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

Research paper by **Tej-Eddine Ghoul, Van Tien Nguyen, Hatem Zaag**

Indexed on: **08 Nov '16**Published on: **08 Nov '16**Published in: **arXiv - Mathematics - Analysis of PDEs**

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#### Abstract

We consider the following exponential reaction-diffusion equation involving a
nonlinear 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 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 $\alpha =
0$. 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 \textit{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.