New life-span results for the nonlinear heat equation

We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation ∂tu−Δu=|u|αu,α>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...

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Bibliographic Details
Published in:Journal of Differential Equations 2023-11, Vol.373, p.564-625
Main Authors: Tayachi, Slim, Weissler, Fred B.
Format: Article
Language:English
Subjects:
Blowup
Hardy-Hénon parabolic equations
Life-span
Local well-posedness
Nonlinear heat equation
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Summary:We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation ∂tu−Δu=|u|αu,α>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-Hénon parabolic equations.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2023.07.011