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.