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New dissipated energy for the unstable thin film equation

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  • The fluid thin film equation $h_t = - (h^n h_{x x x})_x - a_1 (h^m h_x)_x$ is known to conserve mass $\int h dx$, and in the case of $a_1 \leq 0$, to dissipate entropy $\int h^{3/2 - n} dx$ (see [8]) and the $L^2$-norm of the gradient $\int h_x^2 dx$ (see [3]). For the special case of $a_1 = 0$ a new dissipated quantity $\int h^{\alpha} h_x^2 dx $ was recently discovered for positive classical solutions by Laugesen (see [15]). We extend it in two ways. First, we prove that Laugesen's functional dissipates strong nonnegative generalized solutions. Second, we prove the full $\alpha$-energy $\int (\frac{1}{2} h^\alpha h_x^2 - $ $ \frac {a_1 h^{\alpha + m - n + 2}}{(\alpha + m - n + 1)(\alpha + m - n + 2)} ) dx $ dissipation for strong nonnegative generalized solutions in the case of the unstable porous media perturbation $a_1> 0$ and the critical exponent $m = n+2$.
    Mathematics Subject Classification: Primary: 35K55, 35K35, 35Q35; Secondary: 76D08.


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