Effect of Weight Function in Nonlinear Part on Global Solvability of Cauchy Problem for Semi-Linear Hyperbolic Equations

In this paper, we investigate the effect of weight function in the nonlinear part on global solvability of the Cauchy problem for a class of semi-linear hyperbolic equations with damping.


Introduction
Consider the Cauchy problem for the semi-linear wave equation with damping         0 1 0, , 0, , where In the case when is independent of   .a x x , the existence and nonexistence of the global solutions was investigated in the papers [1][2][3][4][5][6][7][8].The authors interests are focused on so called critical exponent , which is the number defined by the following property: if  In the present paper we investigate the effect of the weight function on global solvability of Cauchy problems (1) and (2).

Statement of Main Results
We consider the Cauchy problem for a class of semilinear hyperbolic equation where 1, 2, l   Throughout this paper, we assume that the nonlinear term   , , f t x u satisfies the following conditions: where In the sequel, by .q , we denote the usual   q L norm.For simplicity of notation, in particular, we write .instead of 2 .The constants C, c used throughout this paper are positive generic constants, which may be different in various occurrences.

Proof of Theorem 1
It is well known that if then max , i.e. problem ( 3) and ( 4) have a global solution (see for example [9]).

T  
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˚ 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 ,. ,. d ,. ,.

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 where .
The weak solution of inequality (32) with initial data (33) where which, and   , u t x satisfies the following inequality: From Theorem 1 it follows that if and then there exists 0 0   such that for any , problems (30) and ( 31) have a unique solution , 0 , ; 0, ; and Then problems (32) and (33) have no nontrivial solutions.

Proof of Theorem 2
We assume that   , u t x is a global solution of (32) and (33).Let (see [8]).
Taking such a   , t x  as the test function in Definition 1, we get that .
The choose of   .


. Again, by the choice of , it is easy to show that ,n Take scaled variables , then we have 2 , , 1 , where    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

x u x h x u t x t x x t x u t x x t x x t x
Letting    in (47), owing to (45), we get positive on blow up in finite time regardless of the smallness of the data.
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.(42)