Quasilinearization Methods for Nonlocal Fully-Nonlinear Parabolic Systems

TL;DR

This paper introduces quasilinearization techniques to transform complex nonlocal fully-nonlinear parabolic systems into more manageable quasilinear forms, facilitating analysis and solution of models in behavioral economics.

Contribution

It establishes the equivalence in solvability and well-posedness between nonlocal fully-nonlinear and quasilinear parabolic systems, extending existing methods to the fully-nonlinear case.

Findings

Proves equivalence in solvability between the systems.

Establishes well-posedness for both systems.

Highlights the mathematical and modeling significance of quasilinear systems.

Abstract

In this paper, we propose quasilinearization methods that convert nonlocal fully-nonlinear parabolic systems into the nonlocal quasilinear parabolic systems. The nonlocal parabolic systems serve as important mathematical tools for modelling the subgame perfect equilibrium solutions to time-inconsistent dynamic choice problems, which are motivated by the study of behavioral economics. Different types of nonlocal parabolic systems were studied but left behind the fully-nonlinear case and the connections among them. This paper shows the equivalence in solvability between nonlocal fully-nonlinear and the associated quasilinear systems, given their solutions are regular enough. Moreover, we establish the well-posedness results for the nonlocal quasilinear parabolic systems, so do that for the nonlocal fully-nonlinear parabolic systems. The quasilinear case alone is interesting in its own right from mathematical and modelling perspectives.