Large self-similar solutions of a convection--diffusion equation

We construct self-similar solutions to a nonlinear convection–diffusion equation posed on the whole Euclidean space, for a class of homogeneous initial data. The corresponding self-similar profiles solve a nonlinear elliptic equation on ${\mathbb{R}}^N$. We obtain these profiles as limits of solutions to Dirichlet problems on bounded domains with truncated nonlinearities, using a priori estimates that are uniform with respect to both the domain size and the truncation level. Our approach is inspired by recent constructions of large self-similar solutions for the Navier–Stokes system. The convection–diffusion equation considered here shares some structural features with the Navier–Stokes equations, but its scalar parabolic nature allows us to exploit tools such as the maximum principle and thereby obtain a more direct construction of self-similar solutions.