Semilinear Parabolic Equations on the Heisenberg Group with a Singular Potential

Houda Mokrani,Fatimetou Mint Aghrabatt

doi:10.1155/2012/749683

Abstract

We discuss the asymptotic behavior of solutions for semilinear parabolic equations on the Heisenberg group with a singular potential. The singularity is controlled by Hardy's inequality, and the nonlinearity is controlled by Sobolev's inequality. We also establish the existence of a global branch of the corresponding steady states via the classical Rabinowitz theorem.

Highlights

In this paper, we study a class of parabolic equations on the Heisenberg group Hd
H1 Ω, Hd f ∈ L2 Ω ; ∇Hd f ∈ L2 Ω, 1.5 and H01 Ω, Hd is the closure of C0∞ Ω in H1 Ω, Hd
We introduce the Banach space X H01 Ω, Hd, and the inner product in X is given by u, v X ≡

Summary

The left invariant vector fields are

In the sequel we will denote, we will denote Zj Xj and Zj d Yj for j ∈ {1, . . . , d}. In the sequel we will denote, we will denote Zj Xj and Zj d Yj for j ∈ {1, . Abstract and Applied Analysis where ρ is the Heisenberg distance. The Laplacian-Kohn operator on Hd and Heisenberg gradient is given by n. Let Ω be an open and bounded domain of Hd, we define the associated Sobolev space as follows. H1 Ω, Hd f ∈ L2 Ω ; ∇Hd f ∈ L2 Ω , 1.5 and H01 Ω, Hd is the closure of C0∞ Ω in H1 Ω, Hd. We are concerned in the following semilinear parabolic problem

ΔHd u μ

Define un t

Then we have t