Incomplete Self‐Similar Blow‐Up in a Semilinear Fourth‐Order Reaction‐Diffusion Equation
Incomplete Self‐Similar Blow‐Up in a Semilinear Fourth‐Order Reaction‐Diffusion Equation
复制标题
半线性四阶反应扩散方程中的不完全自相似放大
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
V. Galaktionov
中科院分区：
文献类型：
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作者：
V. Galaktionov
研究了四阶半线性反应扩散方程1的爆破行为.对于燃烧理论中的经典半线性热方程2，自20世纪70年代以来，人们研究了各种Blow-up模式，而对于高阶扩散的情形研究得较少。这与（2）的blow-up有着显著的区别，（2）的blow-up总是完备的，因为对于t> T，blow-up之外的最小（适当）扩张是u（x，t）∞ +∞。 困难的四阶动力系统的扩展对{f（y），F（y）}的各种分析，形式和数值方法的组合进行了研究。 我们还讨论了（1）在非一般完全爆破下的其他非相似模式。
Blow‐up behavior for the fourth‐order semilinear reaction‐diffusion equation 1 is studied. For the classic semilinear heat equation from combustion theory 2 various blow‐up patterns were investigated since 1970s, while the case of higher‐order diffusion was studied much less. Blow‐up self‐similar solutions of (1) of the form are constructed. These are shown to admit global similarity extensions for t > T: The continuity at t = T is preserved in the sense that This is in a striking difference with blow‐up for (2) , which is known to be always complete in the sense that the minimal (proper) extension beyond blow‐up is u(x, t) ≡+∞ for t > T. Difficult fourth‐order dynamical systems for extension pairs {f(y), F(y)} are studied by a combination of various analytic, formal, and numerical methods. Other nonsimilarity patterns for (1) with nongeneric complete blow‐up are also discussed.