New life-span results for the nonlinear heat equation

TL;DR

This paper provides new estimates for the lifespan of solutions to the nonlinear heat equation with various initial data types, including non-regular and sign-changing data, using dilation methods and local solution constructions.

Contribution

It introduces novel lifespan estimates for solutions to the nonlinear heat equation, extending results to non-polynomial decay, measures, and Hardy-Hénon equations.

Findings

New lower and upper bounds for solution lifespan.

Lifespan estimates for non-regular and sign-changing initial data.

Extension of results to Hardy-Hénon parabolic equations.

Abstract

We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation $\partial_tu-\Delta u=|u|^\alpha u,\;\alpha>0$ with initial values in Lebesgue, weighted Lebesgue spaces or measures. Non-regular, sign-changing, as well as non polynomial decaying initial data are considered. The proofs of the lower-bound estimates of life-span are based on the local construction of solutions. The proofs of the upper-bounds exploit a well-known necessary condition for the existence of nonnegative solutions. In addition, we establish new results for life-span using dilation methods and we give new life-span estimates for Hardy-H\'enon parabolic equations.