Behaviour in time of solutions to a class of fourth order evolution equations

Abstract

We consider some initial-boundary value problems for a class of nonlinear parabolic equations of the fourth order, whose solution u(x,t) may or may not blow up in finite time. Under suitable conditions on data, a lower bound for t⁎ is derived, where [0,t⁎) is the time interval of existence of u(x,t). Under appropriate assumptions on the data, a criterion which ensures that u cannot exist for all time is given, and an upper bound for t⁎ is derived. Some extensions for a class of nonlinear fourth order parabolic systems are indicated.