M. Fila et al., Linear and nonlinear heat equations in L-delta(q) spaces and universal bounds for global solutions, MATH ANNAL, 320(1), 2001, pp. 87-113
We develop a theory of both linear and nonlinear heat equations in the weig
hted Lebesgue spaces L-delta(q) where delta is the distance to the boundary
. In particular, we prove an optimal L-delta(q) - L-delta(r) estimate for t
he heat semigroup, and we establish sharp results on local existence-unique
ness and local nonexistence of solutions for semilinear heat equations with
initial values in those spaces. This theory enables us to obtain new types
of results concerning positive global solutions of superlinear parabolic p
roblems. Namely, under certain assumptions, we prove that any global soluti
on is uniformly bounded for t less than or equal to tau > 0 by a universal
constant, independent of the initial data. In all previous results, the bou
nds for global solutions were depending on the initial data.