2026/07/31 by Huyuan Chen, Rui Chen, Daniel Hauer +1
Mathematics · #math.AP #msc:35K58 #msc:35A01 #msc:35B44 #msc:35R11
arxiv created 2026/07/31 · arxiv updated 2026/08/03
We study nonnegative mild solutions of ∂tu+(-Δ)lnu=f(u) in (0,T)×\mathbb RN, with initial datum \(u(0,⋅)=μu0\), \(μ>0\). 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 \(u0\). The behavior of \(f\) near zero determines local solvability: if ∫0+(dσ)/(f(σ))<∞, then no finite nonnegative solution exists on any positive time interval. Under \((\mathcal Uα)\) 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 ∫s_*∞(dσ)/(f(σ))<∞ implies that the maximal existence time tends to zero as \(μ→∞\). 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 \((\mathcal F_∞)\), 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.