a semilinear PDE
Nonlinear equations are not all equally wild, and mathematicians sort them by where the nonlinearity hides. A semilinear PDE is the mildest step beyond linear: the highest-order derivatives still appear linearly, with coefficients that depend only on the independent variables — but the lower-order part (u itself and possibly its first derivatives) is allowed to enter through a nonlinear function. The 'leading edge' of the equation is well-behaved; only the trailing, lower-order terms misbehave.
Make it concrete with a second-order example. An equation like u_xx + u_yy = f(x, y, u) is semilinear: the principal part u_xx + u_yy is exactly the linear Laplacian, with constant coefficients, and all the nonlinearity sits in the source f, which may bend u through a cube, an exponential, anything — but u^3, say, contains no derivatives. Compare u_xx + u_yy = u^3 (semilinear, since the cube is on u alone) with u_xx + u_yy = (u_x)^2 (not semilinear, because the nonlinearity (u_x)^2 involves a derivative). The rule of thumb: in a semilinear equation, the coefficients multiplying the highest derivatives are functions of x and t only, never of u or its derivatives.
Why bother with this fine print? Because the type and the qualitative theory of a PDE are governed almost entirely by its highest-order (principal) part. Since a semilinear equation has a linear principal part, it 'inherits' the classification — elliptic, parabolic, or hyperbolic — of the corresponding linear equation, and many tools (energy estimates, maximum principles) carry over with care. Reaction-diffusion equations like u_t = D Laplacian u + f(u), which model combustion, nerve impulses, and pattern formation, are semilinear — and the nonlinear reaction term f(u) is exactly where the interesting biology and chemistry live.
Semilinear: u_t = u_xx + u(1 - u) (Fisher-KPP, the reaction term u(1-u) involves no derivatives), Laplacian u = e^u (Liouville's equation). Not semilinear: u_t + u u_x = u_xx (the term u u_x has u multiplying a first derivative — that is quasilinear instead).
Semilinear = linear principal part with constant or x,t-dependent coefficients; nonlinearity only in u and possibly lower derivatives.
The boundary between semilinear and quasilinear is exactly whether a derivative appears inside the nonlinearity at the top order. If the coefficient of the highest derivative depends on u (e.g. u u_xx), it is quasilinear, not semilinear — the distinction is precise, not a matter of taste.