Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations

Abstract In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: { u t = α u x x + β [ φ ( u ) ] x x + f ( u ) in Q : = Ω × ( 0...

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Autores principales:	Quincy Stévène Nkombo, Fengquan Li, Christian Tathy
Formato:	article
Lenguaje:	EN
Publicado:	SpringerOpen 2021
Materias:	Radon measure-valued solution Nonlinear parabolic equations Young measure Asymptotic behavior Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/523a93466f1643d9b0f724b41f03ddbd
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spelling	oai:doaj.org-article:523a93466f1643d9b0f724b41f03ddbd2021-11-28T12:08:27ZExistence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations10.1186/s13662-021-03668-31687-1847https://doaj.org/article/523a93466f1643d9b0f724b41f03ddbd2021-11-01T00:00:00Zhttps://doi.org/10.1186/s13662-021-03668-3https://doaj.org/toc/1687-1847Abstract In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: { u t = α u x x + β [ φ ( u ) ] x x + f ( u ) in Q : = Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , $$ \textstyle\begin{cases} u_{t}=\alpha u_{xx}+\beta [\varphi (u) ]_{xx}+f(u) &\text{in} \ Q:=\Omega \times (0,T), \\ u=0 &\text{on} \ \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) &\text{in} \ \Omega , \end{cases} $$ where T > 0 $T>0$ , Ω ⊂ R $\Omega \subset \mathbb{R}$ is a bounded interval, u 0 $u_{0}$ is nonnegative bounded Radon measure on Ω, and α , β ≥ 0 $\alpha , \beta \geq 0$ , under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.Quincy Stévène NkomboFengquan LiChristian TathySpringerOpenarticleRadon measure-valued solutionNonlinear parabolic equationsYoung measureAsymptotic behaviorMathematicsQA1-939ENAdvances in Difference Equations, Vol 2021, Iss 1, Pp 1-34 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Radon measure-valued solution Nonlinear parabolic equations Young measure Asymptotic behavior Mathematics QA1-939
spellingShingle	Radon measure-valued solution Nonlinear parabolic equations Young measure Asymptotic behavior Mathematics QA1-939 Quincy Stévène Nkombo Fengquan Li Christian Tathy Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations
description	Abstract In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: { u t = α u x x + β [ φ ( u ) ] x x + f ( u ) in Q : = Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , $$ \textstyle\begin{cases} u_{t}=\alpha u_{xx}+\beta [\varphi (u) ]_{xx}+f(u) &\text{in} \ Q:=\Omega \times (0,T), \\ u=0 &\text{on} \ \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) &\text{in} \ \Omega , \end{cases} $$ where T > 0 $T>0$ , Ω ⊂ R $\Omega \subset \mathbb{R}$ is a bounded interval, u 0 $u_{0}$ is nonnegative bounded Radon measure on Ω, and α , β ≥ 0 $\alpha , \beta \geq 0$ , under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.
format	article
author	Quincy Stévène Nkombo Fengquan Li Christian Tathy
author_facet	Quincy Stévène Nkombo Fengquan Li Christian Tathy
author_sort	Quincy Stévène Nkombo
title	Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations
title_short	Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations
title_full	Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations
title_fullStr	Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations
title_full_unstemmed	Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations
title_sort	existence and asymptotic behavior of radon measure-valued solutions for a class of nonlinear parabolic equations
publisher	SpringerOpen
publishDate	2021
url	https://doaj.org/article/523a93466f1643d9b0f724b41f03ddbd
work_keys_str_mv	AT quincystevenenkombo existenceandasymptoticbehaviorofradonmeasurevaluedsolutionsforaclassofnonlinearparabolicequations AT fengquanli existenceandasymptoticbehaviorofradonmeasurevaluedsolutionsforaclassofnonlinearparabolicequations AT christiantathy existenceandasymptoticbehaviorofradonmeasurevaluedsolutionsforaclassofnonlinearparabolicequations
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Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations

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