THE GLOBAL EXISTENCE OF STRONG SOLUTIONS TO THERMOMECHANICAL CUCKER-SMALE-STOKES EQUATIONS IN THE WHOLE DOMAIN

We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale(for short,TCS) model coupled with Stokes equations in the whole space.The coupled system consists of the kinetic TCS equation for a particle ensemble and the Stokes equations for a fluid via a drag force.In this paper,we present a complete analysis of the existence of global-in-time strong solutions to the coupled model without any smallness restrictions on the initial data.

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.

Global Existence of the Solution for a Reduced Model of the Vectorial Quantum Zakharov System

In this paper, we study the global existence of the smooth solution for a reduced quantum Zakharov system in two spatial dimensions. Using energy estimates and the logarithmic type Sobolev inequality, we show the global existence of the solution to this system without any small condition on the initial data.

THE EXISTENCE OF WEAK SOLUTIONS AND PROPAGATION REGULARITY FOR A GENERALIZED KDV SYSTEM

This paper examines the existence of weak solutions to a class of the high-order Korteweg-de Vries(KdV)system in Rn.We first prove,by the Leray-Schauder principle and the vanishing viscosity method,that any initial data N-dimensional vector value function u0(x)in Sobolev space H^(s)(R^(n))(s≥1)leads to a global weak solution.Second,we investigate some special regularity properties of solutions to the initial value problem associated with the KdV type system in R^(2)and R^(3).

GLOBAL SOLUTIONS IN THE CRITICAL SOBOLEV SPACE FOR THE LANDAU EQUATION

The Landau equation is studied for hard potential with-2≤γ≤1.Under a perturbation setting,a unique global solution of the Cauchy problem to the Landau equation is established in a critical Sobolev space H_(x)^(d)L_(v)^(2)(d>3/2),which extends the results of[11]in the torus domain to the whole space R_(x)^(3).Here we utilize the pseudo-differential calculus to derive our desired result.

一类退化抛物方程高初始能量下解的有限时刻爆破及整体存在性

We consider the initial-boundary value problem for finitely degenerate parabolic equation. We first give sufficient conditions for the blow-up and global existence of the parabolic equation at high initial energy level. Then, we establish the existence of solutions blowing up in finite time with initial data at arbitrary energy level. Finally, we estimate the upper bound of the blow-up time under certain conditions.

A Modified Transitional Korteweg-De Vries Equation: Posed in the Quarter Plane

This paper is concerned with a modified transitional Korteweg-de Vries equation ut+f(t)u2ux+uxxx=0, (x,t)∈R+×R+with initial value u(x,0)=g(x)∈H4(R+)and inhomogeneous boundary value u(0,t)=Q(t)∈C2([ 0,∞ )). Under the conditions either 1) f(t)≤0, f′(t)≥0or 2) f(t)≤−αwhere α>0, we prove the existence of a unique global classical solution.

Global Well-Posedness of the Fractional Tropical Climate Model

In this paper, we consider the Cauchy problem of 3-dimensional tropical climate model. This model reflects the interaction and coupling among the barotropic mode u, the first baroclinic mode v of the velocity and the temperature θ. The systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances. Mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization. The aim here is the global strong solution with the least dissipation. By energy estimate and delicate analysis, we prove the existence of global solution under three different cases: first, with the help of damping terms, the global strong solution of the system with Λ2au, Λ2βv and Λ2γ θ for;and second, the global strong solution of the system for with damping terms;finally, the global strong solution of the system for without any damping terms, which improve the known existence theory for this system.

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.

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.

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.

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.

ON THE GLOBAL EXISTENCE OF SMOOTH SOLUTIONS TO THE MULTI-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDING DAMPING IN HALF SPACE

This paper is a continue work of [4, 5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term --u/（1＋t）λpu, where λ≥ 0 and μ 〉 0 are constants. We have showed that, for all λ ≥ 0 andμ 〉 0 the smooth solution to the Cauchy problem exists globally or blows up in finite time. In the present paper, instead of the Cauchy problem we consider the initial- boundary value problem in the half space R＋^d with space dimension d = 2, 3. With the help of the special structure of the equations and the fluid vorticity, we overcome the difficulty arisen from the boundary effect. We prove that there exists a global smooth solution for 0 ≤λ 〈 1 when the initial data is close to its equilibrium state. In addition, exponential decay of the fluid vorticity will also be established.

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.

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.

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.

GLOBAL EXISTENCE AND EXPONENTIAL STABILITY OF SOLUTIONS TO THE QUASILINEAR THERMO-DIFFUSION EQUATIONS WITH SECOND SOUND

This article is devoted to the study of global existence and exponential stability of solutions to an initial-boundary value problem of the quasilinear thermo-diffusion equations with second sound by means of multiplicative techniques and energy method provided that the initial data are close to the equilibrium and the relaxation kernel is strongly positive definite and decays exponentially.

GLOBAL EXISTENCE OF SOLUTIONS FOR QUADRATIC QUASI-LINEAR KLEIN-GORDON SYSTEMS IN ONE SPACE DIMENSION

Consider quadratic quasi-linear Klein-Gordon systems with eventually different masses for small, smooth, compactly supported Cauchy data in one space dimension. It is proved that the global existence holds when a convenient null condition is satisfied by nonlinearities.