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 $M$: \begin{equation*} \begin{cases} u_{t}-\mathfrak{L}{M} u=f(u), ;x\in M, ;t>0, \u(0,x)=u{0}(x), ;x\in M, \end{cases} \end{equation*} for $u_0\ge 0$, where ${\mathfrak{L}}_M$ is a sub-Laplacian of $M$. In the case when $M$ is a connected unimodular Lie group $\mathbb{G}$, which has polynomial volume growth, we obtain a critical Fujita exponent, namely, we prove that all solutions of the Cauchy problem with $u_0\equiv ̸0$, blow up in finite time if and only if $1<p\leq p_{F}:=1+2/D$when$f(u)\simeq u^{p}$,where$D$istheglobaldimensionof$\mathbb G$\mathrm{.}Inthecase$1<p<p_{F}$andwhen$f:[0,\infty)\to [0,\infty)$ is a locally integrable function such that $f(u)\ge K_2{u}^p$ for some $K_2>0$, we also show that the differential inequality $$u_t-{\mathfrak{L}}_{M}u\ge f(u)$$ does not admit any nontrivial distributional (a function $u\in L^{p}{loc}(Q)$whichsatisfiesthedifferentialinequalityin$$\mathcal{D}$^{\prime}(Q)$)solution$u\geq 0$in$Q:=(0,\infty)\times\mathbb G$\mathrm{.}Furthermore,inthecasewhen$\mathbb G$hasexponentialvolumegrowthand$f:[0,\infty)\to[0,\infty)$isacontinuousincreasingfunctionsuchthat$f(u)\leq K{1}u^{p}$forsome$K_{1}>0$, we prove that the Cauchy problem has a global, classical solution for $1<p<\mathrm{\infty }$ and some positive $u_{0}\in L^{q}(\mathbb G)$with$1\leq q<\infty$. Moreover, we also discuss all these results in more general settings of sub-Riemannian manifolds $M$.