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This paper presents an integrated framework to construct local-energysolutions to fairly general nonlinear diffusion equations for initial datagrowing at infinity under suitable assumptions on local-energy estimates forapproximate solutions. A delicate issue for constructing local-energy solutionsresides in the identification of weak limits of nonlinear terms for approximatesolutions in a limiting procedure. Indeed, such an identification process oftenneeds the maximal monotonicity of nonlinear elliptic operators (involved in thedoubly-nonlinear equations) as well as uniform estimates for approximatesolutions; however, even the monotonicity is violated due to a localization ofthe equations, which is also necessary to derive local-energy estimates forapproximate solutions. In the present paper, such an inconsistency will besystematically overcome by reducing the original equation to a localized one,where a (no longer monotone) localized elliptic operator will be decomposedinto the sum of a maximal monotone one and a perturbation, and by integratingall the other relevant processes. Furthermore, the general framework developedin the present paper will also be applied to the Finsler porous medium and fastdiffusion equations, which are variants of the classical PME and FDE and alsoclassified as a doubly-nonlinear equation.

General framework to construct local-energy solutions of nonlinear diffusion equations for growing initial data