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I'm trying to solve the IVP: $u^2u_x + u_y = 0, u(x,0) = \frac{1}{1+x^2}$ and have to determine whether a gradient catastrophe develops.

My attempt: I used the method of characteristic to solve this IVP, and at the end got $u(x,y) = h(x - (u(x,y)^2y)$, where $h(x)= \frac{1}{1+x^2}$. However, I'm stuck at determining whether a gradient catastrophe develops. Is there any general way to determine whether gradient catastrophe happens? Any help would be appreciated.

kkkkstein
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  • You correctly found : $u(x,y) = h(x - (u(x,y)^2y)$, where $h(x)= \frac{1}{1+x^2}$ . Thus : $$u=\frac{1}{1+(x-u^2y)^2}$$ This is the solution on the form of implicit equation.

    $u(x,y)$ is always positive and never tends to infinity.

     – JJacquelin Feb 01 '22 at 07:57
  • @JJacquelin But will gradient catastrophe develop? Does "never tends to infinity" means there is no gradient catastrophe? – kkkkstein Feb 01 '22 at 17:31
  • Hint: follow the steps in this post – EditPiAf Feb 05 '22 at 14:39

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