Well-posedness and dynamic properties of solutions to a class of fourth order parabolic equation with mean curvature nonlinearity

Abstract

The well-posedness and dynamic properties of solutions to a class of fourth order parabolic equations with mean curvature nonlinearity are studied. More specifically, we first prove the existence and uniqueness of strong solutions by semigroup method. Then, we establish conditions for the global existence and finite time blow-up of solutions with subcritical, critical and supercritical initial energies. At the same time, we study the exponential decay and $ \omega $-limit set of the global solutions, and the upper bound of the blow-up time of the blow-up solutions. In addition, we also study the existence and regularity of ground state solutions for the steady state problems.