We prove the existence of solutions for some semilinear elliptic equations in the appropriate H^4 spaces using the fixed point technique where the elliptic equation contains fourth order differential operators with and without Fredholm property, generalizing the results of the preceding work [22].

A system of boundary-domain integral equations (BDIEs) is obtained from the Dirichlet problem for the diffusion equation in non-homogeneous media defined on an exterior two-dimensional domain. We use a parametrix different from the one employed by in . The system of BDIEs is formulated in terms of parametrix-based surface and volume potentials whose mapping properties are analysed in weighted Sobolev spaces. The system of BDIEs is shown to be equivalent to the original boundary value problem and uniquely solvable in appropriate weighted Sobolev spaces suitable for unbounded domains.

In the paper, we devote to broadening the current global regularity results for the two-dimensional magnetic B\’{e}nard fluid equations. We study three cases: (i) fractional Laplacian dissipation $(-\Delta)^\alpha{u}$, partial magnetic diffusion $(\partial_{x_2x_2}b_1,\partial_{x_1x_1}b_2)$ and Laplacian thermal diffusivity $\Delta\theta$; (ii) partial fractional dissipation $(\Lambda^{2\alpha}_{x_2}u_1,\Lambda^{2\alpha}_{x_1}u_2)$, partial magnetic diffusion $(\partial_{x_2x_2}b_1,\partial_{x_1x_1}b_2)$ and Laplacian thermal diffusivity $\Delta\theta$; (iii) partial fractional magnetic diffusion $(\Lambda^{2\beta}_{x_2}b_1,\Lambda^{2\beta}_{x_1}b_2)$, Laplacian thermal diffusivity $\Delta\theta$ and without Laplacian dissipation $\Delta{u}$ (i.e., $\mu=0$)), and establish the global regularity for each cases.

In this paper we consider a class of chemotaxis models with two arbitrary constitutive functions g(u) and f(v). After having performed a complete symmetry group classification with respect to them the reduced systems are derived. By considering g(u) of the logistic form wide classes of exact solutions are found.

This paper analyzes a SIS model for infectious diseases with two classes of individuals with different susceptibilities. It focuses in a transition function between both classes of susceptible individuals depending on the density of the infected population. A classification of all the possible bifurcation diagrams that the model can present is done. Specifically, some conditions for the simultaneous existence of backward bifurcation and multiple endemic states are shown.

In this paper, we define a non-iterative transformation method for an Extended Blasius Problem. The original non-iterative transformation method, which is based on scaling invariance properties, was defined for the classical Blasius problem by T\”opfer in 1912. This method allows us to solve numerically a boundary value problem by solving a related initial value problem and then rescaling the obtained numerical solution. In recent years, we have seen applications of the non-iterative transformation method to several problems of interest. The obtained numerical results are improved by both a mesh refinement strategy and Richardson’s extrapolation technique. In this way, we can be confident that the computed six decimal places are correct.

in this work, we introduce an efficient meshless technique for solving the two-dimensional variable-order time-fractional mobile/immobile advection-diffusion model with Dirichlet boundary conditions. The main advantage of this scheme is to obtain a global approximation for this problem which reduces such problems to a system of algebraic equations. To approximate the first and fractional variable-order against the time, we use the finite difference relations. The proposed method is based on the Moving Kriging (MK) interpolation shape functions. To discretization this model in space variables, we use the MK interpolation. Duo to the fact that the shape functions of MK have Kronecker’s delta property, boundary conditions are imposed directly and easily. To illustrate the capability of the proposed technique on regular and irregular domains, several examples are presented in different kinds of domains and with uniform and nonuniform nodes. Also, we use this scheme to simulating anomalous contaminant diffusion in underground reservoirs.

The current work is devoted for operating the Lie symmetry approach, to coupled complex short pulse equation. The method reduces the coupled complex short pulse equation to a system of ordinary differential equations with the help of suitable similarity transformations. Consequently, these systems of nonlinear ordinary differential equations under each subalgeras are solved for traveling wave solutions. Further, with the help of similarity variable, similarity solutions and traveling wave solutions of nonlinear ordinary differential equation, complex soliton solutions of the coupled complex short pulse equation are obtained which are in form of sinh, cosh, sin and cos functions.

The fifth order short pulse equation models the nonlinear propagation of optical pulses of a few oscillations duration in dielectric media. In particular, it models the propagation of circularly and elliptically polarized few-cycle solitons in a Kerr medium. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem associated with this equation.

Using monotonicity methods, the Lagrange multiplier rule and some variational arguments, we consider a type of localization results pertaining to the existence of critical points to action functionals on a closed ball. A variant of the Schechter critical point theorem on a ball in Hilbert and Banach spaces is obtained. Applications to nonlinear Dirichlet problem and to partial difference equations are given in the final part of this paper.

In the paper, we consider a three-dimensional model of fluid-solid interaction when a thermo-electro-magneto-elastic body occupying a bounded region Ω+ is embedded in an inviscid fluid occupying an unbounded domain $\Omega^{-}=^3 \setminus $. In this case, we have a six-dimensional thermo-electro-magneto-elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function) in the domain Ω+, while we have a scalar acoustic pressure field in the unbounded domain Ω−. The physical kinematic and dynamic relations are described mathematically by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, we prove the uniqueness and existence theorems for the corresponding boundary-transmission problems in appropriate Sobolev-Slobodetskii and Hölder continuous function spaces.

In the paper, we consider the local-in-time and the global-in-time infinity-ion-mass convergence of bipolar Euler-Maxwell systems by setting the mass of an electron me=1 and letting the mass of an ion mi→∞. We use the method of asymptotic expansions to handle the local-in-time convergence problem and find that the limiting process from bipolar models to unipolar models is actually decoupling, but not the vanishing of equations for the corresponding the other particle. Moreover, when the initial data is sufficiently close to the constant equilibrium state, we establish the global-in-time infinity-ion-mass convergence.

In the present paper initial problem for the integro-differential diffusion system with nonlocal nonlinear source is considered. The results on existence of local mild solution and nonexistence of global weak solution to the nonlinear integro-differential diffusion system are presented.

The purpose of this paper is to study the fluid-structure interaction (FSI) problem which is a simplified model to describe high frequency and small displacement oscillation of elastic structure in fluids. The elastic structure displacement is modeled by a fourth order nonlinear hyperbolic square equations, the motion of fluid is modeled by the time-dependent incompressible Navier-Stokes equations. we prove the existence of at least one weak solutions (global in time) to this problem by compactness method. The result both holds for two-dimensional and three-dimensional cases.

A document by Gongbao LI. Click on the document to view its contents.

This article focuses on studying mild solutions of an original frac- tional partial differential equation disturbed by multiplicative white noise. We employ techniques of semi group theory, Hausdorff measure, and Darbo xed point theorem.

In this paper, we consider an electromagnetic obstacle scattering problem in a piecewise homogeneous medium. We assume that the incident field is the electromagnetic plane wave and the obstacle consists of metallic and nonmetallic parts. We firstly establish uniqueness and existence for the solution to the electromagnetic obstacle scattering problem. Then we prove that the electric far field pattern satisfies the reciprocity relation and the set of electric far field patterns is complete in a Hilbert space. These results have an important bearing on the solution of the inverse electromagnetic scattering problem i.e. determination of the shape of the obstacle from the knowledge of the electric far field patterns.

The lower bound decay rate of global solution to the compressible viscous quantum magnetohydrodynamic model in three-dimensional whole space under the $H^5\times H^4\times H^4$ framework is investigated in this paper. We firstly show that the lower bound of decay rate for the density, velocity and magnetic field converging to the equilibrium state (1,0,0) in $L^2$-norm is $(1+t)^{-\frac{3}{4}}$ when the initial data satisfies some low frequency assumption. Moreover, we prove that the lower bound of decay rate of $k(k\in [1,3])$ order spatial derivative for the density, velocity and magnetic field converging to the equilibrium state (1,0,0) in $L^2$-norm is $(1+t)^{-\frac{3+2k}{4}}$. Then we show that the lower bound of decay rate for the time derivatives of density and velocity converging to zero in $L^2$-norm is $(1+t)^{-\frac{5}{4}}$, but the lower bound of decay rate for the time derivative of magnetic field converging to zero in $L^2$-norm is $(1+t)^{-\frac{7}{4}}$.