Infinitely many Solutions for a Nonlinear Cauchy Problem

Consider the Cauchy problem

$\displaystyle u_t = \varphi(u_x)_x,\, u(\cdot,0)=f, $

with the non-monotone constitutive function $ \varphi$ shown below and smooth initial data $ f$ with compact support. If $ f_x\in(1,2)$ on an interval $ D\subset\mathbb{R}$, then there exist infinitely many solutions $ u(x,t)$ with bounded gradient for small $ t>0$.

\includegraphics[width=12cm]{CauchyProblemFig.eps}

For any solution, $ u_x(\cdot,t)$ has an increasing number of oscillations as $ t\to0$. The corresponding pattern of discontinuities is not unique; an example, suggested by Gilbert Strang, is shown in the right figure.


$ \square$ Trans. Amer. Math. Soc. 278 (1983), 299-316.

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