ABSTRACT
Let Ω be a bounded sufficiently smooth domain in IRN . Taking the original equation (129) and setting, as usual,
|u|nu = v =⇒ u = |v|− nn+1 v, where − nn+1 > 0 for n ∈ (−1, 0), we arrive at the following initial boundary value problem:
{ ∂ ∂tψ(v) = (−1)m+1Δmv + v in Q = Ω× IR+, v = Dv = . . . = Dm−1v = 0 on ∂Ω× IR+, v(x, 0) = v0(x) in Ω,
(166)
where v0 is an initial function from an appropriate space to be specified. Here, the only nonlinearity is ψ(v) = |v|− nn+1 v. We examine problem (166) in the “native” energy Sobolev space. We introduce the following functionals associated with (166):
Φ(t) := 12 ∫ Ω |v(t, x)|
E(t) := − ∫Ω |D˜mv(t, x)|2 dx− ∫ Ω v
2 dx.
