Existence and non-existence of global solutions for semilinear heat equations and inequalities on sub-Riemannian manifolds, and Fujita exponent on unimodular Lie groups

In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold M: {u(t) -LMu =f (u), x is an element of M, t > 0, u(0, x) = u(0)(x), x is an element of M, for u(0) >= 0, where L-M is a sub-Laplacian of M. In the case when M is a connected unimodular Lie group G, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with u(0) equivalent to 0, blow up in finite time if and only if 1 < p [0, infinity) is a locally integrable function such that f (u) > K2up for some K2 > 0, we also show that the differential inequality ut - ,Mu pound >= f (u) does not admit any nontrivial distributional (a function u E Lploc(Q) which satisfies the differential in- equality in D'(Q)) solution u > 0 in Q := (0, oo) x G. Furthermore, in the case when G has exponential volume growth and f : [0, oo) ->[0, oo) is a continuous increasing function such that f (u) < K(1)u(p) for some K1 > 0, we prove that the Cauchy problem has a global, classical solution for 1 < p < oo and some positive u(0) E Lq(G) with 1 < q < oo. Moreover, we also discuss all these results in more general settings of sub-Riemannian manifolds M. (C) 2021 The Authors. Published by Elsevier Inc.