A diffusion-convection equation in several space dimensions. (English) Zbl 0791.35059

Escobedo, Miguel; Vázquez, Juan Luis; Zuazua, Enrike

Indiana Univ. Math. J. 42, No. 4, 1413-1440 (1993).

We study the large-time behaviour of nonnegative solutions \(u(x,t)\) of the diffusion-convection equation (1) \(u_ t= \Delta u- {\mathbf a}\cdot \nabla(u^ q)\), defined in the whole space \(\mathbb{R}^ N\) for \(t>0\), with initial data \(u_ 0\in L^ 1(\mathbb{R}^ N)\). The direction a is assumed to be constant.
We concentrate in the exponent range \(1<q< (N+1)/N\) and show that for very large times the effect of diffusion is negligible as compared to convection in the direction a, while in the directions orthogonal to a the motion is explained by diffusion. More precisely, the asymptotic behaviour of the solutions to (1) is given by the fundamental entropy solutions of the reduced equation \(u_ t= \Delta' u+{\mathbf a}\cdot \nabla(u^ q)\), where \(\Delta'\) denotes the \((N-1)\)-dimensional Laplacian in the hyperplane orthogonal to a. Existence and uniqueness of such special solutions, which have a selfsimilar form, is proved here, previous to establishing the asymptotic convergence. Such a phenomenon does not occur for \(q=1\) or for \(q\geq (N+1)/N\).

Reviewer: M.Escobedo (Bilbao)

MSC:

35K55	Nonlinear parabolic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

existence; Cauchy problem; nonlinear diffusion-convection equation; large-time behaviour; asymptotic behaviour; fundamental entropy solutions; uniqueness