### Abstract

We study the following one-dimensional evolution equation: frac(∂ u, ∂ t) (x, t) = ∫_{A+ u (x, t)} λ_{1} (ξ, t) (u (ξ, t) - u (x, t)) d ξ - ∫_{A- u (x, t)} λ_{2} (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 λ_{1}, λ_{2} 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 language | English (US) |
---|---|

Pages (from-to) | 1702-1710 |

Number of pages | 9 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 70 |

Issue number | 4 |

DOIs | |

State | Published - Feb 15 2009 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Nonlinear Analysis, Theory, Methods and Applications*,

*70*(4), 1702-1710. https://doi.org/10.1016/j.na.2008.02.053

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

Research output: Contribution to journal › Article

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 70, no. 4, pp. 1702-1710. https://doi.org/10.1016/j.na.2008.02.053

}

TY - JOUR

T1 - Existence and unicity of solutions for a non-local relaxation equation

AU - Paparella, Francesco

AU - Pascali, E.

PY - 2009/2/15

Y1 - 2009/2/15

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.

AB - 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.

KW - Existence and uniqueness

KW - Integro-differential equation

KW - Relaxation equation

UR - http://www.scopus.com/inward/record.url?scp=57049162056&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57049162056&partnerID=8YFLogxK

U2 - 10.1016/j.na.2008.02.053

DO - 10.1016/j.na.2008.02.053

M3 - Article

VL - 70

SP - 1702

EP - 1710

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 4

ER -