The properties of solutions for the coupled 4th-order parabolic equations

Abstract

We consider a 4th-order nonlinear parabolic system involving $p$-Laplacians. First, we prove the local existence of weak solutions. Then, we investigate the behavior of these solutions depending on their initial data. In particular, we show that for initial data with small positive energy, if the Nehari functional energy is positive, then these solutions exist globally, whereas if the Nehari functional energy is negative, they blow up in finite time. In the process, we are able to obtain estimates for the blow-up time, the blow-up rate and the decay of global solutions.