Persistence properties for a family of nonlinear partial differential equations

We examine the persistence of decay properties for a family of dispersive nonlinear partial differential equations. We show that certain decay properties of the initial data persist for as long as the solution exists. On the other hand, for a Subset of the family certain decay rates are possible only for the trivial solution. For example, the only solution that remains with compact Support for any further time is the trivial Solution. (C) 2008 Elsevier Ltd. All rights reserved.