APM346-2016F > Chapter 1

How to determine what is semilinear and what is quasilinear?

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Tianyi Zhang:
In the textbook, the definitions are not very clear. I have trouble doing my week1 homework now.

Our TA said that if an equation is linear in its highest order, it's semilinear.

If coefficients of the derivatives depend on u, it's quasilinear.

But I think these definitions are different from what the textbook said.

Can anyone help me?

Victor Ivrii:
There is no contradiction but Textbook description is clearer: consider only highest order derivatives. If expression is not linear with respect to them, we get (completely) non-linear; if it is linear with respect to them albeit coefficients depend on lower order derivateeves, it is quasilinear. If coefficients at highest order derivatives do not depend on solution it is semilinear.

Example
$a(x,y)u_{xx}+ 2b(x,y)u_{xy}+c(x,y)u_{yy} + d(x,y) u_x + e(x,y)u_y + f(x,y)u=F(x,y)$ is linear.

$a(x,y)u_{xx}+ 2b(x,y)u_{xy}+c(x,y)u_{yy} =F(x,y,u,u_x,u_y)$  is semilinear (unless $F$ is linear with respect to $(u,u_x,u_y)$ plus $g(x,y)$ in which case we are in the previous framework).

$a(x,y,u,u_x,u_y)u_{xx}+ 2b(x,y,u,u_x,u_y)u_{xy}+c(x,y,u,u_x,u_y)u_{yy} =F(x,y,u,u_x,u_y)$ is quasilinear.

$F(x,y,u,u_x,u_y,u_{xx},u_{xy},u_{yy})=0$ is nonlinear.

Tianyi Zhang:
Thank you SO MUCH! Now it's much clearer to me.

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