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.

在本文中，我们考虑了Heisenberg群\（\ mathbf {H} _n \）中具有功率非线性的热方程的半线性柯西问题。在这种情况下，热运算符由\（\ partial _t- \ varDelta _ {{{\，\ mathrm {H} \，}}} \\）给出，其中\（\ varDelta _ {{{\\\ mathrm { H} \，}}}} \）是\（\ mathbf {H} _n \）上的所谓次拉普拉斯式。我们证明藤塔指数\（1 + 2 / Q \）是关键的，其中\（Q = 2n + 2 \）是\（\ mathbf {H} _n \）的齐次维。此外，我们证明了在亚临界情况下和在临界情况下对本地及时解决方案的使用寿命的精确估计。为了获得寿命的上限估计值（特别是在紧急情况下），我们采用了由池田-Sobajima最近开发的重新测试函数方法。另一方面，为了找到寿命的下界估计值，我们证明了加权\（L ^ \ infty \）空间中的局部时间结果。