Positivity 8: 187–208, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Positivity, Trotter Products, and Blow-up
, JEROME A. GOLDSTEIN
, and MARKUS WACKER
1980 South Dexter Street, Denver, CO 80222, USA. E-mail: firstname.lastname@example.org;
University of Memphis, Memphis, TN 38152, USA. E-mail: email@example.com;
Mathematisches Institut, Auf der Morgenstelle 10, D-72076 Tübingen, Germany. E-mail:
(Received 6 November 2001; accepted 5 January 2003)
Abstract. We give a new approach to blow-up results for nonlinear evolution equations of the
form ˙ut=Aut+ut on ordered Banach spaces, using the Lie–Trotter product formula
for obtaining lower and upper bounds for the solution.
1991 Mathematics Subject Classiﬁcations. 47H20, 35G25, 47D06
Key words: ordered Banach space, blow-up, ﬂow semigroup, sum of operators, Trotter product
In the mid 1960s, H. Fujita [18,19] was the ﬁrst to obtain blow-up results for non-
linear parabolic equations using the following simple idea. Consider the differential
where Kc>0 and q>1. It can be solved explicitly by
This formula shows that utc is increasing in t and blowing up as
On the other hand, consider the initial value problem for the heat equation
=kutx t 0
on one of the Banach spaces X=C
, with 1p<
, where x ∈
f∈ Xk>0, and