# PDEMOC13

Solve the initial value problem $\displaystyle a(u)u_x+u_y=0\,$ with $\displaystyle u(x,0)=h(x)\,$ and show the solution becomes singular for some $\displaystyle y>0\,$ unless $\displaystyle a(h(s))\,$ is a nondecreasing function of $\displaystyle s\,$ .

The solution is $\displaystyle u(x,y)=h(x-a(u)y)\,$ .

Suppose $\displaystyle x=s\,$ at $\displaystyle y=0\,$ so that $\displaystyle u(s,0)=h(s)\,$ .

Let $\displaystyle u=u_x\,$ along a characteristic line $\displaystyle \Gamma_\xi\,$ .

$\displaystyle \frac{du}{dy} = \frac{\partial u_x}{\partial x}x'(y) + \frac{\partial u_x}{\partial y} = u_{xx}a(u)+u_{xy}\,$

From the equation $\displaystyle u_y+a(u)u_x=0\,$

we get $\displaystyle u_{xy}+a(u)u_{xx}+a'(u)u_xu_x=0\,$

$\displaystyle \implies \frac{dw}{dt} = -a'(u)u_x^2 = -a'(h(s))w^2\,$

Therefore,

$\displaystyle w(y) = \frac{h'(s)}{1+a'(h(s))h'(s)y}\,$

Thus, $\displaystyle u_x\,$ becomes $\displaystyle \infty\,$ on the characteristic line for some $\displaystyle y>0\,$ unless $\displaystyle a'(h(s))h'(s)y\ge 0\,$ which is equivalent to saying that $\displaystyle a(h(s))\,$ is a non-decreasing function of $\displaystyle s\,$ .