Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

==== The heat semigroup smooths the argument because: ====
# The Duhamel term is a Bochner integral.
# Bochner integrability requires F∈L1F\in L^1F∈L1 in time.
# Heat kernel estimates convert L1L^1L1-in-time forcing into spatial smoothing.
# Loss of L1L^1L1-integrability is precisely the mechanism by which blowup can occur.

If you want, next I can:
* prove the Bochner-integrability claim rigorously,
* show how the Picard iteration guarantees F∈L1F\in L^1F∈L1 locally,
'' or show explicitly how loss of L1L^1L1 near T∗T^''T∗ obstructs smoothing.

Just say which.