Blow-up of solutions of parabolic equations with nonlinear memory
Blow-up of solutions of parabolic equations with nonlinear memory
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DOI：
10.1016/0022-0396(87)90168-9
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发表时间：
1987-10
影响因子：
2.4
通讯作者：
H. Bellout
中科院分区：
文献类型：
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作者：
H. Bellout
（1.17）本身不足以建立同样的结果。在公式1.16中，如果u爆破，u也爆破。在（1.17）式中，先验地，情况不一定如此。取决于u的爆破速率，u可能爆破并且Jb（u+ A）（（p+“P-”dz）是有限的。我们的主要困难是排除最后一种可能性。为此，我们证明了u，<（u+ n）（Tt ′ +~，（1.18）），由此我们推出：s ′（u+~）（（“+”）′ 2）~“d~（u+ E. t '+”）或者我们可以开始一个归纳，证明u（x，T）是有界的。在第二节中，我们将证明存在性、唯一性和有限时间爆破。然后我们证明了u的爆破速率的一个粗略估计，由此我们将推出不等式（1.18）。
(1.17) by itself is insufficient to establish the same result. In (1.16), if u blows up, so does u,. In (1.17), a priori, that is not necessarily the case. Depending on the rate of blow-up of u it is possible for u to blow-up and Jb (u+ A)((p+“P-” dz to be finite. Our main difficulty is to rule out this last possibility. For this purpose we show that u,<(u+ n)(Tt)‘+~,(1.18) from which we deduce that either s ‘(u+~)((“+‘)‘2)~“d~~(u+ E.)‘+” 0 or we can start an induction that shows that u (x, T), is bounded. In Section 2 we will prove existence, uniqueness and finite time blow-up. Then we prove a crude estimate on the rate of blow-up of u from which we will deduce the inequality (1.18).