So far once we have discontinuous initial data, we might have shocks. This also occurs even the given initial data is smooth. Let us still consider the following problem
subject to the following initial condition
where , and .
From a discussion of this entry we consider the case when is non-increasing as we need singularities (because ). Moreover, the solution can be given implicitly as
Since is non-increasing, on . We now consider the characteristics which are straight lines, issuing from two points and on the axis with have speeds and respectively. Because is decreasing and is increasing, it follows that
In other words, the characteristic emanating from is faster than the one emanating from .
Therefore the characteristics cross so a smooth solution cannot exist for all .
We are now interested in calculating the breaking time. To this purpose, we calculate along a characteristic which has equation
Let denote the gradient of along the characteristic given as above. Then
By differentiating the PDE with respect to we also have
along the characteristic. Solving this ODE gives us
where is the initial gradient at . Thus
The fact that and have different sign implies that will blow up at a finite time along the characteristic. What we need to do is to examine along all characteristics to find such so that first blows up.
A simple calculation show by denoting that the time of the first breaking is
where is such that is maximum. An in-depth observation shows that if the initial data is not monotone, breaking will first occur on the characteristic for which and is maximum.
Source: J.D. Logan, An introduction to nonlinear partial differential equations, 2nd, 2008; Section 3.3.