A Type II Blowup for the Six Dimensional Energy Critical Heat Equation
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A Type II Blowup for the Six Dimensional Energy Critical Heat Equation Junichi Harada1 Received: 10 March 2020 / Accepted: 14 September 2020 © Springer Nature Switzerland AG 2020
Abstract We study blowup solutions of the 6D energy critical heat equation u t = u + |u| p−1 u in Rn × (0, T ). A goal of this paper is to show the existence of type II blowup solutions predicted by Filippas et al. (R Soc Lond Proc Ser A Math Phys Eng Sci 456(2004):2957–2982, 2000). The dimension six is a border case whether a type II blowup can occur or not. Therefore the behavior of the solution is quite different from other cases. In fact, our solution behaves like λ(t)−2 Q(λ(t)−1 x) in the inner region: |x| ∼ λ(t), u(x, t) ≈ 1 1 √ − −( p − 1) p−1 (T − t) p−1 in the selfsimilar region: |x| ∼ T − t 5
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with λ(t) = (1 + o(1))(T − t) 4 | log(T − t)|− 8 . Particularly the local energy defined p+1 1 by E loc (u(t)) = 21 ∇u(t)2L 2 (|x| 0 and e1 (z) is a quadratic polynomial in |z| (see Section 4.2 for details). This function coincides with the ODE solution near the singular point. (x, t) ≈ ( p − 1)
1 − p−1
1 − p−1
(T − t)
in |z| < R
for any fixed R > 0. Furthermore we define a cut off function to connect two solutions. χ1 =
1 if |z| < τ −1 , 0 if |z| > 2τ −1 ,
z=√
x T −t
, τ = | log(T − t)|.
Theorem 1 Let n = 6 and p = pS . There exist T > 0 and a radially symmetric solution u(x, t) ∈ C(R6 ×[0, T ))∩C 2,1 (R6 ×(0, T )) of (1.1) satisfying the following properties:
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J. Harada
(i) The function u(x, t) is given by 5 4
u(x, t) = Qλ(t) (x) −
+
15 8| log(T −t)|
T1 (y)χ1 − (x, t)(1 − χ1 ) +v(x, t),
T −t
y=
x , λ(t)
=:u app (x,t)
(2.1) where T1 (y) is a bounded function defined in Section 4.1. 5 15 (ii) λ(t) = (1 + o(1))(T − t) 4 | log(T − t)|− 8 . (iii) There exist c, K > 0 such that ⎧ √ ⎨ 1 + |z|2 τ − 23 (T − t)−1 for |x| < K √τ T − t, 7 |v(x, t)| < c √ √ ⎩τ − 16 (T − t)−1 |z|− 18 for |x| > K τ T − t,
z= √
x T −t
, τ = | log(T − t)|.
(iv) Let E loc (u) be the local energy defined by E loc (u) =
1 2
|x| 0).
A Type II Blowup for the Six Dimensional Energy Critical Heat Equation
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However this fact is not mentioned in the above theorem, this is guaranteed by Theorem 2.1 [13]. Remark 2.3 As a consequence of (2.2), the property (iv) is formally derived from E loc (u(t)) ∼ E loc (Q) + E(−(t)),
E loc (Q) > 0,
lim E loc (−(t)) = −∞.
t→T
In our construction, there might be other possibilities of (x, t) to obtain a finite energy solution. However by the classification results for the ODE type blowup solutions (see [9,17,30]), the possible (x, t) must have the same asymptotic form and the same energy value. Therefore we have no hope of constructing a type II blowup solution with finite energy for n = 6. Furthermore since our formal argument discussed in Section 4 fails in dimensions n ≥ 7, we believe that a type II blowup with infinite energy occurs only for n = 6. Remark 2.4 We have no idea whether there exist solutions with a different blowup rate for n = 6. O
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