Construction of a blow-up solution for a perturbed nonlinear heat equation with a gradient term

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. We prove the existence of a blow-up solution, and give its blow-up profile. Our method relies on the two-step method: we first linearize the equation (in similarity variables) around the expected profile, then we use a topological argument to controle the positive directions of the spectrum