Sharp lifespan estimates for heat equations with Hörmander operators

Abstract

In this paper, we study the Cauchy problem $u_t-\mathcal L u=h(t)u^p$, $u(0)=\varepsilon u_0\ge0$, where $p>1$, $h\ge0$ is only locally integrable, and $\mathcal L=\sum_{j=1}^mX_j^2$ is a homogeneous Hörmander sum of squares on $\mathbb R^n$ of homogeneous dimension $q$. Diffusion spreads the solution out and weakens the nonlinearity, while the reaction $h(t)u^p$ may force blow-up. We prove that the outcome is decided by the single quantity $\mathcal{H}_{p,h}(T)=\int_0^Th(t)(1+t)^{-\frac q2(p-1)}\,d t$, in which the reaction at time $t$ is weighted by the factor by which diffusion has weakened the solution: small data are global if $\mathcal{H}_{p,h}(\infty)<\infty$, while otherwise the lifespan $T_\varepsilon$ satisfies $c\,\varepsilon^{-(p-1)}\le\mathcal{H}_{p,h}(T_\varepsilon)\le C\,\varepsilon^{-(p-1)}$ for small $\varepsilon$. Our technique of proof is based on a backward-semigroup functional combined with a Gaussian-envelope energy and an ODE cap telescoped along interlaced geometric grids. The result contains the Euclidean, Heisenberg, Carnot and Grushin cases, yields explicit lifespan laws for power, logarithmic, exponential, periodic and intermittent coefficients, and exhibits effects with no autonomous counterpart, such as a lifespan with no power law in $\varepsilon$. We also prove regularity and a priori estimates for classical solutions and discuss what remains true for a general nonlinearity $f(u)$.