linear PDE

A PDE where the unknown function and its partial derivatives combine only by addition and multiplication by known functions of the independent variables.

A PDE is linear if it can be written as $\sum a_\alpha(x)\,D^\alpha u = g(x)$ , a sum of the unknown's partial derivatives $D^\alpha u$ each multiplied by a coefficient depending only on the independent variables -- no products of $u$ with itself or its own derivatives, and no $u$ inside a nonlinear function.