Self-similar boundary blow-up for higher-order quasilinear parabolic equations
Self-similar boundary blow-up for higher-order quasilinear parabolic equations
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高阶拟线性抛物型方程的自相似边界爆炸
DOI：
10.1017/s0308210500004339
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
A. Shishkov
中科院分区：
文献类型：
--
作者：
V. Galaktionov;A. Shishkov
研究了一类2阶拟线性抛物型方程边界爆破的演化性质，其中对于齐次幂非线性，其典型渐近行为由精确或近似自相似解描述。通过能量估计和流的收缩性，证明了这种相似解的存在性和渐近稳定性。利用与圣维南原理有关的能量估计，进一步证明了更一般方程的渐近结果。通过与各种自相似模式的比较，证明了所建立的由边界爆炸机制产生的奇点传播估计是尖锐的。
We study evolution properties of boundary blow-up for 2mth-order quasilinear parabolic equations in the case where, for homogeneous power nonlinearities, the typical asymptotic behaviour is described by exact or approximate self-similar solutions. Existence and asymptotic stability of such similarity solutions are established by energy estimates and contractivity properties of the rescaled flows. Further asymptotic results are proved for more general equations by using energy estimates related to Saint-Venant's principle. The established estimates of propagation of singularities generated by boundary blow-up regimes are shown to be sharp by comparing with various self-similar patterns.