We classify the blow up self-similar profiles for the following reaction-diffusion
equation with weighted reactionu(t) = (u(m))(xx) + vertical bar x vertical bar(sigma)
u(m),posed for (x, t) is an element of R x (0, T), with m > 1 and sigma > 0. In strong
contrast with the well-studied equation without the weight (that is sigma = 0), on
the one hand we show that for sigma > 0 sufficiently small there exist multiple self-similar
profiles with interface at a finite point, more precisely, given any positive integer
k, there exists delta(k) > 0 such that for sigma is an element of (0, delta(k)), there
are at least k different blow up profiles with compact support and interface at a
positive point. On the other hand, we also show that for sigma sufficiently large,
the blow up self-similar profiles with interface cease to exist. This unexpected balance
between existence of multiple solutions and non-existence of any, when sigma > 0 increases,
is due to the effect of the presence of the weight vertical bar x vertical bar(sigma),
whose influence is the main goal of our study. We also show that for any sigma > 0,
there are no blow up profiles supported in the whole space, that is with u(x, t) >
0 for any x is an element of R and t is an element of (0, T). (C) 2019 Elsevier Ltd.
All rights reserved.
