Nonexistence results for a class of evolution equations in the Heisenberg group

Abstract

Abstract We establish a nonexistence result of global solutions to the nonlinear evolution equation (|u|β)tt − |ϑ|α HΔH(|u|β) + h(ϑ)(|u|q)t = f(t, ϑ)|u|p + w(t, ϑ), ϑ ∈ H, where ΔH is the Kohn-Laplace operator on the (2N+1)-dimensional Heisenberg group H, |ϑ|H is the distance from ϑ to the origin, β, p, q > 0, α ≥ 0, f(t, v) ≥ 0, h(ϑ) and w(t, ϑ) are given functions. Next, we extend this result to the case of systems. Our technique of proof is based on Pohozaev’s nonlinear capacity method.