Infinitely Multiple Eigenvalues of The Dirichlet Problem for Quasilinear Elliptic Equations

In this paper, We consider the following Dirichlet problem for quasilinearelliptic equation ( /x_i)F_i(x,Du)=-λu-p(x,u) x∈Ω(1) u| Ω=0     

Blow-up,quenching, aggregation and collapse in a chemotaxis model with reproduction term

In this paper,we consider the following chemotaxis model with ratio-dependent logistic reaction term u/t=D▽(▽u-u▽ω/ω)+u(α-bu/ω),(x,t)∈QT,ω/t=βu-δω,(x,t)∈QT,u▽㏑(u/w)·=0,x ∈Ω,0tT,u(x,0)=u0(x)0,x ∈,w(x,0)=w0(x)0,x ∈,It is shown that the solution to the problem exists globally if b+β≥0 and will blow up or quench if b+β0 by means of function transformation and comparison method.Various asymptotic behavior related to different coefficients and initial data is also discussed.     

ON BLOW-UP PHENOMENON OF THE SOLUTION TO SOME WAVE-HARTREE EQUATION IN d≥5

This article mainly considers the blow up phenomenon of the solution to the wave-hartree equation u_(tt)-?u =(|x|~(-4)*|u|~2)u in the energy space for high dimensions d ≥ 5. The main result of this article is that: if the initial data(u_0, u_1) satisfy the conditions E(u_0, u_1) E(W, 0) and ||?u0||_2~2 ||?W|| _2~2 for some ground state W, then the corresponding solution must blows up in finite time.     

退缩拟线性椭圆型方程某类边值问题的无穷多解

文[1]证明了如下D氏问题 -D_i(g|Du|~2)D_iu=f(x,u),x∈Ω, u=0,x∈Ω存在非平凡解,本文讨论上述方程的另一类边界问题 -D_i(g|Du|~2)D_iu=f(x,u),x∈Ω, g(|Du|~2)D_iu(0)(n,x_i)+h(x,u)=0,x∈Ω, (1)其中Ω∈R~n是具有光滑边界的有界区域,n(x)是Ω在x点的外法向,D_iu=u/x_i,Du=gradu=u,重复指标表示求和,与问题(1)相应的泛函为:     

双非线性抛物方程初始迹与Harnack不等式

在最优的初始值条件下考虑如下拟线性抛物方程的柯西问题u_t-diva(x,t,u,Du)=b(x,t,u,Du),(x,t)属于S_T=R~N×(0,T).令a(x,t,u,Du)={a_i(x,t,u,Du)},假设a_i(x,t,u,Du)与b(x,t,u,Du)皆为Caratheodory函数,并且假设它们满足Du的单调性,关于u,|Du|等一定的增长阶条件下,得到了解的比较定理,证明了解的存在性,并得到了相关的Harnack不等式.     

二阶拟线性椭圆型方程振动性的积分平均方法

§ 1.　IntroductionThepurposeofthispaperistostudytheoscillatorybehaviorofsolutionsofcertainquasi linearellipticequationsdiv( ｜Du｜m -2 A(x)Du) + p(x) ｜u｜m -2 u=0 ,x∈Ω Rn,(E)whereΩisanexteriordomain ,m >1 ,andfunctionsA(x) ,p(x)aretobespecifiedinthefollowingtext.Recently ,USAMI [6]consideredEq .(E)whenA(x)≡I (identitymatrix) ,andob tainedoscillationcriteriaforEq .(E)with“infiniteintegral”coefficient [cf.[6],Theorem 4].However,asfarasthepresentreferencesisconcerned ,therearefewo…     

REGULARITY OF H~1 ∩L~(n((γ-1)/(2-γ))) WEAK SOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS

Let us consider the following elliptic systems of second order-D_α(A_i~α(x, u, Du))=B_4(x, u, Du), i=1, …, N, x∈Q(?)R~n, n≥3 (1) and supposeⅰ) |A_i~α(x, u, Du)|≤L(1+|Du|);ⅱ) (1+|p|)~(-1)A_i~α(x, u, p)are H(?)lder-continuous functions with some exponent δ on (?)×R~N uniformly with respect to p, i.e.ⅲ) A_i~α(x, u, p) are differentiable function in p with bounded and continuous derivativesⅳ)ⅴ) for all u∈H_(loc)~1(Ω, R~N)∩L~(n(γ-1)/(2-γ))(Ω, R~N), B(x, u, Du)is ineasurable and |B(x, u, p)|≤a(|p|~γ+|u|~τ)+b(x), where 1+2/n<γ<2, τ≤max((n+2)/(n-2), (γ-1)/(2-γ)-ε), (?)ε>0, b(x)∈L2n/(n+2), n~2/(n+2)+e(Ω), (?)ε>0.Remarks. Only bounded open set Q will be considered in this paper; for all p≥1, λ≥0, which is clled a Morrey Space.Let assumptions ⅰ)-ⅳ) hold, Giaquinta and Modica have proved the regularity of both the H~1 weak solutions of (1) under controllable growth condition |B|≤α(|p|~γ+|u|~((n+2)/(n-2))+b, 0<γ≤1+2/n and the H~1∩L~∞ weak solutions of (1) under natural     

障碍问题解的内正则性

对非幂次增长的障碍问题 :∫Ωai(x,u,Du) φ xidx + ∫Ωb(x,u,Du)φ dx≥ 0　　这里φ(x)≥ψ(x) - u(x) ,u(x)≥ψ(x) ,而φ∈ W1 0 LM(Ω ) ,ψ为局部 Holder连续的 ,我们得到其在 W1 LM(Ω)中弱解的 C0 ,αloc 正则性     

一类椭圆型方程障碍问题很弱解的局部正则性

本文研究一类非齐次二阶椭圆型方程-divA(x,u,Du)=B(x,u,Du)的Krψ,θ(Ω)-障碍问题的很弱解.利用Hodge分解的方法及逆Hlder不等式,给出非齐次方程的障碍问题很弱解的局部正则性.由于B(x,u,Du)中u和Du的增长指数为次临界,为了得到局部正则性,我们对同一积分项使用了两次Young和Hlder不等式的技巧.     

具加权非局部源反应扩散方程解的渐近行为（英文）

In this paper,we deal with the blow-up property of the solution to the diffusion equation u_t = △u + a(x)f(u) ∫_Ωh(u)dx,x∈Ω,t0 subject to the null Dirichlet boundary condition.We will show that under certain conditions,the solution blows up in finite time and prove that the set of all blow-up points is the whole region.Especially,in case of f(s) = s~p,h(s) = s~q,0 ≤ p≤1,p + q 1,we obtain the asymptotic behavior of the blow up solution.