Error analysis of splitting methods for semilinear evolution equations

summary:We consider a Strang-type splitting method for an abstract semilinear evolution equation $$\partial _t u = Au+F(u).$$ Roughly speaking, the splitting method is a time-discretization approximation based on the decomposition of the operators $A$ and $F.$ Particularly, the Strang method is a popular splitting method and is known to be convergent at a second order rate for some particular ODEs and PDEs. Moreover, such estimates usually address the case of splitting the operator into two parts. In this paper, we consider the splitting method which is split into three parts and prove that our proposed method is convergent at a second order rate

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.Comment: 10 page

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.Comment: 42 pages. In the second version, the main modification in the second version is the improvement on the regularity of the solution for 1<p,q<2, which was caused by trivial oversight. We have the solution in the classical sense all the time. This version is accepted for the publication in the journal, Communications on Pure and Applied Analysis on 15/02/202

非線形波動方程式の爆発現象に関する数値・数学解析

学位の種別: 課程博士審査委員会委員 : （主査）東京大学准教授 齊藤 宣一, 東京大学教授 俣野 博, 東京大学教授 儀我 美一, 東京大学教授 時弘 哲治, 東京大学准教授 宮本 安人, 東京理科大学教授 太田 雅人, 芝浦工業大学准教授 石渡 哲哉University of Tokyo(東京大学

The blow-up curve for a weakly coupled system of semilinear wave equations with nonlinearities of derivative-type (Mathematical structures of integrable systems, their developments and applications)

In this paper, we study a blow-up curve for a weakly coupled system of semilinear wave equations with nonlinearities of derivative type in one space dimension. Employing the idea of Caffarelli and Friedman [1], we prove the blow-up curve becomes Lipschitz continuous under suitable initial conditions. Moreover, we show the blow-up rates of the solution of the wave equations

Blow-Up of Finite-Difference Solutions to Nonlinear Wave Equations

Finite-difference schemes for computing blow-up solutions of one dimensional nonlinear wave equations are presented. By applying time increments control technique, we can introduce a numerical blow-up time which is an approximation of the exact blowup time of the nonlinear wave equation. After having verified the convergence of our proposed schemes, we prove that solutions of those finite-difference schemes actually blow up in the corresponding numerical blow-up times.Then, we prove that the numerical blow-up time converges to the exact blow-up time as the discretization parameters tend to zero.Sev eral numerical examples that confirm the validity of our theoretical results are also offered

The blow-up curve for a weakly coupled system of semilinear wave equations with nonlinearities of derivative-type (Mathematical structures of integrable systems, their developments and applications)

In this paper, we study a blow-up curve for a weakly coupled system of semilinear wave equations with nonlinearities of derivative type in one space dimension. Employing the idea of Caffarelli and Friedman [1], we prove the blow-up curve becomes Lipschitz continuous under suitable initial conditions. Moreover, we show the blow-up rates of the solution of the wave equations