# quasi- linear equation(curve of discontinuous solution )

Help me please to solve this equation: $u_{t}+c\left [ u(1-u) \right ]_{x}=0$ in [a,b]

$c$ is constant. Find smooth boundary and initial conditions when the solution is discontinuous .

Thanks a lot!

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On which domain do you consider your equation? Is there any correspondance between $q$ and $u$? Is $c$ constant? –  martini Jul 18 '12 at 10:23
@martini Oh, thanks. It's no q just u, in[a,b], and indeed c is constant –  Lilly Jul 18 '12 at 10:40
Does the expanded form of your PDE is $u_t+cu_x-2cuu_x=0$ ? –  doraemonpaul Jul 18 '12 at 21:19
@doraemonpaul yes, it's the expand form –  Lilly Jul 22 '12 at 8:44

$u_t+c[u(1-u)]_x=0$

$u_t+c[u-u^2]_x=0$

$u_t+c(u_x-2uu_x)=0$

$u_t+c(1-2u)u_x=0$

This belongs to a PDE of the form http://eqworld.ipmnet.ru/en/solutions/fpde/fpde2203.pdf

So the general solution is $x=ct(1-2u)+C(u)$

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