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We consider the semilinear heat equation, to which we add a nonlineargradient term, with a critical power. We construct a solution which blows up infinite time. We also give a sharp description of its blow-up profile. The proofrelies on the reduction of the problem to a finite dimensional one, and usesthe index theory to conclude.  Thanks to the interpretation of the parameters of the finite-dimensionalproblem in terms of the blow-up time and point, we also show the stability ofthe constructed solution with respect to initial data. This note presents theresults and the main arguments. For the details, we refer to our paper\cite{TZ15}.

Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term