Systems of semilinear higher-order evolution inequalities on the Heisenberg group

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