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Local well‐posedness and blow‐up for an inhomogeneous nonlinear heat equation
Author(s) -
Alessa Rasha,
Alshehri Aisha,
Altamimi Haya,
Majdoub Mohamed
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6266
Subject(s) - mathematics , nonlinear system , standard probability space , lebesgue integration , heat equation , polynomial , mathematical analysis , exponential growth , space (punctuation) , lp space , exponential function , function (biology) , banach space , linguistics , philosophy , physics , quantum mechanics , evolutionary biology , biology
In this paper, we consider the nonlinear heat equation with inhomogeneous nonlinearityu t − Δ u = a ( x ) f ( u ) where f : R → R having either a polynomial growth or exponential growth, and a : R N → R is a function satisfying some assumptions to be stated later. We first prove the local well‐posedness in suitable Lebesgue spaces when a belongs to some Lebesgue space and f has polynomial growth. We also obtain some blow‐up results.