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

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  • We consider a fourth order degenerate equation describing thin films over an inclined plane in this paper. A new approximating problem is introduced in order to obtain the local energy estimate of the solution. Based on combined use of local entropy estimate, local energy 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 of weak 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 of convergence in the $ L^\infty $ norm of the solution, and this improves and extends the previous results.
    Mathematics Subject Classification: 35K35, 76A20, 35K55, 35K65, 35B40.


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