Effect of Weight Function in Nonlinear Part on Global Solvability of Cauchy Problem for Semi-Linear Hyperbolic Equations ()
1. Introduction
Consider the Cauchy problem for the semi-linear wave equation with damping
, (1)
, (2)
where
, 
In the case when 
is independent of
, the existence and nonexistence of the global solutions was investigated in the papers [1-8]. The authors interests are focused on so called critical exponent
, which is the number defined by the following property: if 
then all small data solutions of corresponding Cauchy problem have a global solution, while 
all solutions with data positive on blow up in finite time regardless of the smallness of the data.
In the present paper we investigate the effect of the weight function 
on global solvability of Cauchy problems (1) and (2).
2. Statement of Main Results
We consider the Cauchy problem for a class of semilinear hyperbolic equation
, (3)
, (4)
where 
Throughout this paper, we assume that the nonlinear term 
satisfies the following conditions:
1)
and 
are continuous functions in the domain
.
2)
, and
(5)
where
, (6)
, (7)
. (8)
In the sequel, by
, we denote the usual 
- norm. For simplicity of notation, in particular, we write 
instead of
. The constants C, c used throughout this paper are positive generic constants, which may be different in various occurrences.
Theorem 1. Suppose that the conditions (5)-(8) are satisfied. Then there exists a real number 
such that, if
Then problem (3) and (4) admit a unique solution
satisfied the decay property
(9)
(10)
where
,
.
3. Proof of Theorem 1
It is well known that if
, (11)
then
, i.e. problem (3) and (4) have a global solution (see for example [9]).
Using the Fourier transformation, Plancherel theorem and the Hausdorff-Young inequality, for the solution 
we have the following inequalities (see [1]):
(12)
(13)
(14)
where,
(15)
On the other hand, by virtue of condition 2˚
(16)
and
. (17)
Using the Holder inequality, from (16) we have
.
By virtue of condition (7), (8) and the multiplicative inequality of Gagliardo-Nirenberg type, we have
(18)
where
, (see [10]). (19)
Analogously from (17) we have
(20)
where
. (21)
From (12), (16) and (20) we have the following estimates
,(22)
. (23)
It follows from (22) and (23) that
(24)
(25)
where 
and 
are defined by
, (26)
, (27)
and
. (28)
Then, we have from (19), (21) and (28) that
, (29)
. (30)
It is clear from conditions (7), (8) and (29), (30) that
.
Allowing for (24), (25) we obtain that
(31)
Thus the a priori estimate (9) is satisfied, so
. From (14) and (31) we yield the inequality (10).
4. Nonexistence of Global Solutions
Next let us discus the counterpart of the conditions (7) and (8). To this end we considered the Cauchy problem for the semi-linear hyperbolic inequalities
(32)
, (33)
where
.
The weak solution of inequality (32) with initial data (33) where
is called a function 
which, and 
satisfies the following inequality:
for any function
, where
.
From Theorem 1 it follows that if 
and
, (34)
then there exists 
such that for any
, problems (30) and (31) have a unique solution
.
Theorem 2. Let
, (35)
and
. (36)
Then problems (32) and (33) have no nontrivial solutions.
5. Proof of Theorem 2
We assume that 
is a global solution of (32) and (33). Let 
be such that
and, choose
(see [8]).
Taking such a 
as the test function in Definition 1, we get that
(37)
The choose of 
implies that
. (38)
Define
. Again, by the choice of
, it is easy to show that
Take scaled variables
, then we have
(39)
where
(40)
(41)
(42)
, (43)
. (44)
Letting 
in (39), owing to (35), (40), (41) we get
(45)
Taking into account condition (36), from (45) it follows that
(46)
Further, by applying the Holder inequality, from (37) we obtain
(47)
Letting 
in (47), owing to (45), we get
Finally, taking into condition (36), we have that
.
6. Acknowledgments
This work was supported by the Science Development Foundation under the President of the Republic of Azerbaijan Grant No EIF-2011-1(3)-82/18-1.