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

Abstract

In this paper we study the global well-posedness of the following Cauchy problem on a sub-Riemannian manifold M:{ut−LMu=f(u),x∈M,t>0,u(0,x)=u0(x),x∈M, for u0≥0, where LM 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 u0≢0, blow up in finite time if and only if 1<p≤pF:=1+2/D when f(u)≃up, where D is the global dimension of G. In the case 1<p<pF and when f:[0,∞)→[0,∞) is a locally integrable function such that f(u)≥K2up for some K2>0, we also show that the differential inequalityut−LMu≥f(u) does not admit any nontrivial distributional (a function u∈Llocp(Q) which satisfies the differential inequality in D′(Q)) solution u≥0 in Q:=(0,∞)×G. Furthermore, in the case when G has exponential volume growth and f:[0,∞)→[0,∞) is a continuous increasing function such that f(u)≤K1up for some K1>0, we prove that the Cauchy problem has a global, classical solution for 1<p<∞ and some positive u0∈Lq(G) with 1≤q<∞. Moreover, we also discuss all these results in more general settings of sub-Riemannian manifolds M.