Type II Blow-up in the 5-dimensional Energy Critical Heat Equation

• Published in: Acta mathematica Sinica. English series Vol. 35; no. 6; pp. 1027 - 1042
• Main Authors: del Pino, Manuel, Musso, Monica, Wei, Jun Cheng
• Format: Journal Article
• Language: English
• Published: Beijing Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01.06.2019; Springer Nature B.V
• Subjects: Bubbling; Cauchy problems; Mathematics; Mathematics and Statistics; Thermodynamics; Singularity formation; bubbling phenomena; critical parabolic equations; 35K58; 35B40
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-019-8341-5

We consider the Cauchy problem for the energy critical heat equation { u t = Δ u + | u | 4 n − 2 u in R n × ( 0 , T ) u ( ⋅ , 0 ) = u 0 in R n in dimension n = 5. More precisely we find that for given points q 1 , q 2 ,..., q k and any sufficiently small T > 0 there is an initial condition u 0 such that the solution u ( x , t ) of (0.1) blows-up at exactly those k points with rates type II, namely with absolute size ~( T - t ) -α for α > 3 4 . The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin–Talenti bubbles.