共查询到20条相似文献,搜索用时 156 毫秒
1.
Let M^n be a smooth, compact manifold without boundary, and F0 : M^n→ R^n+1 a smooth immersion which is convex. The one-parameter families F(·, t) : M^n× [0, T) → R^n+1 of hypersurfaces Mt^n= F(·,t)(M^n) satisfy an initial value problem dF/dt (·,t) = -H^k(· ,t)v(· ,t), F(· ,0) = F0(· ), where H is the mean curvature and u(·,t) is the outer unit normal at F(·, t), such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton. 相似文献
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Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations 总被引:6,自引:0,他引:6
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(4):881-892
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
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谷晓明 《应用数学学报(英文版)》1993,(1)
A boundary problem for the Klein-Gordon equation in the strip O≤t≤T is considered with the boundary condition:the initial state at t=O and the final state at t=T.It is proven that the problem admits of an infinite number of solutions.The same result holds for a generic 2nd order hyperbolic equation in 2-variables.Using the result for the wave operator in 3-space dimensions we give a method to reconstruct functions whose integral on all unit spheres in R~3 is a given function. 相似文献
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In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x = 0. First, we prove a simple but useful property of Lax-Oleinik formula(Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)-′qF(q) + L(′F(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t T~*. Meanwhile, we can give the equation of the shock and an explicit value of T~*. 相似文献
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Local and Global Existence of Solutions to
Initial Value Problems of Modified Nonlinear
Kawahara Equations 总被引:3,自引:0,他引:3
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(5):1035-1044
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. 相似文献
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In this paper we prove an ε0-regularity theorem for mean curvature flow from surface to a flat Riemannian manifold. More precisely, we prove that if the initial energy ∫Σ0 |A|2 ≤ε0 and the initial area μ0(Σ0) is not large, then along the mean curvature flow, we have ∫Σt|A|2 ≤ε0. As an application, we obtain the long time existence and convergence result of the mean curvature flow. 相似文献
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Zhou Fujun Cui Shangbin 《高校应用数学学报(英文版)》2005,20(4):441-447
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. 相似文献
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Shang Quan Bu 《数学学报(英文版)》2010,26(7):1223-1232
We study the well-posedness of the equations with fractional derivative D^αu(t) = Au(t) + f(t),0≤ t ≤ 2π, where A is a closed operator in a Banach space X, α 〉 0 and D^α is the fractional derivative in the sense of Weyl. Using known results on LP-multipliers, we give necessary and/or sufficient conditions for the LP-well-posedness of this problem. The conditions we give involve the resolvent of A and the Rademacher boundedness. Corresponding results on the well-posedness of this problem in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces are also obtained. 相似文献
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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
10.
In this paper we consider the construction of solutions to the Cauchy problem of Burgers' equationsut-γ△u + u·▽u = 0, t∈R+,x∈R3, (1)u(0,x)= u0(x), x ∈ R3, (2)in pseudomeasure spaces, where γ(?) 0 is a small parameter that plays the role of the viscosity and u = u(t,x) is a velocity-like vector field defined on R+×R3. The initial datum u0(x) is a vector-valued function defined on R3, In one space dimension Burgers' equation is 相似文献
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We prove the so-called conjecture: for every real-monic polynomial of degree there exists an n by n matrix with sign patternwhose characteristic polynomial is . The proof converts the problem of determining the nonsingularity of a certain Jacobi matrix to the problem of proving the non-existence of a nonzero matrix B that commutes with a nilpotent matrix with sign pattern and has zeros in positions , and for . 相似文献
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The quasilinear chemotaxis–haptotaxis system is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain . Here , and , for all with some and for all . It is shown that if the ratio is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m. 相似文献
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In this paper we study the existence of distributional solutions of Dirichlet problems whose simplest example is 相似文献
19.
Michael Winkler 《Journal of Differential Equations》2018,264(10):6109-6151
A class of chemotaxis-Stokes systems generalizing the prototype is considered in bounded convex three-dimensional domains, where is given.The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that
(0.1)
Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state in the large time limit.This extends previous results which either relied on different and apparently less significant energy-type structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than (0.1). 相似文献