ABSTRACT
This book focuses attention into the problem of ascertaining the asymptotic behavior of the solutions of the parabolic problem
∂u ∂t − d∆u = λu+ a(x)f(x, u)u in Ω× (0,∞), u = 0 on ∂Ω× (0,∞), u(·, 0) = u0 > 0 in Ω,
(1.1)
where d > 0 is a constant, ∆ stands for the Laplace operator in RN ,
∆ :=
∂2
∂x2j , x = (x1, ..., xN ) ∈ RN ,
Ω is a bounded domain of RN , N ≥ 1, with smooth boundary ∂Ω of class C2+ν , for some ν ∈ (0, 1], λ ∈ R is regarded as a parameter, and
a ∈ Cν(Ω¯), f ∈ Cν,1+ν(Ω¯× [0,∞)),
although these regularity requirements can be substantially relaxed to assume ∂Ω is of class C2, a ∈ L∞(Ω) and f ∈ C0,1(Ω¯× [0,∞)). However the first eight chapters of this book will study the special situation when
a < 0 (a ≤ 0, a 6= 0),
which is usually referred to as the sublinear case if f ≥ 0, because in such circumstances we have
λu+ a(x)f(x, u)u ≤ λu for all u ≥ 0,
the general case when a(x) changes sign will be dealt with in Chapter 9.
