Existence and unicity of solutions for a non-local relaxation equation

Research output: Contribution to journalArticle

Abstract

We study the following one-dimensional evolution equation: frac(∂ u, ∂ t) (x, t) = ∫A+ u (x, t) λ1 (ξ, t) (u (ξ, t) - u (x, t)) d ξ - ∫A- u (x, t) λ2 (ξ, t) (u (x, t) - u (ξ, t)) d ξ, where A+ u (x, t) = {ξ ∈ [0, 1] {divides} u (ξ, t) > u (x, t)}, A- u (x, t) = [0, 1] {set minus} A+ u (x, t), and λ1, λ2 are non-negative functions. We prove the existence of solutions for a particular class of initial data u (x, 0). We also prove that the solutions are unique. Finally, under additional constraints on the initial data, we give an explicit expression for the solution.

Original languageEnglish (US)
Pages (from-to)1702-1710
Number of pages9
JournalNonlinear Analysis, Theory, Methods and Applications
Volume70
Issue number4
DOIs
StatePublished - Feb 15 2009

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Evolution Equation
Divides
Existence of Solutions
Non-negative
Class

Keywords

  • Existence and uniqueness
  • Integro-differential equation
  • Relaxation equation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Existence and unicity of solutions for a non-local relaxation equation. / Paparella, Francesco; Pascali, E.

In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 70, No. 4, 15.02.2009, p. 1702-1710.

Research output: Contribution to journalArticle

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N2 - We study the following one-dimensional evolution equation: frac(∂ u, ∂ t) (x, t) = ∫A+ u (x, t) λ1 (ξ, t) (u (ξ, t) - u (x, t)) d ξ - ∫A- u (x, t) λ2 (ξ, t) (u (x, t) - u (ξ, t)) d ξ, where A+ u (x, t) = {ξ ∈ [0, 1] {divides} u (ξ, t) > u (x, t)}, A- u (x, t) = [0, 1] {set minus} A+ u (x, t), and λ1, λ2 are non-negative functions. We prove the existence of solutions for a particular class of initial data u (x, 0). We also prove that the solutions are unique. Finally, under additional constraints on the initial data, we give an explicit expression for the solution.

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