Semilinear PDE

Definition

Let $\Omega$ be an open set in $\R^n$ and let $m \ge 1$. A partial differential equation of order $m$ in $\Omega$ is semilinear if it has the following form. $$\sum_{\abs{\alpha} = m} a_\alpha(x) \, \partial^\alpha u + a_0\left(D^{m-1} u, \dots, D u, u, x\right) = 0.$$ Here the coefficients $a_\alpha$ of the highest-order derivatives depend on $x$ only, and $a_0$ is a given function, which may be nonlinear. Here $\alpha$ runs over multi-indices, and $D^j u$ denotes the collection of all partial derivatives of order $j$.