Blowup solutions for a nonlinear heat equation involving a critical power nonlinear gradient term

We consider the following exponential reaction diffusion equation involving a nonlinear gradient term: partial derivative U-t = Delta U + alpha vertical bar del U vertical bar(2) + e(U), (x, t) is an element of R-N x [0, T), alpha > -1. 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 sense, resulting in the change of the final blowup profile in comparison with the case alpha = 0. The proof of the construction is inspired by the method of Merle and Zaag in 1947. It 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 in the initial data. (C) 2017 Elsevier Inc. All rights reserved.