一类非线性Volterra积分不等式

本文简要讨论Gronwall不等式的研究进展,并给出关于如下的一类非线性Volterra积分不等式的一个结果：w(u(t))≤g(t))+∑_i=1ⁿ∫_αi(t0)^α_i(t)f_i(t,s)∏_j=1^mH_ij(u(s))G_ij(max_s-h≤ξ≤s u (ξ))ds.     

A resumé of nonlocal transport theories

Analysis from a number of different perspectives has shown diffusion and dispersion in natural porous formations to generally be nonlocal in character, i.e., the mass balance involves integro-partial differential equations. Only in certain asymptotic limits do these laws localize to classical partial differential equations. Compiled within is a resume of nonlocal laws that our group has developed over the last few years for systems with physical, chemical and biological heterogeneity. Analytical tools used to obtain these laws are nonequilibrium and equilibrium statistical mechanics, and first-order spectral-perturbation methods. This paper is an expansion of the material presented at the Waterloo conference held in the memory of Dr. Unny.