Lifespan estimates for local in time solutions to the semilinear heat equation on the Heisenberg group

Abstract

In this paper, we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group \(\mathbf {H}_n\). The heat operator is given in this case by \(\partial _t-\varDelta _{{{\,\mathrm{H}\,}}}\), where \(\varDelta _{{{\,\mathrm{H}\,}}}\) is the so-called sub-Laplacian on \(\mathbf {H}_n\). We prove that the Fujita exponent \(1+2/Q\) is critical, where \(Q=2n+2\) is the homogeneous dimension of \(\mathbf {H}_n\). Furthermore, we prove sharp lifespan estimates for local in time solutions in the subcritical case and in the critical case. In order to get the upper bound estimate for the lifespan (especially, in the critical case), we employ a revisited test function method developed recently by Ikeda–Sobajima. On the other hand, to find the lower bound estimate for the lifespan, we prove a local in time result in weighted \(L^\infty \) space.