluid in an electro-magnetohydrodynamics system. We can prove that the solutions to the compressible planar magnetohydrodynamics system tend time-asymptotically to the

We demonstrate the existence of solutions to Signorini’s problem for the Timoshenko’s beam by using a hybrid disturbance. This disturbance enables the use of semigroup theory to show the existence and asymptotic stability. We show that stability is exponential, when the waves speed of propagation is equal. When the waves speed is different, we show that the solution decays polynomially. This result

The correct analysis of heat transport at nanoscale is one of the main reasons of new developments in physics and nonequilibrium thermodynamic theories beyond the classical Fourier law. In this paper, we provide a two-temperature model which allows to describe the different regimes which electrons and phonons can undergo in the heat transfer phenomenon. The physical admissibility of that model is showed

We investigate mathematical properties of the system of nonlinear partial differential equations that describe, under certain simplifying assumptions, evolutionary processes in water-saturated granular materials. The unconsolidated solid matrix behaves as an ideal plastic material before the activation takes place and then it starts to flow as a Newtonian or a generalized Newtonian fluid. The plastic

In this paper, we study a reaction–diffusion problem in a thin domain with varying order of thickness. Motivated by the applications, we assume the oscillating behavior of the boundary and prescribe the Robin-type boundary condition simulating the reaction catalyzed by the upper wall. Using the appropriate functional setting and the unfolding operator method, we rigorously derive lower-dimensional

In this paper, we prove the uniform regularity of the compressible full Navier–Stokes–Maxwell system in \(\mathbb {T}^3\).

The results obtained in this article aim at analyzing Bogdanov–Takens bifurcation in a predator–prey model with an age structure for the predator. Firstly, we give the existence result of the Bogdanov–Takens singularity. Then we describe the bifurcation behavior of the parameterized predator–prey model with Bogdanov–Takens singularity.

In this paper, we study the local and global existence of solution and the blow-up phenomena for a class of heat equation involving the p(x)-Laplacian with triple regime.

In mathematics and physics, the problem of the stability of perturbations near the hydrostatic balance is very important. Due to the classical tools designed for the fully dissipated systems are no longer apply, stability and global regularity problems on partially dissipated magnetic Bénard fluid equations can be extremely challenging. This paper considers the stability problem on perturbations near

We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the dynamics capillarity equation $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u_{\varepsilon ,\delta } +\mathrm {div} {\mathfrak f}_{\varepsilon ,\delta }(\mathbf{x}, u_{\varepsilon ,\delta })=\varepsilon \Delta u_{\varepsilon ,\delta }+\delta (\varepsilon ) \partial _t \Delta u_{\varepsilon ,\delta

We consider a swelling porous-elastic system with a single memory term as the only damping source. The coupling gives new contributions to the theory associated with asymptotic behaviors of swelling porous elastic soils. Unexpectedly, using the multiplier method, we establish a general decay result irrespective of the wave speeds of the system.

We derive the formulas that describe the exact solution of the boundary value problem in the theory of elasticity for a rectangle in which two opposite (horizontal) sides are free and stresses are specified (all cases of symmetry relative to the central axes) on the other two sides (rectangle ends). The formulas for a half-strip are also given. The solutions are represented as series in Papkovich–Fadle

In this paper, we consider a viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term. Under suitable assumptions on relaxation functions, we establish general decay result for energy.

We consider the existence of spatially localized traveling wave solutions of the mass-in-mass lattice. Under an anti-resonance condition first discovered by Kevrekidis, Stefanov and Xu, we prove that such solutions exist in two distinguished limits; the first where the mass of the internal resonator is small and the second where the internal spring is very stiff. We then numerically simulate the solutions

In this paper, the geometric dislocation density tensor and Burgers vector are studied using an elastic–plastic decomposition of Laplace stretch \(\varvec{\mathcal {U}}\). The Laplace stretch arises from a \(\mathbf {QR}\) decomposition of the deformation gradient and is very useful, as one can directly and unambiguously measure its components by performing experiments. The geometric dislocation density

This work establishes the existence of a unique global mild solution for the 3D generalized magnetohydrodynamics equations in Lei–Lin–Gevrey and Lei–Lin spaces; provided that the initial data are assumed to be small enough.

This paper is devoted to investigating the wave propagation and its stability for a class of two-component discrete diffusive systems. We first establish the existence of positive monotone monostable traveling wave fronts. Then, applying the techniques of weighted energy method and the comparison principle, we show that all solutions of the Cauchy problem for the discrete diffusive systems converge

In this article, the non-linear anti-symmetric shear motion for some comparative studies between the non-homogeneous and homogeneous plates, having two free surfaces with stress-free, is considered. Assuming that one plate contains hyper-elastic, non-homogeneous, isotropic, and generalized neo-Hookean materials and the other one consists of hyper-elastic, homogeneous, isotropic, and generalized neo-Hookean

Existence and uniqueness of a solution to the nonstationary Navier–Stokes equations having a prescribed flow rate (flux) in the infinite cylinder \(\Pi =\{x=(x^\prime , x_n)\in {{\mathbb {R}}}^n:\; x^\prime \in \sigma \subset {{\mathbb {R}}}^{n-1},\; -\infty

In this manuscript, we deal with an equation involving a combination of quasi-linear elliptic operators of local and non-local nature with p-structure, and concave–convex nonlinearities. The prototypical model is given by $$\begin{aligned} \left\{ \begin{array}{rclcl} -\Delta _p u + (-\Delta )^s_p u &{} = &{} \lambda _p u^q(x) + u^r(x) &{} \text{ in } &{} \Omega , \\ u(x)&{}>&{}0&{}\text{ in }&{} \Omega

In this paper, we study the blowup of smooth solutions to the compressible Euler equations with radial symmetry on some fixed bounded domains (\(B_{R}=\{x\in {\mathbb {R}}^{N}:\ |x|\le R\}\), \(N=1,2,\ldots \)) by introducing some new averaged quantities. We consider two types of flows: initially move inward and initially move outward on average. For the flow initially moving inward on average, we

In this paper, we are concerned with an SIRS epidemic reaction–diffusion system with standard incidence infection mechanism in a spatially heterogeneous environment. We first establish the uniform bounds of solutions and then derive the threshold dynamics in terms of the basic reproduction number \(\mathcal {R}_0\). Our main focus is on the asymptotic profile of endemic equilibria (when exists) if

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.