Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation

Abstract

We consider a fourth order degenerate equation describingthin films over an inclined plane in this paper. A new approximatingproblem is introduced in order to obtain the local energy estimateof the solution. Based on combined use of local entropy estimate, localenergy estimate and the suitable extensions of Stampacchia's Lemma to systems,we obtain the finite speed of propagation property of strong solutions,which has been known for the case of strong slippage $ n<2, $ in the case ofweak slippage $ 2 \leq n < 3. $The long time behavior of positive classical solutions is also discussed.We apply the entropy dissipation method to quantify the explicit rate ofconvergence in the $ L^\infty $ norm of the solution, and thisimproves and extends the previous results.