+∞] 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 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.

We consider the following exponential reaction-diffusion equation involving a nonlinear gradient term:We construct for this equation a solution which blows up in finite time T > 0 and satisfies some prescribed asymptotic behavior. We also show that the constructed solution and its gradient blow up in finite time T simultaneously at the origin, and find precisely a description of its final blowup profile. It happens that the quadratic gradient term is critical in some senses, resulting in the change of the final blowup profile in comparison with the case α = 0. The proof of the construction inspired by the method of Merle and Zaag in 1997, relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. One of the major difficulties arising in the proof is that outside the blowup region, the spectrum of the linearized operator around the profile can never be made negative. Truly new ideas are needed to achieve the control of the outer part of the solution. Thanks to a geometrical interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we obtain the stability of the constructed solution with respect to perturbations of the initial data.2010 Mathematics Subject Classification. Primary: 35K58, 35K55; Secondary: 35B40, 35B44.

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We construct solutions to the two‐dimensional parabolic‐elliptic Keller‐Segel model for chemotaxis that blow up in finite time T. The solution is decomposed as the sum of a stationary state concentrated at scale λ and of a perturbation. We rely on a detailed spectral analysis for the linearised dynamics in the parabolic neighbourhood of the singularity performed by the authors in [10], providing a refined expansion of the perturbation. Our main result is the construction of a stable dynamics in the full nonradial setting for which the stationary state collapses with the universal law λ ∼ 2 e − 2 + γ 2 T − t e − ∣ ln T − t ∣ 2 where γ is the Euler constant. This improves on the earlier result by Raphael and Schweyer and gives a new robust approach to so‐called type II singularities for critical parabolic problems. A by‐product of the spectral analysis we developed is the existence of unstable blowup dynamics with speed λ ℓ ∼ C 0 T − t normalℓ 2 ln T − t − ℓ 2 () normalℓ − 1 1em for normalℓ ≥ 2 integer . © 2021 Wiley Periodicals LLC.