Bandle, Catherine (ed.) et al., Elliptic and parabolic problems. A special tribute to the work of Haim Brezis. Basel: Birkhäuser (ISBN 3-7643-7249-4/hbk). Progress in Nonlinear Differential Equations and their Applications 63, 43-50 (2005).
The quasilinear parabolic problem
where and is a bounded open subset of , , , is studied. Furthermore is assumed to be of class and
where is the inner product in and . Let when , while when . Suppose that , , and
where is the best constant of the embedding . The authors prove that then the energy of the solution decays exponentially if , and polynomial when .