# Infinitely many solutions for a first order Cauchy problem.

Is this correct that the following Cauchy problem has infinitely many solutions?

$$\begin{cases}‎ ‎xu_t+u_x=0 \\‎ ‎u(x,0)=\cos x‎ ‎\end{cases}$$

Using the method of characteristics it is obvious that it has a local solution around the curve $$t=0$$. But I am puzzled why it should have infinitely many solutions.

It seems that we can construct many solutions by Laplace transform method but I am not confident.

## 1 Answer

Using the method of characteristics we obtain that $$u$$ is constant along the parametrised by $$s$$ curves: $$(t,x)=\left(\frac{s^2}{2}+c,s\right),$$ and $$u$$ is of the form $$u=f(x^2-2t)$$.

This means that the initial data cover only the region: $$\{(t,x): 0\le t\le x^2/2\}.$$ In other words, the characteristics which start from the $$x-$$axis never arrive in the region. $$\{(t,x): t> x^2/2\}.$$ In particular, if $$f(x)=\left\{ \begin{array}{lll} \cos(x^{1/2}) & \text{if} & x\ge 0, \\ 1+xg(x) & \text{if} & x< 0, \end{array} \right.$$ where $$g$$ is an arbitrary continuously differentiable function, then $$u(t,x)=f(x^2-2t)$$ satisfies the given initial value problem.

• @ Yiorgos S. Smyrlis: thanks. I have a similar broblem which says If in the above cauchy problem we replace $\cos x$ by $\sin x$. there is no solution. How to handle it? – Finish Apr 8 at 18:59
• @Finish Because, due to the fact that $u=f(x^2-2t)$, the solution satisfies $$u(t,-x)=u(t,x), \text{for all}\,\, t,x\in\mathbb R.$$ – Yiorgos S. Smyrlis Apr 8 at 19:02
• @YiorgosS.Smyrlis Out of curiosity, when using the method of characteristics, how do you know whether to parametrize as $(x(t),t)$ or $x(s),t(s)$? If you try the former on this you get $\frac{dt}{dt} = x$ and hence $t=tx+C$, which causes issues – Hushus46 Apr 8 at 19:06
• @Hushus46 You always parametrise with respect to a new parameter, say $s$. Next, it might be possible to replace $s$ by one of the arguments. In this case, the $s$ CANNOT be replaced by $t$, but it can be replaced by $x$. – Yiorgos S. Smyrlis Apr 9 at 8:46