Takiko Sasaki

Akitoshi Takayasu, Kaname Matsue, Takiko Sasaki et al.

This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.

Takiko Sasaki, Tetsuji Tokihiro

Nonlocal MEMS equations exhibit finite-time quenching, or touchdown, which is difficult to capture numerically. We study a stagewise rescaling algorithm for a two-dimensional nonlocal MEMS equation in an asymptotically constant-feedback touchdown regime. The equation is not exactly invariant under the $A^{3/2}$--$A^3$ scaling used here; the scaling is justified when the reciprocal-integral feedback $K(t)=1+\int_Ω(1-u)^{-1}dx$ remains bounded and converges to a finite positive limit, as in the single-point touchdown profiles of Duong--Zaag. In this regime the leading-order core dynamics reduce to a local MEMS equation with an asymptotically constant coefficient. Using a fixed-stage scaling of the deficit variable, we obtain a gradient flow for a rescaled energy at frozen amplitude and prove an exact energy dissipation identity within each stage. We introduce a minimizing-movement stage solver and derive a discrete energy inequality. Since strict monotonicity need not hold across stage transitions, we separate the switch and outer-update defects and prove an exact defect balance. Under a uniform switch-defect estimate, this yields quantitative almost monotonicity and a defect-aware criterion for nonexistence of a global admissible continuation. The numerical section is organized around reproducible two-dimensional reference computations: a full-domain stagewise run showing trigger detection, fixed-stage energy decay, and geometric accumulation of physical time, and a direct fixed-domain energy check. These tests are not used as proof of the bounded-window criterion; instead, they report finite-feedback diagnostics and identify the ideal-transfer switch-energy diagnostics required for a posteriori verification.