Transient Bounds and Time-Asymptotic Behavior of Solutions to Nonlinear Equations of Fisher Type

A study is made of the transient behavior of solutions to certain classes of semilinear parabolic initial value problems. First, families of upper and lower bounds for these solutions are explicitly constructed and compared. Secondly, the question of time-asymptotic convergence of these solutions towards certain traveling wave solutions (of the underlying parabolic equation) is discussed. It is shown, in particular, that this long time behavior depends crucially and in an interesting manner on the behavior of initial data at infinity.