Sharp Lifespan Dichotomies and Threshold Phenomena for Semilinear Heat Equations Driven by the Logarithmic Laplacian

Abstract

We study nonnegative mild solutions of \[ \partial_tu+(-Δ)^{\ln}u=f(u) \qquad \text{in }(0,T)\times\mathbb R^N, \] with initial datum \(u(0,\cdot)=μu_0\), \(μ>0\). Unlike the classical and fractional heat kernels, the positive logarithmic heat kernel exists only for \(0<t<N/2\). The associated linear evolution may become singular at its terminal time, and both its maximal lifespan and terminal growth depend on the spatial decay of \(u_0\). The behavior of \(f\) near zero determines local solvability: if ${\int }_{0^{+}}\frac{d\ddot{I}ƒ}{f(\ddot{I}ƒ)}<\infty ,$ then no finite nonnegative solution exists on any positive time interval. Under \((\mathcal U_α)\) and \((\mathcal F)\), we develop a well-posedness theory adapted to the nonintegrable logarithmic heat kernel. We also prove two complementary lifespan criteria: globally at most linear growth of \(f\) yields the full linear lifespan, whereas ${\int }_{s_{\ast }}^{\infty }\frac{d\ddot{I}ƒ}{f(\ddot{I}ƒ)}<\infty$ implies that the maximal existence time tends to zero as \(μ\to\infty\). We then establish lifespan dichotomies in the slow-decay, fast-decay, and critical-tail regimes. In the noncritical regimes, the weighted Osgood tail condition forces premature blow-up, while its failure, under \((\mathcal F_\infty)\), yields an amplitude threshold. For critical-tail data, an analogous dichotomy holds with the square-root weighted Osgood condition, and the dividing power becomes \(3/2\). Finally, we derive terminal-time estimates and sharp blow-up rates for power nonlinearities.