LOCAL EXISTENCE OF SOLUTION TO FREE BOUNDARY VALUE PROBLEM FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS

This paper is concerned with the free boundary value problem for multidimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local （in time） existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.

LOCAL EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO THE TWO DIMENSIONAL NONHOMOGENEOUS INCOMPRESSIBLE PRIMITIVE EQUATIONS

In this article,we study the initial boundary value problem of the two-dimensional nonhomogeneous incompressible primitive equations and obtain the local existence and uniqueness of strong solutions.The initial vacuum is allowed.

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS

In this paper,we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations-(ε^(2)a+εb∫_(R^(3))|■u|^(2))△u+V(x)u=u^(p),u>0 in R^(3),which concentrate at non-degenerate critical points of the potential function V(x),where a,b>0,1<p<5 are constants,andε>0 is a parameter.Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity,we establish the existence and local uniqueness results of multi-peak solutions,which concentrate at{a_(i)}1≤i≤k,where{a_(i)}1≤i≤k are non-degenerate critical points of V(x)asε→0.

Local Existence for Inhomogeneous Schrdinger Flow into Kihler Manifolds

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the inhomogeneous Schrdinger flow for maps from a compact Riemannian manifold M with dim(M)≤3 into a compact Khler manifold(N, J)with nonpositive Riemannian sectional curvature

On Local Existence and Blow-up of a Moving Boundary Problem in 1-D Chemotaxis Model

In this paper, the local existence and uniqueness of a chemotaxis model with a moving boundary are considered by the contraction mapping principle, and the explicit expression for the moving boundary is formulated. In addition, the finite-time blowup and chemotactic collapse of the solution for such kind of problem are discussed.

具有奇异势和一般非线性的伪抛物方程解的局部存在性和爆破

In this paper,a semilinear pseudo-parabolic equation with a general nonlin-earity and singular potential is considered.We prove the local existence of solution by Galerkin method and contraction mapping theorem.Moreover,we prove the blow-up of solution and estimate the upper bound of the blow-up time for J(u0)≤0.Finally,we prove the finite time blow-up and estimate the upper bound of blow-up time for J(u0)>0.

LOCAL WELL-POSEDNESS OF STRONG SOLUTIONS FOR THE NONHOMOGENEOUS MHD EQUATIONS WITH A SLIP BOUNDARY CONDITIONS

This article is concerned with the 3 D nonhomogeneous incompressible magnetohydrodynamics equations with a slip boundary conditions in bounded domain.We obtain weighted estimates of the velocity and magnetic field,and address the issue of local existence and uniqueness of strong solutions with the weaker initial data which contains vacuum states.

LOCAL CLASSICAL SOLUTION OF FREE BOUNDARY PROBLEM FOR A COUPLED SYSTEM

This paper considers a two-phase free boundary problem for coupled system including one parabolic equation and two elliptic equations. The problem comes from the discussion of a growth model of self-maintaining protocell in multidimensional case. The local classical solution of the problem with free boundary (?) y = g(x,t) between two domains is being seeked. The local existence and uniqueness of the problem will be proved in multidimensional case.

Existence for Semilinear Wave Equations with Low Regularity

In this paper, we study how much regularity of initial data is needed to ensure existence of a local solution to the following semilinear wave equations utt-△u=F（u,Du） u（0,x）=f（x）∈H^s,δtu（0,x）=g（x）∈H^s-1,where F is quadratic in Du with D = （δr, δx1,…, δxn）.We proved that the range of s is s ≥n＋1/2 ＋ δ, respectively, with δ 〉 1/4 if n = 2, and δ 〉 0 if n = 3, and δ ≥0 if n ≥ 4. Which is consistent with Lindblad＇s counterexamples [3] for n = 3, and the main ingredient is the use of the Strichartz estimates and the refinement of these.

Global Existence and Decay of Solution to Parabolic-Parabolic Keller-Segel Model in R^(d)

We consider the global existence and decay of integral solutions to the parabolic-parabolic Keller-Segel system in d-dimension.On the one hand,by Banach fixed point theorem and some properties of heat kernel,we prove the local existence and the global existence of integral solutions for the different initial data under some conditions that involve the size of the initial data.On the other hand,in the case of global solutions,we obtain their optimal time decay by Gronwall’s lemma.

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.

Local and Global Existence of Solutions to Initial Value Problems of Modified Nonlinear Kawahara Equations

This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third ＋α the second partial dervative of u to x ,the second the third ＋β the third partial dervative of u to x ,the second the thire ＋γ the fifth partial dervative of u to x = 0,（x,t）∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo（x） ∈ H^s（R） with s ≥ 1/4, and a global solution exists if s ≥ 2.

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.

SOLVABILITY OF A PARABOLIC-HYPERBOLIC TYPE CHEMOTAXIS SYSTEM IN 1-DIMENSIONAL DOMAIN

In this paper, we use contraction mapping principle, operator-theoretic approach and some uniform estimates to establish local solvability of the parabolic-hyperbolic type chemotaxis system with fixed boundary in 1-dimensional domain. In addition, local solvability of the free boundary problem is considered by straightening the free boundary.

On Existence of Local Solutions of a Moving Boundary Problem Modelling Chemotaxis in 1-D

we prove the local existence and uniqueness of a moving boundary prob- lem modeling chemotactic phenomena. We also get the explicit representative for the moving boundary in a special case.

LOCAL AND GLOBAL EXISTENCE OF SOLUTIONS OF THE GINZBURG-LANDAU TYPE EQUATIONS

This paper is devoted to studying the initial value problem of the Ginzburg-Landau type equations. We treat the case where the nonlinear interaction function is a general continuous function, not required to satisfy any smoothness conditions. Local and global existence results of solutions of the problem are given. Decay estimates are also shown.

A Dirichlet Inhomogenous Boundary Value Problem for 1D Nonlinear Schrödinger Equation

Pure initial value problems for important nonlinear evolution equations such as nonlinear Schrödinger equation (NLS) and the Ginzburg-Landau equation (GL) have been extensively studied. However, many applications in physics lead to mathematical models where boundary data is inhomogeneous, e.g. in radio frequency wave experiments. In this paper, we investigate the mixed initial-boundary condition problem for the nonlinear Schrödinger equation iut = uxx – g|u|p-1u, g ∈R, p > 3 on a semi-infinite strip. The equation satisfies an initial condition and Dirichlet boundary conditions. We utilize semi-group theory to prove existence and uniqueness theorem of a strong local solution.

Local Schrodinger flow into Kahler manifolds

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the Schrodinger flow for maps from a compact Riemannian manifold into a complete Kahler manifold, or from a Euclidean space Rm into a compact Kahler manifold. As a consequence, we prove that Heisenberg spin system is locally well-posed in the appropriate Sobolev spaces.

The Regularity of Local Solutions for a Generalized Camassa-Holm Type Equation

The regularity of the Cauchy problem for a generalized Camassa-Holm type equation is investigated. The pseudoparabolic regularization approach is employed to obtain some prior estimates under certain assumptions on the initial value of the equation. The local existence of its solution in Sobolev space Hs （R） with 1 〈 s ≤ 3/2 is derived.

Blow-up of Solutions for a Time-space Fractional Evolution System

In this paper, firstly, we study the local existence and uniqueness of mild solutions for fractional evolution systems with nonlocal in time nonlinearity. Then, we claim that such a mild solution is weak solution of this system. Finally, we prove a blow-up result under some conditions.