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2000 Uniqueness of solutions to the initial value problem for an integro-differential equation
Kôhei Uchiyama
Differential Integral Equations 13(4-6): 401-422 (2000).

Abstract

We establish uniqueness of a solution to the initial-value problem for the integro-differential equation $$ \frac{d}{dt} \int_0^1J(x)\mu_t(dx) = \frac1{2}\int_0^1\int_0^1 \frac{J'(y)-J'(x)}{y -x} \cdot\frac{\mu_t(dx)\mu_t(dy)}{|y-x|^{\gamma}}, \quad\;\; t>0 $$ where the equality is required to hold for every smooth testing function $J$ with $J'(0) =J'(1) = 0$, and the solution $\mu_t=\mu_t(dx)$ is a finite measure on the unit interval $[0,1]$ for each $t$ and ${\gamma}$ a constant from the open interval $(-1,1).$ Stationary solutions are given explicitly and the convergence to them of general time-dependent solutions is proved.

Citation

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Kôhei Uchiyama. "Uniqueness of solutions to the initial value problem for an integro-differential equation." Differential Integral Equations 13 (4-6) 401 - 422, 2000.

Information

Published: 2000
First available in Project Euclid: 21 December 2012

zbMATH: 0972.45006
MathSciNet: MR1750033

Subjects:
Primary: 45K05
Secondary: 60K35

Rights: Copyright © 2000 Khayyam Publishing, Inc.

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Vol.13 • No. 4-6 • 2000
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