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I know a PDE is linear when the dependent variable $u$ and its derivatives appear only to the first power. So, $u_t + u_x +5u = 1$ would be linear. However, I do not quite understand the other two.

My professor described "semilinear" PDE's as PDE's whose highest order terms are linear, and "quasilinear" PDE's as PDE's whose highest order terms appear only as individual terms multiplied by lower order terms. No examples were provided, only equivalent statements involving sums and multiindeces which I do not think I could decipher by tomorrow.

Can someone provide some examples of "semilinear" and "quasilinear" PDE's?

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    Another good reference book is from the book of Zachmanoglou"introduction to PDE with applications" – 2018-07-29
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    You can find a formal definition in 'Partial Differential Equations': Second Edition written by Evans. It is on the second page of chapter 1. – 2018-07-29

2 Answers 2

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I think this will help you to understand the PDE $:$

Linear PDE: $a(x,y)u_x+b(x,y)u_y+c(x,y)u=f(x,y)$

Semi-linear PDE: $a(x,y)u_x+b(x,y)u_y=f(x,y,u)$

Quasi-linear PDE: $a(x,y,u)u_x+b(x,y,u)u_y=f(x,y,u)$

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    could you please give a similar explanation for the case of second order pde ? – 2018-09-30
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I hope these examples will help you.

Semilinear/Almost Linear PDE:

1) $a(x,y)u_x+b(x,y)u_y+c(x,y,u)=0$

2) $U_{tt}-U_{xx}+U^3=0$

Qausi Linear PDE:

1) $a(x,y,u)u_x+b(x,y,u)u_y-c(x,y,u)=0$

2) $U_x+UV_y=0$

3) $U_{tt}-UU_{xx}+U^3=0$

4) $U_{tt}-UU_{xx}+U=0$

5) Navier Stokes equation is also Qausi Linear Equation