PDEMOC13

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

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

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

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

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

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

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

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

Therefore,

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

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

Main Page : Partial Differential Equations : Characteristics

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