1. General framework to construct local-energy solutions of nonlinear diffusion equations for growing initial data.
- Author
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Akagi, Goro, Ishige, Kazuhiro, and Sato, Ryuichi
- Subjects
- *
BURGERS' equation , *ELLIPTIC operators , *NONLINEAR operators , *HEAT equation , *MONOTONE operators , *POROUS materials , *ELLIPTIC equations , *SCHRODINGER operator , *MONOTONIC functions - Abstract
This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of nonlinear terms for approximate solutions in a limiting procedure. Indeed, such an identification process often needs the maximal monotonicity of nonlinear elliptic operators (involved in the doubly-nonlinear equations) as well as uniform estimates for approximate solutions; however, even the monotonicity is violated due to a localization of the equations, which is also necessary to derive local-energy estimates for approximate solutions. In the present paper, such an inconsistency is systematically overcome by reducing the original equation to a localized one, where a (no longer monotone) localized elliptic operator is decomposed into the sum of a maximal monotone operator and a perturbation, and by integrating all the other relevant processes. Furthermore, the general framework developed in the present paper is also applied to the Finsler porous medium and fast diffusion equations , which are variants of the classical PME and FDE and also classified as a doubly-nonlinear equation. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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