Consider the Cauchy problem for the semi-linear wave equation with damping

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).

We consider the Cauchy problem for a class of semilinear hyperbolic equation

where

Throughout this paper, we assume that the nonlinear term satisfies the following conditions:

1) and are continuous functions in the domain.

2), and

where

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

where

,.

It is well known that if

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]):

where,

On the other hand, by virtue of condition 2˚

and

Using the Holder inequality, from (16) we have

.

By virtue of condition (7), (8) and the multiplicative inequality of Gagliardo-Nirenberg type, we have

where

Analogously from (17) we have

where

From (12), (16) and (20) we have the following estimates

,(22)

It follows from (22) and (23) that

where and are defined by

and

Then, we have from (19), (21) and (28) that

It is clear from conditions (7), (8) and (29), (30) that

.

Allowing for (24), (25) we obtain that

Thus the a priori estimate (9) is satisfied, so. From (14) and (31) we yield the inequality (10).

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

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

then there exists such that for any

, problems (30) and (31) have a unique solution

.

Theorem 2. Let

and

Then problems (32) and (33) have no nontrivial solutions.

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

The choose of implies that

Define. Again, by the choice of, it is easy to show that

Take scaled variables, then we have

where

Letting in (39), owing to (35), (40), (41) we get

Taking into account condition (36), from (45) it follows that

Further, by applying the Holder inequality, from (37) we obtain

Letting in (47), owing to (45), we get

Finally, taking into condition (36), we have that

.

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.