On the low regularity of the Benney-Lin equation

We consider the low regularity of the Benney-Lin equation ut+uux+uxxx+β(uxx+uxxxx)+ηuxxxxx=0. We established the global well posedness for the initial value problem of Benney-Lin equation in the Sobolev spaces Hs(R) for 0?s>−2, improving the well-posedness result of Biagioni and Linares [H.A. Biaginoi, F. Linares, On the Benney-Lin and Kawahara equation, J. Math. Anal. Appl. 211 (1997) 131-152]. For s<−2 we also prove some ill-posedness issues.     

Well-posedness of the Cauchy problem for Ostrovsky, Stepanyams and Tsimring equation with low regularity data

In this paper we prove that the initial value problem of the OST equation ut+uxxx+η(Hux+Huxxx)+uux=0 (x∈R, t?0), where η>0 and H denotes the usual Hilbert transformation, is locally well-posed in the Sobolev space Hs(R) when , and globally well-posed in Hs(R) when s?0.     

On the global Cauchy problem for the Hartree equation with rapidly decaying initial data

This paper is concerned with the Cauchy problem for the Hartree equation on ${\mathbb{R}}^n,n\in \mathbb{N}$ with the nonlinearity of type $(|?{|}^{?\gamma }?|u{|}^2)u,0<\gamma <n$. It is shown that a global solution with some twisted persistence property exists for data in the space $L^p\cap L^2,1\le p\le 2$ under some suitable conditions on γ and spatial dimension $n\in \mathbb{N}$. It is also shown that the global solution u has a smoothing effect in terms of spatial integrability in the sense that the map $t?u(t)$ is well defined and continuous from $\mathbb{R}?\{0\}$ to $L^{p^{\prime }}$, which is well known for the solution to the corresponding linear Schrödinger equation. Local and global well-posedness results for hat $L^p$-spaces are also presented. The local and global results are proved by combining arguments by Carles–Mouzaoui with a new functional framework introduced by Zhou. Furthermore, it is also shown that the global results can be improved via generalized dispersive estimates in the case of one space dimension.     

On low regularity of the Ostrovsky, Stepanyams and Tsimring equation

In this paper we prove, by using the Fourier restriction norm method, that the initial value problem of the Ostrovsky, Stepanyams and Tsimring equation ut+uxxx+η(Hux+Huxxx)+uux=0 (x∈R, t?0), where η>0 and H denotes the usual Hilbert transformation, is locally well-posed in the Sobolev space Hs(R) for any , which improve our former result in Zhao and Cui (2009) [5].     

Local solvability of the cauchy problem of a fifth-order nonlinear dispersive equation

The solvability of the fifth-order nonlinear dispersive equation δtu＋au （δxu）^2＋βδx^3u＋γδx^5u = 0 is studied. By using the approach of Kenig, Ponce and Vega and some Strichartz estimates for the corresponding linear problem,it is proved that if the initial function u0 belongs to H^5（R） and s〉1/4,then the Cauchy problem has a unique solution in C（[-T,T],H^5（R）） for some T〉0.     

A note on initial value problem for the generalized Tricomi equation in a mixed-type domain

In this paper, we study the well-posedness of initial value problem for n-dimensional gener-alized Tricomi equation in the mixed-type domain {(t,x):t∈[1,+∞),x∈Rn} with the initial data given on the line t=1 in Hadamard's sense. By taking partial Fourier transformation, we obtain the explicit expression of the solution in terms of two integral operators and further establish the global estimate of such a solution for a class of initial data and source term. Finally, we establish the global solution in time direction for a semilinear problem used the estimate.     

Uniqueness and stability in an inverse problem for a Poisson's equation

Consider the Poisson's equation(?)″(x)=-e~(v-(?)) e~((?)-v)-N(x)with the Diriehlet boundary data,and we mainly investigate the inverse problem of determining the unknown function N(x)from a parameter function family.Some uniqueness and stability results in the inverse problem are obtained.     

Uniqueness and stability in an inverse problem for a Poisson’s equation

Consider the Poisson's equation ψ" (x) ＝ -ev-ψ eψ-v-N(x) with the Dirichlet boundary data, and we mainly investigate the inverse problem of determining the unknown function N(x) from a parameter function family. Some uniqueness and stability results in the inverse problem are obtained.     

The cauchy problem for a nonlinear integro-differential transport equation

In the present paper, we study the Cauchy problem for a nonlinear time-dependent kinetic neutrino transport equation. We prove the existence and uniqueness theorem for the solution of the Cauchy problem, establish uniform bounds int for the solution of this problem, and prove the existence and uniqueness of a stationary trajectory and the stabilization ast→∞ of the solution of the time-dependent problem for arbitrary initial data. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 677–686, May, 1997. Translated by A. M. Chebotarev     

On the optimization of the initial boundary value problem for a conservation law

In the present note, the theory of shift differentiability for the Cauchy problem is extended to the case of an initial boundary value problem for a conservation law. This result allows to exhibit an Euler-Lagrange equation to be satisfied by the extrema of integral functionals defined on the solutions of initial boundary value problems of this kind.     

The initial boundary value problem for quasi-linear wave equation with viscous damping

In this paper, the existence and uniqueness of the local generalized solution and the local classical solution for the initial boundary value problem of the quasi-linear wave equation with viscous damping are proved. The nonexistence of the global solution for this problem is discussed by an ordinary differential inequality. Finally, an example is given.     

The first initial boundary value problem for a compound type equation in a three-dimensional multiply connected region

The method of boundary integral equations is used for solving the first initial boundary value problem for a compound type equation in a three-dimensional multiply connected region. The problem is reduced to a uniquely solvable integral equation. The solution of the problem is obtained in the form of dynamic potentials whose density satisfies this integral equation. Thus the existence theorem is proved. Moreover, the uniqueness of the solution is also studied. All the results are valid for both interior and exterior regions provided that the corresponding conditions at infinity are taken into account. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 249–265, August, 2000.     

The heat equation with singular nonlinearity and singular initial data

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut−Δu=a(x)uq+b(x)up in a bounded domain and with Dirichlet's condition on the boundary. We consider here a∈Lα(Ω), b∈Lβ(Ω) and 0<q?1<p. The initial data u(0)=u₀ is considered in the space Lr(Ω), r?1. In the main result (0<q<1), we assume a,b?0 a.e. in Ω and we assume that u₀?γdΩ for some γ>0. We find a unique solution in the space .     

On a cauchy problem with subharmonic initial values

Summary Investigation of the Euler-Poisson-Darboux equation for special initial values. Generalized transformation ofTricomi-Cibrario. Extension of classical theorems ofF. Riesz andP. Montel on mean values of subharmonic functions. This research was supported in part by the United States Air Force under Contract AF 18(600)573 — monitored by the Office of Scientific Research, Air Research and Development Command.     

On the cauchy problem for a class of degenerate differential equations with Lipschitz right-hand side

This note deals with an application of the fixed point theorem for multivalued contraction mappings to the study of degenerate differential equations with Lipschitz right-hand side and with degeneration determined by a closed surjective linear operator.