Quasilinear

Quasilinear refers to a class of partial differential equations (PDEs) in which the highest-order derivatives appear linearly, but the coefficients may depend nonlinearly on the unknown function and its lower-order derivatives. In other words, the equation is linear with respect to the highest-order derivatives, while the nonlinearity is allowed in the coefficients or in lower-order terms.

In the standard taxonomy of PDEs, quasilinear equations generalize linear equations by permitting the coefficients of the highest-order terms to depend on the solution or its gradient. They contrast with linear equations, where all derivatives enter linearly with fixed coefficients; semilinear equations, which are linear in the highest-order derivatives but have nonlinearities only in the unknown function itself (not its derivatives); and fully nonlinear equations, where the highest-order derivatives can appear nonlinearly.

Examples illustrate the range of quasilinear equations. A first-order quasilinear example is u_t + c(u) u_x = 0, which is linear in the derivatives but has a nonlinear coefficient that depends on u. A second-order quasilinear elliptic example is div( a(x,u,∇u) ∇u ) = f(x,u,∇u), where the second derivatives enter linearly but the coefficient a may depend on u and ∇u. A quasilinear parabolic example is the evolution equation u_t = div(|∇u|^{p-2} ∇u), the p-Laplacian flow, which is nonlinear in ∇u but linear in the second derivatives.

Applications include nonlinear diffusion, gas dynamics, elasticity, and nonlinear acoustics. Analyzing quasilinear PDEs often requires specialized techniques such as monotone operator theory, energy methods, or viscosity solutions, reflecting their blend of linear structure with nonlinear coefficients.