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 $u_{0}\geq 0$, 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 $u_{0}\not\equiv 0$, blow up in finite time if and only if $1<p\leq p_{F}:=1+2/D$ when $f(u)\simeq u^{p}$, where $D$ is the global dimension of $\mathbb G$. In the case $1<p<p_{F}$ and when $f:[0,\infty)\to [0,\infty)$ is a locally integrable function such that $f(u)\geq K_{2}u^{p}$ for some $K_{2}>0$, we also show that the differential inequality $$ u_{t}-\mathfrak{L}_{M} u\geq f(u) $$ does not admit any nontrivial distributional (a function $u\in L^{p}_{loc}(Q)$ which satisfies the differential inequality in $\mathcal{D}^{\prime}(Q)$) solution $u\geq 0$ in $Q:=(0,\infty)\times\mathbb G$. Furthermore, in the case when $\mathbb G$ has exponential volume growth and $f:[0,\infty)\to[0,\infty)$ is a continuous increasing function such that $f(u)\leq K_{1}u^{p}$ for some $K_{1}>0$, we prove that the Cauchy problem has a global, classical solution for $1<p<\infty$ 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 $M$.