Abstract :
This paper is devoted to nonexistence results for solutions to the problem
(Sm
k ) ∂kui
∂tk −ΔH(aiui ) |η|γi+1
H |ui+1|pi+1, η∈ HN, t ∈ ]0,+∞[, 1 i m,
um+1 = u1,
where ΔH is the Laplacian on the (2N +1)-dimensional Heisenberg group HN, |η|H is the distance
from η in H to the origin, m 2, k 1, pm+1 = p1, γm+1 = γ1, and ai ∈ L∞(HN × ]0,+∞[),
1 i m. These nonexistence results hold forQ≡ 2N +2 less than critical exponents which depend
on k, pi and γi, 1 i m. For k = 1 and 2 we retrieve the results, obtained by El Hamidi and
Kirane (Manuscripta Math., submitted), corresponding, respectively, to the parabolic and hyperbolic
systems. In order to show that the obtained exponents are also valid for m = 1, we study the scalar
case
(Ik)
∂ku
∂tk −ΔH(au) |η|γ
H |u|p,
where p >1, γ are real parameters, and a ∈ L∞(HN×]0,+∞[).
2003 Elsevier Science (USA). All rights reserved
