Abstract
We consider , a solution of which blows up at some time , where , and . Define to be the blow-up set of u, that is, the set of all blow-up points. Under suitable non-degeneracy conditions, we show that if S contains an -dimensional continuum for some , then S is in fact a manifold. The crucial step is to make a refined study of the asymptotic behavior of u near blow-up. In order to make such a refined study, we have to abandon the explicit profile function as a first-order approximation and take a non-explicit function as a first-order description of the singular behavior. This way we escape logarithmic scales of the variable and reach significant small terms in the polynomial order for some . Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S.
Published Version
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