Integro-partial differential equation

Remarks

Example: let $J \colon \R \to [0, \infty)$ be even and integrable, and consider $\partial_t u(x, t) = \int_{\R} J(x - y) \big(u(y, t) - u(x, t)\big) \, dy$ on $\R \times (0, \infty)$. Write $\hat J(\omega) = \int_{\R} J(z) \cos(\omega z) \, dz$. Then $u(x, t) = e^{(\hat J(\omega) - \hat J(0)) t} \cos(\omega x)$ is a solution, because $\int_{\R} J(x - y) \cos(\omega y) \, dy = \hat J(\omega) \cos(\omega x)$ for even $J$.

Non-example: the heat equation $\partial_t u = \partial_x^2 u$ contains no integral of $u$, so it fails clause 2 and is a partial differential equation. The equation $x'(t) = \int_0^t x(s) \, ds$ has an unknown of one variable only, so it is an integro-differential equation but not a partial one.