Consideration in the present paper is a weakly dissipative shallow water equation. The parameters take different values, which include several different important shallow water equations, such as CH equation, DP equation, Novikov equation and so on. The wave-breaking phenomena are investigated by three different kinds of method. Due to the presence of high-order nonlinear terms \(u^{2n+1}\) and \(u^{2m}u_{xx}\)

In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution (u, b) becomes regular provided that one velocity and one current density component of the solution satisfy $$\begin{aligned} u_{3}\in L^{\frac{30\alpha }{7\alpha -45}}\left( 0,T;L^{\alpha ,\infty }\left( {\mathbb {R}}^{3}\right) \right) \quad \text { with }\frac{45}{7}

We introduce a one-dimensional stress-rate type nonlinear viscoelastic model for solids that obey the assumptions of the strain-limiting theory. Unlike the classical viscoelasticity theory, the critical hypothesis in the present strain-limiting theory is that the linearized strain depends nonlinearly on the stress and the stress rate. We show the thermodynamic consistency of the model using the complementary

The final purpose of this paper is to show that, by inserting a convexity constraint on the cables of a suspension bridge, the torsional instability of the deck appears at lower energy thresholds. Since this constraint is suggested by the behavior of real cables, this model appears more reliable than the classical ones. Moreover, it has the advantage to reduce to two the number of degrees of freedom

We explore the efficiency of a thermoelectric energy converter constituted by a Si/Ge nanowire of length L. A constitutive equation of thermal conductivity as function of composition and temperature is derived in accordance with experimental data obtained at the constant temperatures \(T=300\,\hbox {K}\), \(T=400\,\hbox {K}\), and \(T=500\,\hbox {K}\) by a nonlinear regression method. A thermodynamic

We consider the large time behavior of weak solutions to the 3D magneto-micropolar fluid system. We establish the rate of decay \(\bigl \Vert \;\!({{\varvec{u}}},{{\varvec{w}}}, {{\varvec{b}}})(\cdot ,t)\,\!\bigr \Vert _{{{\varvec{L}}}^{2}({\mathbb {R}}^3)} \,\le \,C \;\!(t + 1)^{-\;\!\frac{3}{4}}\) for all \(t \ge 0\) through Fourier splitting method. Furthermore, we use a direct method to show how

We are concerned with the Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \rho _t+u\cdot \nabla \rho =\Delta \rho -\nabla \cdot (\rho \mathcal {S}(x,\rho ,c)\nabla c)-\rho m, &{}\quad (x,t)\in \varOmega \times (0,T),\\ m_t+u\cdot \nabla m=\Delta m-\rho m, &{}\quad (x,t)\in \varOmega \times (0,T),\\ c_t+u\cdot \nabla c=\Delta c-c+m, &{}\quad (x,t)\in \varOmega

In this paper, we establish a small data global well-posedness results for 3D incompressible MHD equations with Hall and ion-slip effects. Then, by using Fourier splitting method and iteration, we consider the decay rate on higher order derivative of solutions.

We establish a large deviation principle for the 2D stochastic Cahn–Hilliard–Navier–Stokes equations driven by multiplicative noise in a bounded domain. This system consists of the Navier–Stokes equations for the velocity, coupled with a Cahn–Hilliard model for the order parameter. The proof is completed using a weak convergence approach based on the variational representational of functional of infinite-dimensional

Conjugation of reciprocal transformations is used to solve a class of boundary value problems involving a source term relevant to water transport through a heterogeneous medium with a volumetric extraction mechanism. The main emphasis is the solution method that involves conjugation of reciprocal transformations, as well as other changes of variable, applied to a newly identified integrable model.

In this paper, we formulate a theory for the coupling of accretion mechanics and thermoelasticity. We present an analytical formulation of the thermoelastic accretion of an infinite cylinder and of a two-dimensional block. We develop numerical schemes for the solution of these two problems, and numerically calculate residual stresses and observe a strong dependence of the final mechanical state on

This paper is concerned with a barotropic model of capillary compressible fluids derived by Dunn and Serrin (1985). We consider the initial boundary value problem in bounded domains of \({\mathbb {R}}^d (d=2,3)\) with more general pressure including Van der Waals equation of state. First, the local in time existence and uniqueness of strong solutions is proved for initial data belonging to the Sobolev

It is known that in the case that several constitutive tensors fail to be positive definite the system of the thermoelasticity could become unstable and, in certain cases, ill-posed in the sense of Hadamard. In this paper, we consider the Moore–Gibson–Thompson thermoelasticity in the case that some of the constitutive tensors fail to be positive and we will prove basic results concerning uniqueness

We prove that the internal stresses inside two hyperbolic elastic inhomogeneities can indeed remain uniform when the matrix is subjected to uniform remote anti-plane and in-plane stresses. Conditions leading to internal uniform stresses are established rigorously for both anti-plane and plane elasticity. Once these conditions are satisfied, the internal uniform stresses inside the two hyperbolic inhomogeneities

We investigate a time-harmonic wave problem in a waveguide. We work at low frequency so that only one mode can propagate. It is known that the scattering matrix exhibits a rapid variation for real frequencies in a vicinity of a complex resonance located close to the real axis. This is the so-called Fano resonance phenomenon. And when the geometry presents certain properties of symmetry, there are two

We study the collapse of a many-body system which is used to model two-component Bose–Einstein condensates with attractive intra-species interactions and either attractive or repulsive inter-species interactions. Such a system consists a mixture of two different species for N identical bosons in \({\mathbb {R}}^2\), interacting with potentials rescaled in the mean-field manner \(-N^{2\beta -1}w^{(\sigma

We consider the system with \(\rho \in {\mathbb {R}}, \mu> 0, \chi > 0\) in a bounded domain \(\Omega \subseteq {\mathbb {R}}^2\) with smooth boundary. While very similar to chemotaxis models from biology, this system is in fact inspired by recent modeling approaches in criminology to analyze the formation of crime hot spots in cities. The key addition here in comparison with similar models is the

We study the Cauchy problem for an inhomogeneous Gross–Pitaevskii equation. We first derive a sharp threshold for global existence and blowup of the solution. Then, we construct and classify finite time blowup solutions at the minimal mass threshold. Additionally, using variational techniques, we study the existence, the orbital stability and instability of standing waves.

We consider the vertical motion of a free falling ball bouncing elastically on a racket moving in the vertical direction according to a regular periodic function f. We give a sufficient condition on the second derivative of f giving motions with arbitrarily large amplitude and chaotic dynamics in the sense of positive entropy. We get the results by breaking many invariant curves of the corresponding

In this challenging analytic survey, we present a new approach for solving equations of the dynamics of a charged particle (describing its motion in variable, time-dependent electromagnetic field) which has been applied earlier in various fields of mechanics for solving equations of hydrodynamics, Euler–Poisson equations of rigid body rotation and even in celestial mechanics (solution of equations

We analyze the stability of Maxwell equations in bounded domains taking into account electric and magnetization effects. Well-posedness of the model is obtained by means of semigroup theory. A passitivity assumption guarantees the boundedness of the associated semigroup. Further the exponential or polynomial decay of the energy is proved under suitable sufficient conditions. Finally, several illustrative

We study a nonlinear elliptic system with prescribed inner interface conditions. These models are frequently used in physical system where the ion transfer plays the important role, for example, in modeling of nano-layer growth or Li-on batteries. The key difficulty of the model consists of the rapid or very slow growth of nonlinearity in the constitutive equation inside the domain or on the interface

We compute the natural frequencies for the oscillations of the free boundary of gravity waves in contact with a solid container. First, we study the case of a semicircular shaped container. We deduce an integrodifferential evolutionary equation for the linearized free boundary and impose pinned-end and free-end boundary conditions. For both cases, the natural oscillations frequencies for the free surfaces

Scalar wave propagation across a semi-infinite step or step-like discontinuity on any one boundary of the square lattice waveguides is considered within nearest-neighbour interaction approximation. An application of the Wiener–Hopf method does yield an exact solution of the discrete scattering problem, using which, as the main result of the paper, the transmission coefficients for energy flux are obtained

In this paper, global well-posedness of the non-Markovian Unruh–Zurek and Hu–Paz–Zhang master equations with nonlinear electrostatic coupling is demonstrated. They both consist of a Wigner–Poisson like equation subjected to a dissipative Fokker–Planck mechanism with time-dependent coefficients of integral type, which makes necessary to take into account the full history of the open quantum system under

In this paper, we are concerned with existence and behavior of positive solution to the class of nonlinear elliptic problems with discontinuous nonlinearities of the type $$\begin{aligned} \left\{ \begin{array}{l} -\Delta u+V(x)u = H(u-\beta )f(x,u) \quad \text {in}\,\, {\mathbb {R}}^{N},\\ u\in D^{1,2}({\mathbb {R}}^{N})\cap W_{\hbox {loc}}^{2,2}({\mathbb {R}}^{N}), \end{array} \right. \qquad {(P)_{\beta

The purpose of this work is to investigate the existence and stability of monostable traveling wavefronts for a Lotka–Volterra competition model with partially nonlocal interactions. We first establish an innovative lemma for the existence of positive solutions to a system of linear inequalities. By this lemma, we can construct a pair of sub-super-solutions and derive the existence result by applying

In this paper, a chemotaxis model with bounded chemotactic sensitivity and signal absorption is considered under homogeneous Neumann boundary conditions in the ball \(\Omega =B_R(0)\subset {\mathbb {R}}^n\), where \(R>0\) and \(n\ge 2\). Here, S is a scalar function with \(S(s,t)\in C^2([0,\infty )\times [0,\infty ))\). Moreover, for some positive constant K, \(|S(s,t)|\le K\) for all \(s,t\in [0,\infty

Using the embedded gradient vector field method, we explicitly compute the set of critical points of the shear-stretch strain energy of a Cosserat body model. We also formulate necessary and sufficient conditions for critical points in the abstract case of the special orthogonal group \(\text {SO}(n)\). Each critical point is then characterized using an explicit formula for the Hessian operator of

Radiative heat transfer across nanoscale-thick layers attracts considerable attention because of its importance for modern technology, and also because of the evidence that conventional methods of radiative heat transfer fail at such small scales. This paper analyzes several approaches to this problem that have been proposed since the late 1960s when the first adaptations of the classical Stefan–Boltzmann

Damageable elastic buckling beams with a crack motivate to formulate a differential equation system which can be extended to higher-dimensional problems. Elastic buckling bodies are assumed to evolve their damage and break apart from an edge crack. A nonlinear variational formulation is derived that is equivalent to a partial differential equation, subject to crack conditions and boundary conditions

The goal of this paper is to study the system of rotating fluids between two infinite parallel plates with Dirichlet boundary conditions and with small viscosity which vanishes when the Rossby number goes to zero. We want to improve the convergence result of [18] and show the global in time convergence of the weak solution of the system of rotating fluids toward the solution of a two-dimensional damped

This paper is concerned with a time-periodic and nonlocal system arising from the spread of a deterministic epidemic in multi-types of population by incorporating a seasonal variation. The existence of the critical wave speed of the periodic traveling wavefronts and its coincidence with the spreading speed were proved in Wu et al. (J Math Anal Appl 463:111–133, 2018). In this paper, we prove the uniqueness

We study a parabolic–parabolic chemotactic PDE’s system which describes the evolution of a biological population “u” and a chemical substance “v” in a two-dimensional bounded domain with regular boundary. We consider a growth term of logistic type in the equation of “u” in the form \(u (1-u+f(x,t))\), for a given bounded function “f” which tends to a periodic in time function independent of x when

It is shown that singularity will be developed in finite time for the original (without symmetry) multidimensional compressible Euler equations for generalized Chaplygin gas if the initial momentum is sufficiently strong and an initial averaged quantity is nonnegative.