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

.

Comparing gives

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.

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Comment by nguyenvanphan — April 2, 2010 @ 14:18

Hi bạn, đề CH thì chắc là vẫn thế thôi, cứ ôn luyện đề thi của các năm trước nữa là ổn thôi mà. Chúc may mắn nhé.

Comment by Ngô Quốc Anh — April 2, 2010 @ 14:25