Differential and Integral Equations

Uniqueness of solutions to the initial value problem for an integro-differential equation

Kôhei Uchiyama

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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.

Article information

Differential Integral Equations Volume 13, Number 4-6 (2000), 401-422.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]


Uchiyama, Kôhei. Uniqueness of solutions to the initial value problem for an integro-differential equation. Differential Integral Equations 13 (2000), no. 4-6, 401--422. https://projecteuclid.org/euclid.die/1356061232.

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