Proof of Theorem 1
We divide the proof into steps.
Step 1. In the first step, we show a lemma.
Lemma 9
For the 1-dimensional Euler equations with damping (3) withand, we have the following relation.
(29)
Proof of Lemma 9
It is well known that is always positive if is set to be positive. From (3), we have, for ,
(30)
(31)
Thus, we have
(32)
Taking integration with respect to x, we obtain
(33)
On the other hand, multiplying on both sides of (3), we get
(34)
From (33), we have
(35)
and
(36)
Substituting (36), (32) and (35) into (34), one obtains the relation claimed in the lemma.
Step 2. We set
(37)
where and are functions of t. Then, (29) is transformed to
(38)
where we arrange the terms according to the coefficients of x.
Step 3. We use the Hubble transformation:
(39)
and set the coefficient of (38) to be zero. Thus,
(40)
Note that we have the novel identity
(41)
Multiplying the both sides of (40) by , it becomes
(42)
(43)
for some constant .
Step 4. With (39), we set the coefficient of x in (38) to be zero. Thus, b satisfies
(44)
where
(45)
(46)
Last Step. With (39) and setting the coefficient of 1 in (38) to be zero, we are required to solve
(47)
where
(48)
(49)
(50)
Solving the O.D.E (47) by the method of integral factor, one arrives at the solutions. The proof of Theorem
1 is complete.
Next, we prove Theorem 3 as follows.
Proof of Theorem 3
For , case i) and case ii) of Theorem 3 follow from Case 1. and Case 2. of Lemma 7.
For , by Case 3. of Lemmas 6 and7, there exists a finite such that the one-sided limit of a(t) is zero as t approaches to . It remains to show is not a removable singularity of . To this end, suppose one has
(51)
Then,
(52)
Thus, the singularity is of essential type and case iii) of Theorem 3 is proved.
For , (6) becomes
(53)
which can be solved by using integral factor. The solution is
(54)
Thus, if . Also, if and , where . As
(55)
(T, x) is an essential singularity of u(t, x) for any x. Thus, cases iv) and v) of Theorem 3 are established. The proof is complete.
Proof of Theorems 4 and 5
The corresponding relation of Lemma 9 for is
(56)
With similar steps, one can obtain the family of exact solutions in Theorem 4.
Note that (10) is a special case of (12) and the arguments in the proof of Theorem 3 hold for . Thus, the results for Theorem 5 follows.