On Zero Mass Solutions of Viscous Conservation Laws

Abstract

In the paper, we consider the large time behavior of solutions to the convection-diffusion equation ut−∆u+∇·f(u) = 0 in IRn× [0,∞), where f(u) ∼ uq as u → 0. Under the assumption that q ≥ 1 + 1/(n + β) and the initial condition u0 satisfies: u0 ∈ L1(IRn), ∫ IRn u0(x) dx = 0, and ‖eu0‖L1(IRn) ≤ Ct−β/2 for fixed β ∈ (0, 1), all t > 0, and a constant C, we show that the L1-norm of the solution to the convection-diffusion equation decays with the rate t−β/2 as t → ∞. Moreover, we prove that, for small initial conditions, the exponent q∗ = 1+1/(n+β) is critical in the following sense. For q > q∗ the large time behavior in Lp(IRn), 1 ≤ p ≤ ∞, of solutions is described by self-similar solutions to the linear heat equation. For q = q∗, we prove that the convection-diffusion equation with f(u) = u|u|q∗−1 has a family of self-similar solutions which play an important role in the large time asymptotics of general solutions. 2000 Mathematics Subject Classification: 35B40, 35K55.

Cite this paper

@inproceedings{Karch2008OnZM, title={On Zero Mass Solutions of Viscous Conservation Laws}, author={Grzegorz Karch and Maria E. Schonbek}, year={2008} }