Finite time vs. infinite time gradient blow-up in a degenerate diffusion equation
Finite time vs. infinite time gradient blow-up in a degenerate diffusion equation
复制标题
DOI：
10.1512/iumj.2008.57.3337
复制
发表时间：
2008
影响因子：
1.1
通讯作者：
Christian Stinner;M. Winkler
中科院分区：
文献类型：
--
作者：
Christian Stinner;M. Winkler
本文讨论了Ωx(0，T)中(*)ut=u p u xx+Ku ru x 2+u q在有界区间Ωc R上的Dirichlet问题非负经典解的梯度爆破现象，其中p>；2，1≤q≤p-1，r≥1，K≥0，且初值u o属于C1(Q)且满足u0(X)>；C0 dist(x，δΩ).证明了如果(*)中的梯度项在u=0附近足够弱，则所有有界解都经历无限时间的梯度爆破，而如果这项足够强，则所有解在有限时间内在C1(Ω)中爆破.这里的‘弱’指的是参数满足r=p-1和Kp-1，而‘强’的意思正好相反，即r=p-1和K>；p-2，或者r<；p-1。
This paper deals with the phenomenon of gradient blow-up of nonnegative classical solutions of the Dirichlet problem for (*) u t = u p u xx + Ku r u x 2 + u q in Ω x (0,T) in a bounded interval Ω c R, where p > 2, 1 ≤ q ≤ p- 1, r ≥ 1, K ≥ 0, and the initial data u o are assumed to belong to C 1 (Q) and satisfy u 0 (x) > C 0 dist(x, δΩ) in Q with some C 0 > 0. It is shown that if the gradient term in (*) is weak enough near u = 0, then all bounded solutions undergo an infinite time gradient blow-up, whereas if this term is sufficiently strong, then all solutions blow up in C 1 (Ω) within finite time. Here by 'weak' we mean that the parameters satisfy either r = p - 1 and K p - 1, and by 'strong' the precise opposite, that is, either r = p - 1 and K > p - 2, or r < p - 1.