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 $M$: \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 $u_0\not\equiv 0$, blow up in finite time if and only if $1<p\le p_F:=1+2/D$ 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 $f(u)\ge K_2{u}^p$ 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 $u\in L_{loc}^p(Q)$ which satisfies the differential inequality in ${\mathcal{D}}^{\prime }(Q)$) solution $u\ge 0$ 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 $f(u)\le K_1{u}^p$ 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 $1\le q<\infty$. Moreover, we also discuss all these results in more general settings of sub-Riemannian manifolds $M$.

17 pages, Final version, to appear in J. Differential Equations

References in corpus (4)

• Lifespan estimates for local in time solutions to the semilinear heat equation on the Heisenberg group
• On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups
• A comparison principle for nonlinear heat Rockland operators on graded groups
• Semilinear wave equation on compact Lie groups

Cited by in corpus (7)

• Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity
• Decay estimates for the linear damped wave equation on the Heisenberg group
• Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified Groups
• On the blow-up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups
• Fujita exponent on stratified Lie groups
• Liouville theorems for Kirchhoff-type hypoelliptic Partial Differential Equations and systems. I. Heisenberg group
• Blow-up solutions of damped Klein-Gordon equation on the Heisenberg group