Searcharxiv⌕ Search

arXiv subjects

Takiko Sasaki

Publications and source records attributed to Takiko Sasaki.

12 recordsLinked to original sources
arXiv2026

Lifespan estimate for one dimensional wave equation with semilinear terms of spatial derivative

This paper studies the upper and lower bounds of the lifespan for the classical solutions to the initial value problems of one dimensional wave equations with non-autonomous semilinear terms including the space-derivative of the unknown function.This is a non-trivial business comparing to the analogous results with time-derivative type semilinear terms, especially for the proof to obtain the sharp upper bound of the lifespan as we have to deal with space dependent weights among iteration procedures of the weighted functional of the solution. Also it is surprising that a part of them reaches to the same ordinary differential inequality for classical semilinear damped wave equations introduced by Li and Zhou (Discrete Contin. Dynam. Systems, 1995, 1(4): 503-520), and we show a simple proof for blow up result from this ordinary differential inequality by iteration argument and slicing method in more general situation.

math.AP↗
arXiv2026

Generalized FDNF fuzzification of elementary cellular automata and its nonlinear pattern dynamics

Fuzzy disjunctive normal form (FDNF) gives the canonical multi-affine extension of an elementary cellular automaton (ECA) rule to the unit cube. Although it preserves the Boolean rule on binary states, its multi-affine structure can smooth high-contrast CA patterns and restrict continuous-state dynamics. We introduce generalized FDNF rules \[ \widetilde f_k^{g,u,v,w}(x,y,z) = g\left(f_k(u(x),v(y),w(z))\right), \] where the transformations $g,u,v,w: [0,1] \to [0,1]$ fix the endpoints. The identity maps recover ordinary FDNF, while threshold-like, discontinuous, non-monotone, and expanding choices yield rule-preserving fuzzy ECAs. We demonstrate, in representative rules, that the transformation shape strongly affects pattern dynamics: threshold-like maps promote ECA-like pattern recovery, parameter deformations interpolate toward FDNF-like smoothing, and discontinuities induce gap-generated regimes. Pattern changes are summarized by contrast, fuzziness, and a finite-resolution participation-type support exponent. In three-cell systems, an expanding non-monotone transformation yields stable period-six cycles for rule 210, verified by interval arithmetic, coexisting with an expanding invariant line set; rules 51 and 85 inherit one-dimensional expanding dynamics. The framework provides a rule-preserving bridge from Boolean cellular automata to fuzzy and continuous-state nonlinear dynamics.

nlin.CG↗
arXiv2026

A Structure-Preserving Stagewise Rescaling Algorithm for a Two-Dimensional Nonlocal MEMS Equation in an Asymptotically Constant-Feedback Regime

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.

math.NA↗
arXiv2026

Sector-dominant graph-local drivers for path-window barrier Hamiltonians on the Boolean hypercube

We study finite-size adiabatic state preparation on Boolean hypercubes using graph-local drivers built from sector/path coordinates related to monotone Gray-code representatives. The construction is not presented as a new all-$n$ Gray-code existence theorem; rather, it provides finite representatives, explicitly checked through the cases used in the numerical experiments, for testing problem-dependent graph-local drivers. For ordinary diagonal-cost transverse-field annealing, the ordering does not yield a robust advantage, and we include this negative result as a baseline. For non-diagonal target Hamiltonians whose geometry is expressed in the same sector/path coordinates, hybrid drivers combining sector, path-window, and small transverse-field components can substantially improve the final ground-state fidelity in centered barrier instances. Reproduction runs from the accompanying code confirm a representative centered original-window barrier value of approximately \(0.9799\) for the fixed-control hybrid parameters \((w,α,ε)=(8,0.50,0.15)\), while also showing that the improvement is target-class dependent. Randomized and ablation controls indicate that the dominant contribution is the sector-preserving skeleton, with strict one-bit completion acting as a secondary refinement. We provide code, finite certificates, CSV files, validation logs, and reproduction scripts to make the finite-size claims traceable.

quant-ph↗
arXiv2026

Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping

In this article, we investigate the blow-up behavior of solutions to the one-dimensional damped nonlinear wave equation, namely $$ \partial_t^2 u - \partial_x^2 u + \fracμ{1 + t} \partial_t u = |\partial_t u|^p \quad (p > 1). $$ Under the assumption of sufficiently large and smooth initial data, we establish that the blow-up curve is continuously differentiable ($\mathcal{C}^1$). A key step in our analysis involves the characterization of the blow-up profile of the solution. The proof relies on transforming the equation into a first-order system and adapting the techniques of Sasaki in \cite{Sasaki2018,Sasaki2019} which have elegantly extended the method of Caffarelli and Friedman \cite{Caffarelli1986} to nonlinear wave equations with time derivative nonlinearity, but without the scale-invariant term ($μ=0$).

math.AP↗
arXiv2025

Note on the existence of classical solutions of derivative semilinear models for one dimensional wave equation

This note is a supplement with a new result to the review paper by Takamura [13] on nonlinear wave equations in one space dimension. We are focusing here to the long-time existence of classical solutions of semilinear wave equations in one space dimension, especially with derivative nonlinear terms of product-type. Our result is an extension of the single component case, but it is meaningful to provide models as possible as many to cover the optimality of the general theory. The proof is based on the classical iteration argument of the point-wise estimate of the solution.

math.AP↗
arXiv2023

The combined effect in one space dimension beyond the general theory for nonlinear wave equations

In this paper, we show the so-called "combined effect" of two different kinds of nonlinear terms for semilinear wave equations in one space dimension. Such a special phenomenon appears only in the case that the total integral of the initial speed is zero. It is remarkable that, including the combined effect case, our results on the lifespan estimates are partially better than those of the general theory for nonlinear wave equations.

math.AP↗
arXiv2023

The generalized combined effect for one dimensional wave equations with semilinear terms including product type

We are interested in the so-called "combined effect" of two different kinds of nonlinear terms for semilinear wave equations in one space dimension. Recently, the first result with the same formulation as in the higher dimensional case has been obtained if and only if the total integral of the initial speed is zero, namely Huygens' principle holds. In this paper, we extend the nonlinear term to the general form including the product type. Such model equations are extremely meaningful only in one space dimension because the most cases in higher dimensions possess the global-in-time existence of a classical solution in the general theory for nonlinear wave equations. It is also remarkable that our results on the lifespan estimates are partially better than those of the general theory. This fact tells us that there is a possibility to improve the general theory which was expected complete more than 30 years ago.

math.AP↗
arXiv2023

The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative

This paper is devoted to the lifespan estimates of small classical solutions of the initial value problems for one dimensional wave equations with semilinear terms of the spatial derivative of the unknown function. It is natural that the result is same as the one for semilinear terms of the time-derivative. But there are so many differences among their proofs. Moreover, it is meaningful to study this problem in the sense that it may help us to investigate its blow-up boundary in the near future.

math.AP↗
arXiv2022

Multi-order asymptotic expansion of blow-up solutions for autonomous ODEs. I -- Method and Justification

In this paper, we provide a systematic methodology for calculating multi-order asymptotic expansion of blow-up solutions near blow-up for autonomous ordinary differential equations (ODEs). Under the specific form of the principal term of blow-up solutions for a class of vector fields, we extract algebraic objects determining all possible orders in the asymptotic expansions. Examples for calculating concrete multi-order asymptotic expansions of blow-up solutions are finally collected.

math.CA↗
arXiv2016

Numerical validation of blow-up solutions of ordinary differential equations

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.

math.NA↗