Weak asymptotic methods for scalar equations and systems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:444

Article

Abreu, Eduardo¹  Colombeau, Mathilde²  Panov, Eugeny^3,4 

[1] Univ Estadual Campinas, BR-13081970 Campinas, SP, Brazil

[2] Univ Sao Paulo, BR-05508 Sao Paulo, Brazil

[3] Novgorod State Univ, Veliky Novgorod, Russia

[4] St Petersburg State Univ, St Petersburg 199034, Russia

关键词: Partial differential equations;    Functional analysis;    Maslov asymptotic analysis;    Initial value problem;    Ordinary differential equations;   

DOI  :  10.1016/j.jmaa.2016.06.047

来源: Elsevier

【 摘 要 】

In this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions. This construction produces weak asymptotic methods which are issued from Maslow asymptotic analysis. We obtain a sequence of functions which tend to satisfy the equation(s) in the weak sense in the space variable and in the strong sense in the time variable. To this end we reduce the partial differential equations to a family of ordinary differential equations in a classical Banach space. For scalar equations we prove that the initial value problem is well posed in the L-1 sense for the approximate solutions we construct. Then we prove that this method gives back the widely accepted solutions when they are known. For systems we obtain existence in the general case and uniqueness in the analytic case using an abstract Cauchy-Kovalevska theorem. (C) 2016 Published by Elsevier Inc.

【 授权许可】

Free