Semilinear partial differential equation

A semilinear partial differential equation is linear in its highest-order derivatives, with their coefficients depending only on the independent variables; lower-order terms may depend nonlinearly on the solution and its lower derivatives. A semilinear heat equation such as $u_t=\Delta u+f(u)$ is an example. This differs from a quasilinear equation, whose highest-order derivative coefficients can depend on the solution or its lower derivatives.

A heat equation with a nonlinear reaction term depending on the value of its solution.