一类具有局部化源和吸收项的抛物系统解的整体存在与爆破

在齐次Dirichlet边界条件研究如下抛物系统其中x_0(t):R⁺→(0,a)是Holder连续函数;常数0≤α,β<1,p_1,p_2,q_1,q_2,k_1,k_2>0.利用正则化方法,在一定的假设条件下证明了经典解的存在性.接着利用比较原理证明了该系统正解的整体存在性和爆破性.最后给出了爆破解的精确爆破速率和爆破模式.     

半线性抛物方程可变号解的全局存在和爆破

该文研究了如下柯西问题狌狋＝狌狓狓＋ （狋＋１）－σ／２狘狓狘σ狘狌狘狆－１狘狌狘，　狓∈犚，狋∈犚＋ ，狌（狓，０）＝狌０（狓），　狓∈犚｛ ．其中参数σ≥０，狌０（狓）在犚上犽次变号，满足某种速降条件．证明了：如果ｍａｘ｛σ，１｝＜狆≤１＋２犽＋１，那么所有非零解在有限时间内爆破；如果狆＞ｍａｘ｛σ，１＋２犽＋１｝则存在一个非零全局解．     

Global Existence and Blow-Up Results for a Classical Semilinear Parabolic Equation

The author studies the boundary value problem of the classical semilinear parabolic equations ut-△u = |u|p-1u inΩ×(0, T), and u = 0 on the boundary × [0, T) and u = φ at t = 0, where Rnis a compact C1domain, 1 < p ≤ p S is a fixed constant, and φ∈ C1 0(Ω) is a given smooth function. Introducing a new idea, it is shown that there are two sets W and Z, such that for φ∈ W, there is a global positive solution u(t) ∈ W with H1omega limit 0 and for φ∈ Z, the solution blows up at finite time.     

一类带非线性边界条件的非线性抛物方程的解的整体存在性

该文考虑带非线性边界条件的非线性抛物方程的正整体解的存在性与非存在性。通过使用上下解技巧，得到了所有正解整体存在的充分必要条件。作者所构造的上下解具有相同的形式且计算简便。     

Existence and Nonexistence of Global Solution for Quasilinear Parabolic Equation

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ，ｗｅｓｔｕｄｙｔｈｅｅｘｉｓｔｅｎｃｅａｎｄｎｏｎｅｘｉｓｔｅｎｃｅｏｆｇｌｏｂａｌｐｏｓｉｔｉｖｅｓｏｌｕｔｉｏｎｓｏｆｔｈｅｆｏｌｌｏｗｉｎｇｑｕａｓｉｌｉｎｅａｒｐａｒａｂｏｌｉｃｅｑｕａｔｉｏｎｓｗｉｔｈｎｏｎ－ｌｉｎｅａｒｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎｓ：ｕｔ＝１ｍ△ｕｍ－ｕｎ，　　　ｘ∈Ω，ｔ＞０１ｍ·ｕｍγ＝ｕｐ，ｘ∈Ω，ｔ＞０（１）ｕ（ｘ，０）＝ｕ０（ｘ）＞０，ｘ∈Ω－ｗｈｅｒｅｎ，ｍ，ｐａｒｅｐｏｓｉｔｉｖｅｃｏ…     

Global Existence and Blow-Up for a Class of Degenerate Parabolic Systems with Localized Source

This paper deals with a class of localized and degenerate quasilinear parabolic systems

一类非线性发展方程整体解的存在性、渐近性与解的爆破

研究一类非线性发展方程初边值问题整体弱解的存在性，渐近性和解的爆破问题，证明在关于非线性项的不同条件下，上述初边值问题分别在大初值和小初始能量的情况下存在整体弱解，并且讨论了弱解的渐近性。还证明：在相反的条件下，上述弱解在有限时刻爆破，并且给出了一个实例。     

退缩抛物方程组解的全局存在性

利用上下解方法讨论一类退缩抛物方程组存在全局解的条件,并证明了在一定条件下全局解的收敛性.     

拟线性抛物方程整体解的存在性和不存在性

The paper studies the existence,the exponential decay and the nonexistence of global solution for a class of quasilinear parabolic equations.     

拟线性抛物型方程解的爆破

本文讨论了一类拟线性抛物型方程具有某种非线性边界条件的解的爆破性质,证明了其解在有限时间T_0~-内爆破。     

On Blowup of a Solution to a Sobolev-Type Equation with a Nonlocal Source

We consider the initial-boundary value problem for a dissipative nonlinear wave equation of the Sobolev type with a nonlocal cubic source. For this problem we obtain a sufficient condition for blowup of a strong generalized solution. Moreover, we obtain a two-sided estimate for the blowup time of the solution.Original Russian Text Copyright © 2005 Korpusov M. O. and Sveshnikov A. G.The authors were supported by the Russian Foundation for Basic Research (Grants 02-01-00253; 02-01-06038) and a grant of the Project Junior Scientists of Russia.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 567–578, May–June, 2005.     

Global Existence for a Parabolic Chemotaxis Model with Prevention of Overcrowding

In this paper we study a version of the Keller–Segel model where the chemotactic cross-diffusion depends on both the external signal and the local population density. A parabolic quasi-linear strongly coupled system follows. By incorporation of a population-sensing (or “quorum-sensing”) mechanism, we assume that the chemotactic response is switched off at high cell densities. The response to high population densities prevents overcrowding, and we prove local and global existence in time of classical solutions. Numerical simulations show interesting phenomena of pattern formation and formation of stable aggregates. We discuss the results with respect to previous analytical results on the Keller–Segel model.     

Global Existence and Blow-up for a Parabolic System with Nonlinear Boundary Conditions

This paper deals with the global existence and blow-up of positive solutions to the systems: u_t = ∇(u^∇u) + u¹ + v^a v_t = ∇(v^n∇v) + u^b + v^k in B_R × (0, T) \frac{∂u}{∂η} = u^αv^p, \frac{∂v}{∂η} = u^qv^β on S_R × (0, T) u(x, 0) = u_0(x), v(x, 0} = v_0(x) in B_R We prove that there exists a global classical positive solution if and only if l ≤ l, k ≤ 1, m + α ≤ 1, n + β ≤ 1, pq ≤ (1 - m - α)(1 - n - β),ab ≤ 1, qa ≤ (1 - n - β) and pb ≤ (1 - m - α).     

具有非局部源的退化半线性抛物型方程组解的爆破

本文讨论具有非局部源退化半线性抛物型方程组的初边值问题 .证明了局部解的存在唯一性并且得到当初值充分大时解在有限时刻爆破 .     

非局部的退化抛物型方程组的解的爆破和整体存在性

该文采用弱上下解方法以及正则化的技巧,研究了一类非局部的退化的抛物型方程组的解的爆破和整体存在性,给出了方程组的解的爆破指标p_c=(p₁+p₂)(q₁+q₂)-mn,证得当p_c<0时,对任意的初值,方程组的解整体存在;当p_c>0时,对充分大的初值,解在有限时刻爆破,对充分小的初值,解整体存在;当p_c=0时,若区域充分小,则方程组存在非负整体解,若区域包含了一个充分大的球, 则解在有限时刻爆破.     

具非局部源项的抛物系统解的整体存在与爆破(英文)

考虑一个具有非局部源项的抛物系统非负解的整体存在与爆破问题,通过使用上下解技术,建立了系统的临界指数,而且,相关的分类是最优与最完全的.     

一类带有阻尼项和源项的波动方程的整体解的存在性(英文)

研究带有阻尼项的非线性波动方程(1.1),得出在初值满足一定条件时,方程的整体解存在,且解的能量是指数衰减的.     

Existence Result to a Parabolic Equation with Quadratic Gradient Term and an $L^{1}$ Source

In a number of models for coupled oscillators and nonlinear wave equations primary resonances dominate the phase-space phenomena. A new feature is that in a Hamiltonian framework, the interaction of primary and higher order resonances is shown to be important and can be signaled by using recurrence properties. The interaction may involve embedded double resonance. We will demonstrate these phenomena for the cubic Klein-Gordon equation on a square with Dirichlet boundary conditions using normal form techniques. The results are qualitatively and quantitatively very different from the one-dimensional spatial case.

     

带局部非线性反应项的退化抛物方程解的爆破性质

本文研究带局部非线性反应项的退化抛物方程解的爆破性质ut=△um+up(x0,t)-kuq(x,t),其中p≥q>0,p>1,01),x0是有界区域Ω内的固定点,Ω(?)RN.在一定的假设条件下,证明了解在有限时刻爆破并且爆破点集是整个区域Ω.另外,如果解u(·,t)是径向对称函数且ur≤0,则解在接近爆破时刻的爆破速率在区域Ω上是一致的.在解是非径向对称的情况下,我们用其他技巧证得解的整体爆破性.