Finite time blow-up for the inhomogeneous equation

We consider the inhomogeneous equation

where , and , and give criteria on , and which determine whether for all and all the solution blows up in finite time or whether for and sufficiently small, the solution exists for all time.

     

The asymptotics of blow-up in inviscid Boussinesq flow and related systems

We consider the approach to blow-up in two-dimensional inviscidflows with stagnation-point similitude, in particular a buoyancy-drivenflow resulting from a horizontally quadratic density variationin a horizontally unbounded slab. The blow-up, which is onlypossible because the flow has infinite energy, proceeds by intensificationof the vorticity and density gradient in a layer adjacent tothe upper boundary, while the remainder of the flow tends towardsirrotationality. The governing Boussinesq flow equations arefirst solved numerically, and the results suggest scalings whichare then used in an asymptotic analysis as 0, where is thetime remaining until blow-up. The structure of the asymptoticsolution, involving exponential orders as well as powers andlogarithms of the small parameter, is suggested by the analysisof a simpler related problem for which an exact solution isavailable. The expansion is uniformly valid across the upperboundary layer and the outer region, but there is a layer adjacentto the lower boundary where the flow remains dependent on theinitial conditions and is undetermined by the asymptotics.     

Adaptive numerical schemes for a parabolic problem with blow-up

In this paper we present adaptive procedures for the numericalstudy of positive solutions of the following problem: u_t = u_xx (x, t) (0, 1) x [0, T), u_x(0, t) = 0 t [0, T), u_x(1, t) = u^p(1, t) t [0, T), u(x, 0) = u₀(x) x (0, 1), with p > 1. We describe two methods. The first one refinesthe mesh in the region where the solution becomes bigger ina precise way that allows us to recover the blow-up rate andthe blow-up set of the continuous problem. The second one combinesthe ideas used in the first one with moving mesh methods andmoves the last points when necessary. This scheme also recoversthe blow-up rate and set. Finally, we present numerical experimentsto illustrate the behaviour of both methods.     

Monodromy and Stability of a Class of Degenerate Planar Critical Points

Necessary and also sufficient monodromy conditions for a large class of degenerate singular points of planar differential systems are given. For these systems, we also find a computable expression of the principal term of the asymptotic expansion of the return map, which gives the stability of the point.     

一类三阶方程奇摄动边值问题“啪”解的渐近分析

研究了一类出现所谓“啪”解(对照空间结构)的三阶奇摄动边值问题,给出了上述问题在出现一个“啪”解时的渐近解的构造算法及其条件;采用逐步推算法,利用解在“啪”点的连续性条件,具体地找出了“啪’点,且得到了边界层函数的指数式衰减估计;最后,对在描述“啪”点主项t_0处间断的渐近解进行了修正,使其在讨论区间上二次连续可微,得到了误差估计定理。     

On the Blow-up for Quasilinear Parabolic Equations

This article is concerned with the position of blow-up points, blow up rate and an isoperimetric problem for the equation u_t = Δu^m + u^p(p > m ≥ 1) in a convex bounded domain.     

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).     

BLOW-UP OF THE SOLUTIONS FOR THE INITIAL-BOUNDARY PROBLEMS OF THE NONLINEAR SCHR?DINGER EQUATIONS

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一类边界耦合的扩散系统爆破现象的数值模拟

对一类边界上非平凡耦合的抛物型系统进行了数值模拟．为验证已有的关于是否出现爆破的理论成果,先用固定网格算法针对四种具体的边界条件进行试验,并根据不同的初始数据分组．为了进一步探讨爆破发生的时刻、位置以及爆破速率,再用移动网格算法针对可能出现爆破现象的两组边界条件和初始数据进行试验,并根据不同的监测函数分组．随后对算法的有效性做出说明并分析试验结果．最后对系统的一种特殊情况给出一个算例．     

一个反应——扩散方程组解的整体存在与有限时刻爆破

研究一类带有非局部边界条件的抛物型方程组解的整体存在与有限刻爆破.主要通过构造上、下解,利用比较原理得到定理的证明.