No abstract

+∞] the maximal existence time of the classical solution (u, v) of problem (1.1). If T < +∞, then the solution blows up in finite time T in the sense thatlim t→T ( u(t) L ∞ (R N ) + v(t) L ∞ (R N ) ) = +∞.

We construct a solution for a class of strongly perturbed semilinear heat equations which blows up in finite time with a prescribed blow-up profile. The construction relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude.

We will study the problem under an additional assumption of 1-corotational symmetry, with the following corotational ansatz Φ(x, t) = cos(u(|x|, t))x |x| sin(u(|x|, t)) 2 √ κ , where κ is an upper bound on the sectional curvature of the target manifold M (see Jost [31] and Lin-Wang [33]). Without these assumptions, the solution u(r, t) may develop singularities in some finite time (see for examples, Coron and Ghidaglia [14], Chen and Ding [11] for d ≥ 3, Chang, Ding and Yei [12] for d = 2). In this case, we say that u(r, t) blows up in a finite time T < +∞ in the sense that lim t→T ∇u(t) L ∞ = +∞.Here we call T the blowup time of u(x, t). The blowup has been divided by Struwe [60] into two types: u blows up with type I if: lim sup t→T (T − t) 1 2 ∇u(t) L ∞ < +∞, u blows up with type II if: lim sup t→T (T − t) 1 2 ∇u(t) L ∞ = +∞. 1-COROTATIONAL HARMONIC MAP HEAT FLOW IN SUPERCRITICAL DIMENSIONS 4 d−4−2 √ d−1 for d ≥ 11, correspond to the cases d = 2 and d = 7 in the study of equation (1.4) respectively.