THE GLOBAL EXISTENCE AND UNIQUENESS OF SMOOTH SOLUTIONS TO A FLUID-PARTICLE INTERACTION MODEL IN THE FLOWING REGIME

This paper is concerned with the Cauchy problem for a 3D fluid-particle interaction model in the so-called flowing regime inℝ3.Under the smallness assumption on both the external potential and the initial perturbation of the stationary solution in some Sobolev spaces,the existence and uniqueness of global smooth solutions in H3 of the system are established by using the careful energy method.

Global Existence and Decay of Solutions for a Class of a Pseudo-Parabolic Equation with Singular Potential and Logarithmic Nonlocal Source

This article investigates the well posedness and asymptotic behavior of Neumann initial boundary value problems for a class of pseudo-parabolic equations with singular potential and logarithmic nonlinearity. By utilizing cut-off techniques and combining with the Faedo Galerkin approximation method, local solvability was established. Based on the potential well method and Hardy Sobolev inequality, derive the global existence of the solution. In addition, we also obtained the results of decay.

Existence and Uniqueness of Global Smooth Solutions for Vlasov Maxwell Equations

Global existence of classical solutions to the relativistic Vlasov-Maxwell system, given sufficiently regular initial data, is a long-standing open problem. The aim of this project is to present in details the results of a paper published in 1986 by Robert Glassey and Walter Strauss. In that paper, a sufficient condition for the global existence of a smooth solution to the relativistic Vlasov-Maxwell system is derived. In the following, the resulting theorem is proved by taking initial data , . A small data global existence result is presented as well.

GLOBAL EXISTENCE,UNIQUENESS,AND STABILITY FOR A NONLINEAR HYPERBOLIC-PARABOLIC PROBLEM IN PULSE COMBUSTION

A global existence theorem is established for an initial-boundary value prob- lem, with time-dependent boundary data, arising in a lumped parameter model of pulse combustion; the model in question gives rise to a nonlinear mixed hyperbolic-parabolic sys- tem. Using results previously established for the associated linear problem, a fixed point argument is employed to prove local existence for a regularized version of the nonlinear problem with artificial viscosity. Appropriate a-priori estimates are then derived which imply that the local existence result can be extended to a global existence theorem for the regularized problem. Finally, a different set of a priori estimates is generated which allows for taking the limit as the artificial viscosity parameter converges to zero; the corresponding solution of the regularized problem is then proven to converge to the unique solution of the initial-boundary value problem for the original, nonlinear, hyperbolic-parabolic system.

Global Existence, Uniqueness of Weak Solutions and Determining Functionals for Nonlinear Wave Equations

We consider the initial-boundary value problem for a nonlinear wave equation with strong structural damping and nonlinear source terms in IR. We prove the global existence and uniqueness of weak solutions of the problem and then we will study the determining modes on the phase space by using energy methods and the concept of the completeness defect.

Existence and Uniqueness of Global Solution for Certain Degenerate Reaction-Diffusion Systems

In this paper we discuss the following nonlinear degenerate parabolic systems u_i=△a_i(u_i)+b_i(x,t,u_i)Du_i+f_i(x,t,u)for i = 1,2, …, m and u = (u_1,…, u_m) is a vector function, with Dirichlet boundary condition. Under some structure conditions on a_i,b_i and f_i and initial data u_i^o∈Li(Ω) for some pi>p_i^o = 1,2,…,m, the result on existence and uniquence of global solution is established.

On existence and uniqueness of the solution of elastoplastic contact problems

Contact problems and elastoplastic problems are unified and described by the variational inequality formulation, in which the constraints of the constitutional relations for elastoplastic materials and the contact conditions are relaxed totally. First, the coerciveness of the functional is proved. Then the uniqueness of the solution of variational inequality for the elastoplastic contact problems is demonstrated. The existence of the solution is also demonstrated according to the sufficient conditions for the solution of the elliptic variational inequality. A mathematical foundation is developed for the variational extremum principle of elastoplastic contact problems. The developed variational extremum forms can give an effective and strict mathematical modeling to solve contact problems with mathematical programming.

GLOBAL EXISTENCE AND BLOW-UP PHENOMENA OF CLASSICAL SOLUTIONS FOR THE SYSTEM OF COMPRESSIBLE ADIABATIC FLOW THROUGH POROUS MEDIA

By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result on the global existence and the blow-up phenomena of classical solutions of these systems. These results show that the dissipation is strong enough to preserve the smoothness of ‘small ’ solution.

Novel Investigation of Stochastic Fractional Differential Equations Measles Model via the White Noise and Global Derivative Operator Depending on Mittag-Leffler Kernel

Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this paper,we constructed a stochastic fractional framework of measles spreading mechanisms with dual medication immunization considering the exponential decay and Mittag-Leffler kernels.In this approach,the overall population was separated into five cohorts.Furthermore,the descriptive behavior of the system was investigated,including prerequisites for the positivity of solutions,invariant domain of the solution,presence and stability of equilibrium points,and sensitivity analysis.We included a stochastic element in every cohort and employed linear growth and Lipschitz criteria to show the existence and uniqueness of solutions.Several numerical simulations for various fractional orders and randomization intensities are illustrated.

Global existence and blow up of a degenerate parabolic system

This paper deals with positive solutions of a degenerate parabolic system: u t= Δ u m+ v p ln α(h+u), v t= Δ v n+u q ln β(h+v) with homogeneous Dirichlet boundary conditions and positive initial conditions. This system describes the processes of diffusion of heat and burning in two component continuous media with nonlinear conductivity and volume energy release. We obtain the global existence and blow up results of the solution relying on comparison with carefully constructed upper solutions and lower solutions.

GLOBAL EXISTENCE, UNIFORM DECAY AND EXPONENTIAL GROWTH FOR A CLASS OF SEMI-LINEAR WAVE EQUATION WITH STRONG DAMPING

In this paper, we consider the nonlinearly damped semi-linear wave equation associated with initial and Dirichlet boundary conditions. We prove the existence of a local weak solution and introduce a family of potential wells and discuss the invariants and vacuum isolating behavior of solutions. Furthermore, we prove the global existence of solutions in both cases which are polynomial and exponential decay in the energy space respectively, and the asymptotic behavior of solutions for the cases of potential well family with 0 〈 E（0） 〈 d. At last we show that the energy will grow up as an exponential function as time goes to infinity, provided the initial data is large enough or E（0） 〈 0.

BLOW-UP AND GLOBAL EXISTENCE FOR A COUPLED SYSTEM OF DEGENERATE PARABOLIC EQUATIONS IN A BOUNDED DOMAIN

This article deals with the conditions that ensure the blow-up phenomenon or its absence for solutions of the system ut= △u^l ＋ u^p1v^q1 and vt = △v ^m ＋ u^p2 v^q2 with homogeneous Dirichlet boundary conditions. The results depend crucially on the sign of the difference p2q1 - （l -p1）（m- q2）, the initial data, and the domain Ω.

THE LOCAL AND GLOBAL EXISTENCE OF THE SOLUTIONS OF HYPERBOLIC-PARABOLIC SYSTEM MODELING BIOLOGICAL PHENOMENA

The authors prove the local existence and uniqueness of weak solution of a hyperbolic-parabolic system and establish the global existence of the weak solution for this system for the spatial dimension n = 1.

EXISTENCE AND UNIQUENESS FOR THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

In this paper, the existence and uniqueness of solutions for boundary valueproblem x′′′=f(t, x, x′, x″), x(0)=A, x′(0)=B, g(x′(1), x″(1))=0 are studied byusing Volterra type operator and upper and lower solutions. Our results improve someknown works.

GLOBAL EXISTENCE OF THE SOLUTION FOR THE LANDAU-LIFSHITZ EQUATION OF THE FERROMAGNETIC SPIN CHAIN

In this paper, we establish the global existence of the solution for the Landau-Lifshitz equation of the ferromagnetic spin chain. By Galerkin method, we first show the existence of the local solution for this equation, and then by a priori estimates method, we extend the local solution to a global solution.

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR THE 3D COMPRESSIBLE NON–ISENTROPIC EULER EQUATIONS WITH DAMPING

We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.

EXISTENCE, BOUNDEDNESS AND UNIQUENESS OF WEAK SOLUTION FOR THE THERMISTOR PROBLEM WITH MIXED BOUNDARV CONDITIONS

The thermistor problem is a coupled system of nonlinear PDEs with mixed boundary conditions. The goal of this paper is to study the existence, boundedness and uniqueness of the weak solution for this problem.

Global existence and decay of solutions for the generalized bad Boussinesq equation

In this paper, we consider the global existence of solutions for the Cauchy problem of the generalized sixth order bad Boussinesq equation. Moreover, we show that the supremum norm of the solution decays algebraically to zero as （1 ＋ t）-（1/7） when t approaches to infinity, provided the initial data are sufficiently small and regular.

Existence and Uniqueness of Solutions for Higher Order Nonlinear Multi-point Boundary Value Problems

In this paper, the existence and uniqueness theorems of solutions of k-point boundary value problems for nth-order nonlinear differential equations are established by Leray-Schauder continuation theorem.

ELASTIC MEMBRANE EQUATION WITH MEMORY TERM AND NONLINEAR BOUNDARY DAMPING:GLOBAL EXISTENCE,DECAY AND BLOWUP OF THE SOLUTION

In this paper we consider the Elastic membrane equation：with memory term and nonlinear boundary damping： Under some appropriate assumptions on the relaxation function h and with certain initial data, the global existence of solutions ：and a general decay for the energy are established using the multiplier technique. Also, ＇we show that a nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of a nonlinear damping.