A Dynamical System Approach to the Self-Similar Large Time Behavior in Scalar Convection-Diffusion Equations

Abstract

We present a dynamical system method that provides both the existence of self-similar solutions and the self-similar large time behavior for convection-diffusion equations in R N . This method avoids a direct study of the elliptic problem related to the self-similar profiles. We concentrate our attention on the model example u t − Δu =a · ∇(|u| 1/Nu + g(x,t) in R N × (0, ∞) u(0) = u 0 ∈ L 1( R N) ∩ L ∞( R N), where a ∈ R N and g( x, t) = ( t + 1) −( N/2)−1 h( x/[formula]) with h ∈ L 2( R N ; exp(| x| 2/4)) and L ∞ ( R N ) such that ∫ h( y) dy = 0 and ∫ | h( y)|| y| dy being small enough. In the natural similarity variables the self-similar profiles become stationary solutions of a new convection-diffusion equation. By using Lyapunov type arguments that rely in an essential manner on the L 1-contraction property of the system, we prove that those stationary solutions exists and that any trajectory converges to one of them. The limit of any trajectory is completely determined by its mass which is conserved along the time. In order to ensure the relative compactness of trajectories, we work in the functional framework of the weighted Sobolev spaces introduced by Escobedo and Kavian.