一类具不变性质的变系数偏微分方程的特解

In this paper, a few properties of general patial differential operator P beinginvariant with respect to the form F(|x|_p^p+|y|_q^q-|z|_r^r) are studied, where x∈Rⁿ,y∈ R^m, z∈R^l.and explicit formulas are given for certain solution of the equation Pu=Aδ with P being a differential operator with power function coefficientswhich preserves the form (|x|_p^p+|y|_q^q-|z|_r^r)^1/v for arbitrary even integers p, q, r,and odd integers v.     

含有奇异位势和临界指数的拟线性椭圆方程的G-对称解

本文讨论一类奇异拟线性椭圆型方程
-div(|x|^-ap|▽u|^p-2▽u)=μ+h(x)/|x|^(a+1)p|u|^p-2u+k(x)|u|^p-2u/|x|^bq,x∈R^N,
其中1 < p < N, 0 ≤ a < N-p/p, a ≤ b < a + 1, 0 ≤ μ < μ = (N-p/p-a)^p, q=p^*(a, b) = Np/N-(1+a-b)p,h 和k 是R^N上的连续有界函数, 且关于O(N) 的闭子群G满足某些对称性条件. 应用变分方法和Caffarelli-Kohn-Nirenberg 不等式, 在h与k满足适当条件下, 证得了一些G-对称解的存在性和多重性结果.     

Heisenberg群上无穷远处的集中列紧原理和具有Sobolev临界指数的p - 次Laplace方程多解的存在性

通过建立Heisenberg群上无穷远处的集中列紧原理, 研究了如下$p$ -次Laplace方程 -Δ_{H, p}u=λg(ξ)|u|^q-2u+f (ξ)|u|^p*-2u,在Hⁿ上, u∈ D^{1, p}(Hⁿ), 其中ξ∈Hⁿ,λ∈R,1j, 且m, j为整数.     

类渐近线性p＆q—Laplace方程弱解的全局衰减性

该文研究椭圆型方程{-Δpu+m|u|p-2u-Δqu+n|u|q-2u=g(x,u),x∈RN,u∈ W1,p(RN)∩W1,q(RN)弱解在全空间RN上的衰减性,其中m,n≥0,N≥3,1相似文献   

退化弱拟正则映射的正则性

本文考虑退化弱拟正则映射.利用Hodge分解、逆Holder不等式等工具,证明了其正则性结果:存在指数q₁=q₁（n,l,K）1,q_1loc(Ω,Rⁿ)，都有f∈W^1,l_loc(Ω,Rⁿ) ，即f为退化拟正则映射.     

二阶抛物方程组解的Lp-正则性和Hölder连续性

在可控和自然增长条件下,非线性抛物组 u''_t-D_aA_i^a(x,t,u,Du)= B_i(x,t,u,Du),i=1,…,N,(x,t)∈Q之解。u∈L²(0,T;H¹(Ω,R^N))∩L^∞(0,T;L²(Ω,R^N))(或∩L^∞(Q,R^N))的空间导数D_au事实上属于L_loc^p(Q,R^N),p>2;拟线性抛物组 u''_t-Dα[A_ij^αβ(x,t,u)D_βu^j+a_j^a(x,t,u)]=B_i(x,t,u,Du),i=1,…,N的每一个解都在一开集 Q₁?Q上 Holder连续,且H^n+2-p(Q\Q₁)=0;若当j>i时A_ij^αβ=0,且B_i(x,t,u,p)关于|p|的增长阶小于2,则Q₁=Q;若A_ij^αβ和a_i^a都Holder连续,则D_au也在Q₁上 Holdler连续.     

确定凸多边形可碰撞区域的快速算法

设P=(p₀,p₁,…,p_n-1)与Q=(q₀,q₁,…,q_n-1)是任二互不相交的凸多边形,本文研究了如何快速确定它们的可碰撞区域和可移动区域的问题. 文中提出了可碰撞性判定的新方法,研究了斜支撑线的基本性质,利用这些性质构造出了求斜支撑线的快速算法,其时间复杂度为O(log²(n+m)),在此基础上给出了确定可碰撞区域和可移动区域的时间复杂度为O(log²(n+m))的快速算法.     

强非线性抛物型方程的广义解

对于非线性抛物型方程其中Q_T=Ω×(0,T)是上半空间R₊ⁿ⁺¹中的一个柱形区域,S_T=Ω×[0,T]是Q_T的侧面,Ω是R~n中的一个有界区域,其边界Ω充分光滑,本文着重讨论函数a_i(x,t,u,s)和a(x,t,u,s)关于变元s=(s₁…,s_n)按指数形式快速增长的情形.文中得到了强非线性抛物型方程(1.1)和(1.2)在空间中广义解的存在性.这个结果也包括了a_i(x,t,u,s)及a(x,t,u,s)关于变元s=(s₁..,s_n)按幂次|s|ⁿ的形式增长的情形,改正了Ladyenskaja等的文献中第五章定理6.7的证明中的一个疏忽。     

非参数回归函数最近邻估计Lp收敛的速度

Let (X,Y) be a R^d×R¹-valued random vector with E(|Y|)<∞,m(x)=E(Y|X=x) be the regression funvion of Y with respect to X.Suppose that (X_i, Y_i),i=1, …,n, are iid samples drawn from (X,Y). It is desired to estimate m(x) based on these samples,everoye discussed in 1981 (see [2]) the pointwise L_p-convergence of the nearest neigthoor estimate m_n(x) (see (5) of the present paper). In this article we further study the rate of this convergence.It is shown that if there exists p≥2 such that E |Y|^p<∞,then E|m_n(x)-m(x)|^p=O(n^-p/(d+2))a.s. for suitabte choice of the weighte C_m (see(4) of the present paper).     

临界增长拟线性椭圆型方程的非平凡解

本文给出R^N中有界域Ω上拟线性临界增长椭圆型方程的Dirichlet问题的非平凡W^1,p(Ω)广义解的存在性结果。     

Blow-up of solutions of nonlinear heat equations in unbounded domains for slowly decaying initial data

Consider the following equations: (E)  u_t-Du=u^p{(E)\ \ u_t-\Delta u=u^p}, (E￠)  u_t-Du=u^p-m | ?u | ^q{(E')\ \ u_t-\Delta u=u^p-\mu\mid\nabla u\mid^q}, (E")  u_t-Du=u^p+a.?(u^q){(E')\ \ u_t-\Delta u=u^p+a.\nabla (u^q)}, in W ì IR^d{\Omega\subset I\!\!R^d}. For any unbounded domain W\Omega, intermediate between a cone and a strip, we obtain a sufficient condition on the decay at infinity of initial data to have blow-up. This condition is related to the geometric nature of W{\Omega}. For instance, if W\Omega is the interior of a revolution surface of the form | x￠_d | < f( | x_d | ){\mid x'_d\mid (x) > Cf( | x | )^-2/(p-1){\Phi (x)>Cf(\mid x\mid )^{-2/(p-1)}} at infinity. Moreover, for a large class of domains W{\Omega}, we prove that those results are optimal (i.e. there exist global solutions with the same order of decay at infinity for their initial data).     

R~N上一类p(x)-Laplacian方程的无穷多解问题

该文主要讨论了如下p(x)-Laplacian算子方程的解.其中1P-≤p(x)≤P+N.得到了上述方程在变指数Sobolev空间W~(1,p(x))(R~N)中的一列能量值趋向正无穷的解.     

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

Cocompactness and minimizers for inequalities of Hardy–Sobolev type involving N-Laplacian

The paper studies quasilinear elliptic problems in the Sobolev spaces W ^1,p (Ω), ${\Omega\subset{\mathbb R}^N}The paper studies quasilinear elliptic problems in the Sobolev spaces W ^1,p (Ω), W ì \mathbb R^N{\Omega\subset{\mathbb R}^N} , with p = N, that is, the case of Pohozhaev–Trudinger–Moser inequality. Similarly to the case p < N where the loss of compactness in W^1,p(\mathbb R^N){W^{1,p}({\mathbb R}^N)} occurs due to dilation operators u ?t^(N-p)/pu(tx){u {\mapsto}t^{(N-p)/p}u(tx)} , t > 0, and can be accounted for in decompositions of the type of Struwe’s “global compactness” and its later refinements, this paper presents a previously unknown group of isometric operators that leads to loss of compactness in W₀^1,N{W_0^{1,N}} over a ball in \mathbb R^N{{\mathbb R}^N} . We give a one-parameter scale of Hardy–Sobolev functionals, a “p = N”-counterpart of the H?lder interpolation scale, for p > N, between the Hardy functional ò\frac|u|^p|x|^p dx{\int \frac{|u|^p}{|x|^p}\,{\rm d}x} and the Sobolev functional ò|u|^pN/(N-mp)  dx{\int |u|^{pN/(N-mp)} \,{\rm d}x} . Like in the case p < N, these functionals are invariant with respect to the dilation operators above, and the respective concentration-compactness argument yields existence of minimizers for W ^1,N -norms under Hardy–Sobolev constraints.     

Estimates of the Solutions of Strongly Degenerate Elliptic Equations in Weighted Sobolev Spaces

Let W ì \mathbbRⁿ \Omega \subset \mathbb{R}^n be an open set and l(x) | u |_p,l = ( ò_W l^p (x)| u(x) |^p dx )^1/p \text (1 \leqslant p < + ￥\text),\left| u \right|_{p,l} = \left( {\int\limits_\Omega {l^p (x)\left| {u(x)} \right|^p dx} } \right)^{1/p} {\text{ (1}} \leqslant p < + \infty {\text{),}}     

Existence of solutions for non-periodic superlinear Schrödinger equations without (AR) condition

The existence of solutions is obtained for a class of the non-periodic Schrödinger equation −Δu + V(x)u = f(x, u), x ∈ R^N, by the generalized mountain pass theorem, where V is large at infinity and f is superlinear as |u| → ∞.     

Geometry of unbounded domains,poincaré inequalities and stability in semilinear parabolic equations

We investigate thc close relations existing between certain geometric properties of domains Ω of R^N, the validity of Poincark inequalities in Ω, and the behavior of solutions of semilinear parabolic equations. For the equation u_t-△u=|u|^p-1 we obtain a purely geometric, necessary and sufficient condition on Ω, for the 0 solution to be asymptotically (and exponentially) stable in L^r(ω)1<r<∞ when r is supercritical(r>N(p-1)/2 . The condition is that the inradius of Ω be finite. The result is different for r critical. For the equation u_t-△u=u^p-μ|u|^q,q≥p>1,μ>0 we prove that the finiteness of the inradius is a necessary and sufficient condition for global existence and boundedness of all nonnegative solutions.     

Instability of Standing Wave for the Klein–Gordon–Hartree Equation

The instability property of the standing wave uω(t, x) = eiωtφ(x) for the Klein–Gordon– Hartree equation     

Boundary trace embedding theorems for variable exponent Sobolev spaces

We study boundary trace embedding theorems for variable exponent Sobolev space W1,p(⋅)(Ω). Let Ω be an open (bounded or unbounded) domain in RN satisfying strong local Lipschitz condition. Under the hypotheses that p∈L^∞(Ω), 1?infp(x)?supp(x)<N, |∇p|∈Lγ(⋅)(Ω), where γ∈L^∞(Ω) and infγ(x)>N, we prove that there is a continuous boundary trace embedding W1,p(⋅)(Ω)→Lq(⋅)(∂Ω) provided q(⋅), a measurable function on ∂Ω, satisfies condition for x∈∂Ω.     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.