Linear partial differential operator

A linear partial differential operator of order at most $m$ on an open set $U$ has the form

Lu(x)=\sum_{|\alpha|\le m}a_\alpha(x)\partial^\alpha u(x).

The coefficients $a_\alpha$ are prescribed functions. The map is linear in $u$ , although its coefficients may vary with $x$ . It has order exactly $m$ if at least one coefficient of an order- $m$ derivative is not identically zero.

Systems and nonlinearity

For vector-valued $u$ , the coefficients may be matrices of compatible sizes. If the coefficients depend on the unknown itself, the resulting expression is generally nonlinear. For example, $b(x)\partial_xu$ is linear in $u$ , whereas $u\partial_xu$ is not.

References

John K. Hunter, Notes on Partial Differential Equations.