LOCAL WELL-POSEDNESS AND ILL-POSEDNESS ON THE EQUATION OF TYPE □u= u~k(u)~α

This paper undertakes a systematic treatment of the low regularity local well-posedness and ill-posedness theory in H3 and Hs for semilinear wave equations with polynomial nonlinearity in u and (?)u. This ill-posed result concerns the focusing type equations with nonlinearity on u and (?)tu.     

ON THE EQUATION □φ=｜↓△φ｜^2 IN FOUR SPACE DIMENSIONS

This paper considers the following Cauchy problem for semilinear wave equations in n space dimensions □φ=F(δφ),φ(0,x)=f(x),δtφ(0,x)=g(x),whte □=δt^2-△ is the wave operator,F is quadratic in δεφ with δ=(δt,δx1,…,δxn).The minimal value of s is determined such that the above Cauchy problem is locally wellposed in H^s.It turns out that for the general equation s must satisfy s>max(n/2,n+5/4).This is due to Ponce and Sideris (when n=3)and Tataru (when n≥5).The purpose of this paper is to supplement with a proof in the case n=2,4.     

WELL-POSEDNESS FOR THE CAUCHY PROBLEM TO THE HIROTA EQUATION IN SOBOLEV SPACES OF NEGATIVE INDICES

The local well-posedness of the Cauchy problem for the Hirota equation is established for low regularity data in Sobolev spaces Hs(s≥-1/4). Moreover, the global well-posedness for L2 data follows from the local well-posedness and the conserved quantity. For data in Hs(s > 0), the global well-posedness is also proved. The main idea is to use the generalized trilinear estimates, associated with the Fourier restriction norm method.     

ON THE BOUNDED AND UNBOUNDED SOLUTIONS OF ONE DIMENSIONAL NONLINEAR REACTION－DIFFUSION PROBLEM

ＯＮＴＨＥＢＯＵＮＤＥＤＡＮＤＵＮＢＯＵＮＤＥＤＳＯＬＵＴＩＯＮＳＯＦＯＮＥＤＩＭＥＮＳＩＯＮＡＬＮＯＮＬＩＮＥＡＲＲＥＡＣＴＩＯＮ－ＤＩＦＦＵＳＩＯＮＰＲＯＢＬＥＭ￥ＧＥＷＥＩＧＡＯＲ．Ｏ．ＷＥＢＥＲＡｂｓｔｒａｃｔ：Ｔｈｅｅｘｉｓｔｅｎｃｅｏｆｂ...     

THE ZEROS AND ORDER OF MEROMORPHIC SOLUTIONS OF f^（k）＋Bf=H（z）

1.IntroductionandResultsConsidernon-homogeneouslineardifferentialequationsoftheform1.Lameprovedin[7]TheoremA.LetB(z),PO(z),PI(z)*06epolynomialssuchthatdegB=n21,degPO=p<(n k)/kandH=PI(z)epo('),then(a)IfdegPI相似文献   

OSCILLATORY AND ASYMPTOTIC BEHAVIOR OF A CLASS OF FIRST ORDER NEUTRAL DIFFERENTIAL EQUATION WITH PIECEWISE CONSTANT DELAY

The oscillatory and asymptotic behavior of a class of first order nonlinear neutral differential equation with piecewise constant delay and with diverse deviating arguments are considered. We prove that all solutions of the equation are nonoscillatory and give sufficient criteria for asymptotic behavior of nonoscillatory solutions of equation.     

EXISTENCE OF POSITIVE SOLUTIONS FOR A CLASS OF SEMILINEAR ELLIPTIC EQUATION WITH CONVEX AND CONCAVE NONLINEARITIES

In this paper, we consider the existence of positive solutions of the semlinear elliptic boundary value problem with convex and concave nonlinearities, the results obtained improve and generalize some known results.     

（1.2） INVERSES OF OPERATORS BETWEEN BANACH SPACES AND LOCAL CONJUGACY THEOREM

1.(1.2)InverseandLocalFinePropertyofaFamilyofOperatorsTxLetEandFbebothBanachspaces,andB(E,F)thesetofallboundedlinearoperatorsfromEintoFAnoperatorT B(F,E)issaidtobea(1.2)inverseofTifTT T=TandT TT =T .IfT satisfiesonlythefirstcondition,thenT issaidtobe...     

LOCAL WELL-POSEDNESS AND ILL-POSEDNESS ON THE EQUATION OF TYPE □u = uk((e)u)α

This paper undertakes a systematic treatment of the low regularity local wellposedness and ill-posedness theory in Hs andHs for semilinear wave equations with polynomial nonlinearity in u and (e)u. This ill-posed result concerns the focusing type equations with nonlinearity on u and (e)tu.     

ON THE EQUATION $\square\phi =|\nabla \phi |^2$ IN FOUR SPACE DIMENSIONS

This paper considers the following Cauchy problem for semilinear wave equations in $n$ space dimensions $$\align \square\p &=F(\partial\p ),\\p (0,x)&=f(x),\quad \partial_t\p (0,x)=g(x), \endalign$$ where $\square =\partial_t^2-\triangle$ is the wave operator, $F$ is quadratic in $\partial\p$ with $\partial =(\partial_t,\partial_{x_1},\cdots ,\partial_{x_n})$. The minimal value of $s$ is determined such that the above Cauchy problem is locally well-posed in $H^s$. It turns out that for the general equation $s$ must satisfy $$s>\max\Big(\frac{n}{2}, \frac{n+5}{4}\Big).$$ This is due to Ponce and Sideris (when $n=3$) and Tataru (when $n\ge 5$). The purpose of this paper is to supplement with a proof in the case $n=2,4$.     

ILL-POSEDNESS OF MODIFIED KAWAHARA EQUATION AND KAUP-KUPERSHMIDT EQUATION

In this article,we consider the Cauchy problems for the modified Kawahara equation (6)tu + μ(6)x(u3) + α(6)5xu + β(6)3xu + γ(6)xu =0and the Kaup-Kupershmidt equation (6)tu + μu(6)2xu + α(6)5xu + β(6)3x...     

ON THE EQUATION □φ = |vφ|2 IN FOUR SPACE DIMENSIONS

This paper considers the following Cauchy problem for semilinear wave equations in n spacedimensions□φ = F( φ),φ(0, x) = f(x), tφ(0, x) = g(x),The minimal value of s is determined such that the above Cauchy problem is locally well-posed in Hs. It turns out that for the general equation s must satisfyThis is due to Ponce and Sideris (when n = 3) and Tataru (when n ≥ 5). The purpose of thispaper is to supplement with a proof in the case n = 2, 4.     

关于Hirota-Satsuma系统初值问题的局部和整体适定性

本文利用ＫＤＶ方程所对应的线性方程解所具有的光滑效应及压缩映像原理,得到了Ｈｉｒｏｔａ－Ｓａｔｓｕｍａ系统初值问题的局部和整体适定性结果．     

AN IMPROVEMENT OF A RESULT OF IVOCHKINA AND LADYZHENSKAYA ON A TYPE OF PARABOLIC MONGE-AMP\`ERE EQUATION

For the initial-hotmdary value problem about a type of parabolic Monge-Ampere equation of the form (IBVP): {-Dtu + (deD^2xu)^1/n = f(x,t), (x,t) ∈ Q = Ω&#215;(0,T)}, u(x,t) =Ф(x,t)(x,t) ∈δpQ}, where Ω is a bounded convex domain in R^n, the result in [4] by Ivochkina and Ladyzheokaya is improved in the sense that, under assumptions that the data of the problempossess lower regularity and satisfy lower order compatibility conditions than than in [4], the existence of classical solution to (IBVP) is still established (see Theorem 1.1 below). This cannot be reallzed by only using the method in [4]. The main additional effort the authors have done is a kind of nonlinear perturbation.     

ON THE RELATIVE POSITION OF LIMIT CYCLES FOR THE EQUATION OF TYPE $\[{(II)_{l = 0}}\]$

In this paper, we consider the relative position of limit cycles for the system $$\[\begin{array}{*{20}{c}} {\frac{{dx}}{{dt}} = \delta x - y + mxy - {y^2}}\{\frac{{dy}}{{dt}} = x + a{x^2}} \end{array}\]$$ under the condition $$\[a < 0,0 < \delta \le m,m \le \frac{1}{a} - a\]$$ The main result is as follows: (i)Under Condition (2), if $\[\delta = \frac{m}{2} + \frac{{{m^2}}}{{4a}} \equiv {\delta _0}\]$, then system $\[{(1)_{{\delta _0}}}\] $ has no limit cycles and on singular closed trajectory through a saddle point in the whole plane, (ii)Under condition (2), the foci 0 and R'' cannot be surrounded by the limit cycles of system (1) simultaneously.