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

Lihua Min; Xiaoping Yang

doi:10.3934/cpaa.2014.13.543

Article Contents

2014, Volume 13, Issue 2: 543-566. Doi: 10.3934/cpaa.2014.13.543

This issue Previous Article Well-posedness for the supercritical gKdV equation Next Article Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent

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

Lihua Min^1, and
Xiaoping Yang^1,

1.
School of Science, Nanjing University of Science and Technology, Nanjing, 210094

Received: September 30, 2012

Revised: June 30, 2013

Published: February 2014

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Abstract

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.

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Mathematics Subject Classification: 35K35, 76A20, 35K55, 35K65, 35B40.

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