On the equation ut = ∆uα + uβ
On the equation ut = ∆uα + uβ
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DOI：
10.1017/s0308210500025750
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发表时间：
1993
期刊：
影响因子：
--
通讯作者：
Y. Qi
中科院分区：
文献类型：
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作者：
Y. Qi
研究了Ut，=∆uα+uβ的柯西问题，其中0 1.证明了当1α+2/n时，存在爆破解和整体正解，当t为β时，整体正解以t-1/(→∞-1)衰减为零。这样，著名的Fujita关于ut=∆u+up的结果被推广到现在的快速扩散方程。进一步地，将该方程视为Sobolov空间W1，S(W2.s)上的一个无穷维动力系统，并与S和gt；1建立了一个非唯一性结果，证明了该方程存在一个正解u(x，t)，其中u(.，t)→0在W1.s(W2.s)中为t→0.
Synopsis The Cauchy problem of ut, = ∆uα + uβ, where 0 1, is studied. It is proved that if 1α+ 2/n there exist both blow-up solutions and global positive solutions which decay to zero as t–1/(β–1) when t →∞. Thus the famous Fujita result on ut = ∆u + up is generalised to the present fast diffusion equation. Furthermore, regarding the equation as an infinite dimensional dynamical system on Sobolev space W1,s (W2.s) with S > 1, a non-uniqueness result is established which shows that there exists a positive solution u(x, t) with u(., t) → 0 in W1.s (W2.s) as t → 0.