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The study of solution in Sobolev space for the nonlinear differential equations with nonsmooth source term
Applied Mathematics Letters ( IF 2.9 ) Pub Date : 2021-03-01 , DOI: 10.1016/j.aml.2021.107163
Ying Sheng , Tie Zhang

In this paper, we study the solution theory in the Sobolev space for the nonlinear differential equation: Dtny(t)=f(t,y),n1 with given initial values Dtky(0)=dk,k=0,1,,n1,n1. By assuming that function f(t,y)Lp(0,b) and tβf(t,y) is continuous with respect to y where p>1 and 0β<1, we prove that this problem admits a solution in space Wpn(0,b) and the solution is absolutely stable in the L-norm. Moreover, if tβf(t,y) satisfies the Lipschitz condition on variable y, then the solution is unique. Our unique existence conditions allow that f(t,y) is nonsmooth or discontinuous for t[0,b]. Several examples are provided to illustrate our theoretical analysis.



中文翻译:

Sobolev空间中具有非光滑源项的非线性微分方程解的研究。

在本文中,我们研究Sobolev空间中非线性微分方程的解理论: dŤñÿŤ=FŤÿñ1个 具有给定的初始值 dŤķÿ0=dķķ=01个ñ-1个ñ1个。通过假设该功能FŤÿ大号p0bŤβFŤÿ 关于 ÿ 在哪里 p>1个0β<1个,我们证明这个问题可以解决太空问题 w ^pñ0b 该解决方案在以下方面绝对稳定 大号-规范。而且,如果ŤβFŤÿ 满足Lipschitz条件上的变量 ÿ,那么解决方案是唯一的。我们独特的生存条件使FŤÿ 是不光滑或不连续的 Ť[0b]。提供了几个例子来说明我们的理论分析。

更新日期:2021-03-10
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