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We consider the following exponential reaction-diffusion equation involving anonlinear gradient term: $$\partial_t U = \Delta U + \alpha|\nabla U|^2 +e^U,\quad (x, t)\in\mathbb{R}^N\times[0,T), \quad \alpha > -1.$$ We constructfor this equation a solution which blows up in finite time $T > 0$ andsatisfies some prescribed asymptotic behavior. We also show that theconstructed solution and its gradient blow up in finite time $T$ simultaneouslyat the origin, and find precisely a description of its final blowup profile. Ithappens that the quadratic gradient term is critical in some senses, resultingin the change of the final blowup profile in comparison with the case $\alpha =0$. The proof of the construction inspired by the method of Merle and Zaag in1997, relies on the reduction of the problem to a finite dimensional one, anduses the index theory to conclude. One of the major difficulties arising in theproof is that outside the \textit{blowup region}, the spectrum of thelinearized operator around the profile can never be made negative. Truly newideas are needed to achieve the control of the outer part of the solution.Thanks to a geometrical interpretation of the parameters of the finitedimensional problem in terms of the blowup time and the blowup point, we obtainthe stability of the constructed solution with respect to perturbations of theinitial data.

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