Geometry Driven Type Ii Higher Dimensional Blow-up for the Critical Heat Equation
نویسنده
چکیده
We consider the problem vt = ∆v + |v|p−1v in Ω× (0, T ), v = 0 on ∂Ω× (0, T ), v > 0 in Ω× (0, T ). In a domain Ω ⊂ Rd, d ≥ 7 enjoying special symmetries, we find the first example of a solution with type II blow-up for a power p less than the JosephLundgren exponent pJL(d) = { ∞, if 3 ≤ d ≤ 10, 1 + 4 d−4−2 √ d−1 , if d ≥ 11. No type II radial blow-up is present for p < pJL(d). We take p = d+1 d−3 , the Sobolev critical exponent in one dimension less. The solution blows up on circle contained in a negatively curved part of the boundary in the form of a sharply scaled Aubin-Talenti bubble, approaching its energy density a Dirac measure for the curve. This is a completely new phenomenon for a diffusion setting.
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