Title
Convection diffusion problems with non-integrable initial data
Abstract
In this talk we present some recent result about the long time behavior of the solutions to the one-dimensional diffusion problem with nonlinear convection
ut −uxx +(F(u))x = 0 in Q := (0,∞)×R, and u(0) = u0 on R,
where
F(u) := a|u|q-1u, a ∈ {±1}, q > 1,
and u0 ∈ L∞(RN) but not necessarily integrable.
The prototype of the initial data we have in mind is
u0(x) ≃ k± /|x|γ , as x → ±∞, 0 < γ ≤1.
In contrast with the case of integrable data where critical exponent governing the long time bahaviour is q = 2, here qc = 1+ 1/γ. We obtain the main term in the asymptotic expansion of the solutions for large time.
This is a joint work with Fernando Quiros (Univer sidad Autonoma de Madrid) and Nicola de Nitti (Polytechnic University of Bari)
