# Difference between revisions of "Semilinear"

From DispersiveWiki

Jump to navigationJump to searchm (Clarified antecedent of "these equations") |
|||

Line 8: | Line 8: | ||

<center><math>L u = F(u, Du, \ldots, D^k u)</math></center> | <center><math>L u = F(u, Du, \ldots, D^k u)</math></center> | ||

where ''L'' is a linear operator, ''F'' is a nonlinear function of the first few derivatives <math>u, Du, \ldots, D^k u</math>, with | where ''L'' is a linear operator, ''F'' is a nonlinear function of the first few derivatives <math>u, Du, \ldots, D^k u</math>, with | ||

− | ''k'' strictly less than the order of ''L''. | + | ''k'' strictly less than the order of ''L''. Semilinear-with-derivatives equations are more nonlinear than semilinear equations, but are less nonlinear than [[quasilinear]] or [[fully nonlinear]] equations. |

[[Category:concept]] | [[Category:concept]] |

## Latest revision as of 15:41, 12 December 2007

A **semilinear** equation is a PDE of the form

where *L* is a linear operator and *F* is a nonlinear operator which does not involve any derivatives of *u*.

A **semilinear-with-derivatives** equation is a PDE of the form

where *L* is a linear operator, *F* is a nonlinear function of the first few derivatives , with
*k* strictly less than the order of *L*. Semilinear-with-derivatives equations are more nonlinear than semilinear equations, but are less nonlinear than quasilinear or fully nonlinear equations.