Multiple Blowup for Nonlinear Heat Equations at Different Places and Different Times

We study a blowup problem for the semilinear heat equation ut = ∆u + u posed in R × (0,∞), where p is supercritical in the Sobolev sense. A solution u(x, t) is said to blow up at a time 0 < T < +∞ if there exists a sequence tn ↗ T as n → ∞ such that |u(·, tn)|∞ → +∞ as n → ∞ with supremum norm | · |∞ in RN . We establish the following result: for given times T1, T2 with 0 < T1 < T2 and any small ε > 0, there exist a proper solution u of the problem and times Tε,i with |Tε,i − Ti| < ε, i = 1, 2, such that u blows up at the origin at t = Tε,1, becomes a regular solution for t ∈ (Tε,1, Tε,2) and blows up again on a sphere at t = Tε,2. 2000 Mathematics Subject Classification. 35K20, 35K55, 58K57.