On zero mass solutions of viscous conservation laws
| dc.creator | Karch, Grzegorz | |
| dc.creator | Schonbek, Maria E. | |
| dc.date | 2002-08-14 | |
| dc.date.accessioned | 2026-07-07T04:50:15Z | |
| dc.date.available | 2026-07-07T04:50:15Z | |
| dc.description | In the paper, we consider the large time behavior of solutions to the convection-diffusion equation u_t - Delta u + nabla cdot f(u) = 0 in R^n times [0,infinity), where f(u) ~ u^q as u --> 0. Under the assumption that q >= 1+1/(n+beta) and the initial condition u_0 satisfies: u_0 in L^1(R^n), integral_{R^n} u_0(x) dx = 0, and |e^{t Delta}u_0|_{L^1(R^n)} <= Ct^{-beta/2} for fixed beta in (0,1), all t>0, and a constant C, we show that the L^1-norm of the solution to the convection-diffusion equation decays with the rate t^{-beta/2} as t --> infinity. Moreover, we prove that, for small initial conditions, the exponent q^* = 1+1/(n+beta) is critical in the following sense. For q > q^* the large time behavior in L^p(R^n), 1 <= p <= infinity, 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. | |
| dc.description | To appear in Comm. Partial Diff. Eq | |
| dc.identifier | https://arxiv.org/abs/math/0208111 | |
| dc.identifier | http://arxiv.org/abs/math/0208111 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64721 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Mathematical Physics | |
| dc.subject | 35B40; 35K55 | |
| dc.title | On zero mass solutions of viscous conservation laws | |
| dc.type | text |