1.1. The method of characteristics 5
1
x
t
1
Figure 1.1.1. Characteristics diverge in finite time
As indicated in Figure 1.1.1, the characteristics through (0, ±1) diverge
to ±∞ at time t = 1. Thus all the backward characteristics starting in t ≥ 1
meet {t = 0} in the interval ] − 1, 1[. The data for |x| ≥ 1 does not influence
the solution in t ≥ 1. There has been a loss of information. Another
manifestation of this is that the initial values do not uniquely determine a
solution in t 0.
The characteristics starting at t = 0 meet {t = −1} in the interval
] − 1, 1[. Outside that interval, the values of a solution are not determined,
not even influenced by the initial data. There are many solutions in t 0
which have the given Cauchy data. They are constant on characteristics
which diverge to infinity, but their values on these characteristics is otherwise
arbitrary.
To avoid this phenomenon we make the following assumption that not
only prevents characteristics from diverging, but avoids some technical diffi-
culties that occur for unbounded c with characteristics that do not diverge.
Hypothesis 1.1.1. Suppose that for all T 0
∂t,xc
α
∈
L∞([0,T]
× R) .
The coefficient d(t, x) satisfies analogous bounds.
For arbitrary f ∈
C∞(R2)
and g ∈
C∞(R),
there is a unique solution of
the Cauchy problem
∂t + c(t, x) ∂x + d(t, x) u = f, u(0,x) = g.
