X. Cabre et Y. Martel, Existence versus instantaneous blow-up for linear heat equations with singular potentials, CR AC S I, 329(11), 1999, pp. 973-978
Citations number
9
Categorie Soggetti
Mathematics
Journal title
COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE
In this Note, we consider the linear heat equation u(t) - Delta u = a(x)u i
n (0, T) x Omega, u = 0 on (0, T) x partial derivative Omega, and u(0) = u(
0) on Omega, where Omega subset of R-N is a smooth bounded domain. We assum
e that a is an element of L-loc(1)(Omega), a greater than or equal to 0 and
u(0) greater than or equal to 0. A simple condition on the potential a is
necessary and sufficient for the existence of positive weak solutions that
are global ill time and grow at most exponentially in time. We show that th
is condition, based on the existence of a Hardy type inequality with weight
a(x), is "almost" necessary for the local existence in time of positive we
ak solutions. Applying these results to some "critical" potentials, we find
new results on existence and on instantaneous and complete blow-lip of sol
utions. (C) 1999 Academie des sciences/Editions scientifiques et medicales
Elsevier SAS.