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. In some earlier works [1, 2], we constructed a solution u for that equation such that u and ∇u both blow up at the origin and only there. We also gave the final blow-up profile. In this paper, we refine our construction method in order to get a sharper estimate on the gradient at blow-up.

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. In some earlier work [1], we constructed a blow-up solution for that equation, and showed that it blows up (at least) at the origin. We also derived the so called "intermediate blow-up profile". In this paper, we prove the single point blow-up property and determine the final blow-up profile.

We consider a hyperbolic quasilinear perturbation of the Navier–Stokes equations in three space dimensions. We prove global existence and uniqueness of solutions for initial data and forcing terms, which are larger and less regular than in previous works. Furthermore, we prove the convergence of solutions to relaxed system towards solutions to the classical Navier–Stokes problem.

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We consider the second grade fluid equations on a thin three-dimensional domain with periodic boundary conditions. We prove global existence and uniqueness of the solution for large initial data. We use an appropriate decomposition of solution u into a v part, which is solution of a [Formula: see text] second grade fluid equations and the remaining w part which has an initial data converging to 0 as the thickness of the thin domain goes to 0.