New dissipated energy for nonnegative weak solution of unstable thin-film equations

The fluid thin film equation $h_t=-(h^n{h}_{xxx}{)}_x-a_1(h^m{h}_x{)}_x$ is known to conserve mass $\int hdx$, and in the case of $a_1\le 0$, to dissipate entropy $\int h^{3\mathrm{/}2-n}dx$ (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,\bigl(\frac{1}{2} ,h^\alpha , h_x^2, - \frac{a_1,h^{\alpha + m - n + 2}}{(\alpha + m - n + 1)(\alpha + m - n + 2)} \bigr), dx $ dissipation for strong nonnegative generalized solutions in the case of the unstable porous media perturbation $a_1>0$ and the critical exponent $m=n+2$.