This paper is concerned with the quantitative homogenization of 2m-order elliptic systems with bounded measurable, rapidly oscillating periodic coefficients. We establish the sharp O(ε) convergence rate in W m−1,p0 with p 0 = 2d d−1 in a bounded Lipschitz domain in R d as well as the uniform large-scale interior C m−1,1 estimate. With additional smoothness assumptions, the uniform interior C m−1,1 , W m,p and C m−1,α estimates are also obtained. As applications of the regularity estimates, we establish asymptotic expansions for fundamental solutions.

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We first establish the local well-posedness for the nonuniform weakly dissipative bequation which includes both the weakly dissipative Camassa-Holm equation and the weakly dissipative Degasperis-Procesi equation as its special cases. We then study the blow-up phenomena and the long time behavior of the solutions. Two blow-up results are established for certain initial profiles. Moreover, two sufficient conditions for the decay of the solutions are presented.