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1.
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.  相似文献   

2.
The existence of generalized solution to the initial value problem iu_t △u k/(x_N)u_X_N q(x)u |u|~(p-1)u=0 on R~N is studied, By Galerkin method, we prove that the solution always exists for every initial value in H~1(R~N; k) if 1相似文献   

3.
We first prove that the Cauchy problem of the Kawahara equation, δtu + uδxu +βδx^3u+γδx^5u = 0, is locally solvable if the initial data belong to H^r(R) and r〉 r≥-7/5, thus improving the known local well-posedness result of this equation. Next we use this local result and the method of "almost conservation law" to prove that global solutions exist if the initial data belong to H^r(R) and r〉-1/2.  相似文献   

4.
In this paper,we consider the global well-posedness of smooth solutions for the Cauchy problem of a sixth order convective Cahn-Hilliard equation with small initial data.We first construct a local smooth solution,then by combining some a priori estimates,continuity argument,the local smooth solution is extended step by step to all t>0 provided that the L1 norm of initial data is suitably small and the smooth nonlinear functions f(u)and g(u)satisfy certain local growth conditions at some fixed point■.  相似文献   

5.
The local well-posedness of the Cauchy problem for the fifth order shallow water equation δtu+αδx^5u+βδx^3u+rδxu+μuδru=0,x,t∈R, is established for low regularity data in Sobolev spaces H^s(s≥-3/8) by the Fourier restriction norm method. Moreover, the global well-posedness for L^2 data follows from the local well-posedness and the conserved quantity. For data in H^s(s〉0), the global well-posedness is also proved, where the main idea is to use the generalized bilinear estimates associated with the Fourier restriction norm method to prove that the existence time of the solution only depends on the L^2 norm of initial data.  相似文献   

6.
The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u + 3 x u 5 x u + 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s (T) for s > 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.  相似文献   

7.
This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut+(u^2/2)x+px=εuxx, t〉0,x∈R, -αPxx+P=f(u)+α/2ux^2-1/2u^2, t〉0,x∈R, (E) with the initial data u(0,x)=u0(x)→u±, as x→±∞ (I) Here, u_ 〈 u+ are two constants and f(u) is a sufficiently smooth function satisfying f" (u) 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u+, the above Riemann problem admits a unique global entropy solution u^R(x/t) u^R(x/t)={u_,(f′)^-1(x/t),u+, x≤f′(u_)t, f′(u_)t≤x≤f′(u+)t, x≥f′(u+)t. Let U(t, x) be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0(x) - U(0,x) ∈ H^1(R) and u_ 〈 u+, the above Cauchy problem (E) and (I) admits a unique global classical solution u(t, x) which tends to the rarefaction wave u^R(x/t) as → +∞ in the maximum norm. The proof is given by an elementary energy method.  相似文献   

8.
This paper studies the uniqueness of generalized solutions for the first boundary value problem of the form where A_u≥0.It is proved that if u∈BV_x(Q_T) and A(u, x, t) is strictly increasing with respect to u, the solution is unique.  相似文献   

9.
In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u + λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k(Ω) in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞).  相似文献   

10.
We study the Dirichlet initial-boundary value problem of the generalized Kuramoto-Sivashinsky equation ut+uxxxx+λuxx+f(u)x=0 on the interval [0,l],The nonlinear function f satisfies the conditon |f′(u)|≤c|u|^α-1 for some α>1. We prove that if λ4π^2/t^2,then the strong solution is global and exponentially decays to zero for and initial datum uo∈H0^2(0,l) if 1<α≤7,and for small u0∈H0^2(0,l)if α>7,We the consider the equation ut+uxxxx+λuzz+μu+auxxx+bux=F(u,ux,uxx,uxxx),We prove that if F is twice differentiable,Δ↓F is Lipschitz continuous,and F(0)=Δ↓F(0)=0,and if λand μsatisfu μ+σ(λ)>0(σ(λ)=the first eigenvalue of the operator d^4/dx^4+λd^2/dx^2),then the solution for small initial datum is global and exponentially decays to zero.  相似文献   

11.
For the solutions of the elliptic equation
$ \sum\limits_{k = 0}^n {A_k \frac{{\partial ^n f}} {{\partial x^{n - k} \partial y^k }} = 0} $ \sum\limits_{k = 0}^n {A_k \frac{{\partial ^n f}} {{\partial x^{n - k} \partial y^k }} = 0}   相似文献   

12.
In this paper we apply the method of potentials for studying the Dirichlet and Neumann boundary-value problems for a B-elliptic equation in the form
$ \Delta _{x'} u + B_{x_{p - 1} } u + x_p^{ - \alpha } \frac{\partial } {{\partial x_p }}\left( {x_p^\alpha \frac{{\partial u}} {{\partial x_p }}} \right) = 0 $ \Delta _{x'} u + B_{x_{p - 1} } u + x_p^{ - \alpha } \frac{\partial } {{\partial x_p }}\left( {x_p^\alpha \frac{{\partial u}} {{\partial x_p }}} \right) = 0   相似文献   

13.
In this paper we discuss the fundamental solution of the Keldysh type operator $ L_\alpha u \triangleq \frac{{\partial ^2 u}} {{\partial x^2 }} + y\frac{{\partial ^2 u}} {{\partial y^2 }} + \alpha \frac{{\partial u}} {{\partial y}} $ L_\alpha u \triangleq \frac{{\partial ^2 u}} {{\partial x^2 }} + y\frac{{\partial ^2 u}} {{\partial y^2 }} + \alpha \frac{{\partial u}} {{\partial y}} , which is a basic mixed type operator different from the Tricomi operator. The fundamental solution of the Keldysh type operator with $ \alpha > - \frac{1} {2} $ \alpha > - \frac{1} {2} is obtained. It is shown that the fundamental solution for such an operator generally has stronger singularity than that for the Tricomi operator. Particularly, the fundamental solution of the Keldysh type operator with $ \alpha < \frac{1} {2} $ \alpha < \frac{1} {2} has to be defined by using the finite part of divergent integrals in the theory of distributions.  相似文献   

14.
Abstract In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions: {ut-a(u, v)△u=g(u, v), vt-b(u, v)△v=h(u, v), δu/δη=d(u, v), δu/δη=f(u, v).Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques.  相似文献   

15.
We consider the boundary value problem
. Hereu ∈ ℝ2,D = diag{d 1,d 2},d 1,d 2 > 0, and the functionF is jointly smooth in (u, μ) and satisfies the following condition: for 0 <μ ≪ 1 the boundary value problem has a homogeneous (independent ofx) cycle bifurcating from a loop of the separatrix of a saddle. We establish conditions for stability and instability of this cycle and give a geometric interpretation of these conditions. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 697–708, May, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00207.  相似文献   

16.
Let (M,g(t)), 0 ≤ tT, be an n-dimensional closed manifold with nonnegative Ricci curvature, |Rc| ≤ C/t for some constant C > 0 and g(t) evolving by the Ricci flow
$\frac{{\partial g_{ij} }} {{\partial t}} = - 2R_{ij} . $\frac{{\partial g_{ij} }} {{\partial t}} = - 2R_{ij} .   相似文献   

17.
The regular solutions of generalized axisymmetric potential equation , a>−1/2 are called generalized axisymmetric potentials. In this paper, the characterizations of lower order and lower type of entire GASP in terms of their approximation error {En} have been obtained.  相似文献   

18.
  We obtain a new sharp inequality for the local norms of functions x ∈ L ∞, ∞ r (R), namely,
where φ r is the perfect Euler spline, on the segment [a, b] of monotonicity of x for q ≥ 1 and for arbitrary q > 0 in the case where r = 2 or r = 3. As a corollary, we prove the well-known Ligun inequality for periodic functions x ∈ L r , namely,
for q ∈ [0, 1) in the case where r = 2 or r = 3. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1338–1349, October, 2008.  相似文献   

19.
Considering the positive d-dimensional lattice point Z + d (d ≥ 2) with partial ordering ≤, let {X k: kZ + d } be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $ S_n = \sum\limits_{k \leqslant n} {X_k } $ S_n = \sum\limits_{k \leqslant n} {X_k } , nZ + d . Let σ i 2, i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ 2. Let logx = ln(xe), x ≥ 0. This paper studies the convergence rates for $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) . We show that when l ≥ 2 and b > −l/2, E[‖X2(log ‖X‖) d−2(log log ‖X‖) b+4] < ∞ implies $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} , where Γ(·) is the Gamma function and $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } .  相似文献   

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