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

(1)

Effect of Weight Function in Nonlinear Part on Global

Solvability of Cauchy Problem for Semi-Linear

Hyperbolic Equations

Akbar B. Aliev1, Anar A. Kazimov2 1

Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan

2

Nakhchivan State University, Nakhchivan, Azerbaijan Email: [email protected], [email protected]

Received December 13, 2012; revised January 19, 2013; accepted January 30,2013

ABSTRACT

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.

Keywords: Cauchy Problem; Wave Equation; Global Solvability; Weight Function; Semi-Linear Hyperbolic Equation

1. Introduction

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

 

p,

 

,



0,



n

tt t

u    u u a x u t x   R , (1)

 

0, 0

   

, 0, 1

 

,

n t

u x u x u x u x x R , (2)

where

2 2

2

1 n

x x

 

   

   ,

 

, 1

n q

a x L R q

In the case when a x

 

. is independent of x, the existence and nonexistence of the global solutions was inves- tigated in the papers [1-8]. The authors interests are fo- cused on so called critical exponent , which is the number defined by the following property: if

 

c p n

 

c p p n

then all small data solutions of corresponding Cauchy problem have a global solution, while 1 _c

 

n all solutions with data positive on blow up in finite time re- gardless of the smallness of the data.

p p

 

In the present paper we investigate the effect of the weight function on global solvability of Cauchy problems (1) and (2).

 

a x

2. Statement of Main Results

We consider the Cauchy problem for a class of semilin- ear hyperbolic equation

 

1l l



, ,

  

, ,



0,



n

tt t

u    u u  f t x u t x   R , (3)

 

0, 0

   

, 0, 1

 

,

n t

u x u x u x u x x R , (4)

where l1, 2,_

Throughout this paper, we assume that the nonlinear

term f t x u



, ,



satisfies the following conditions: 1)f t x u



, ,



and f t x u_t



, ,





t x u, ,



t  

 are continu-

ous functions in the domain



0, 



Rn1. 2) f t x



, , 0



0, and









 





1 2

1 1

1 2 1

, , , ,

,

p p

f t x u f t x u

a x u  u  u u



   ₂ (5)

where

 

_q

 

n ,

a x L R q1, (6)

2 1

1 l , for

p n

n q

 

2l

 _  _ 

  , (7)









2

1 4

2 , for 2

2 1

n q lq

p l

q q n l q

  

_  _  n

 

  . (8)

In the sequel, by ._q, we denote the usual Lq

 

 -

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.

.

Theorem 1. Suppose that the conditions (5)-(8) are sat-

isfied. Then there exists a real number 00 such that, if











   

 



0

2 1 2 1

0 1 2 1

2 1

0

, , :

,

l n n n n

l n n

n n

W R L R L R L R

u u W R L R

L R L R

  



   





   



    

(2)

Thenproblem (3) and (4) admit a unique solution

 







 



1







 



2 2

, 0, ; l n 0, ; n

u t x C  W R _С  L R

satisfied the decay property

 

 







2 4

, 1

0, , 0,1, ,

n r l r

D u t c d t

t r l

          



 , (9)

 

,

 

1



,

t

u t  c d t  t0, (10)

where



1



min 1 ,

4 4 2

n p

n n

l l lq

     

 

 , c

 

 C R R



, 



.

3. Proof of Theorem 1

It is well known that if

 

_{ }

 

_{ }





2 2 max

, _{W R}l n t , _{L R}n , 0

u t   u t c t ,T , (11)

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]):

 

,

u t x

 

_{ }













 

3 2 2 4 0 1 2 4 0

, 1 ,

1 d

n l l L R

l

t n l l

D u t c t E u u

с t   ;               





(12)

 

_{ }













 

3 2 4 1 4 0

, 1 ,

1 d n l _o L R t n l

u t c t E u u

с t   ,

     

_

   (13)

 

_{ }













 

3 2 1 4 1 1 4 0

, 1 ,

1 d

n l

t _{L R} o

t n

l

u t c t E u u

с t   

       

_

   (14) where,





_{ }

 



 



_{ }

 





_{ }

3 3 1 1 3 3 2 2 3 1 3 2

1 0 1

0 1

,

, , ,

l k

o L R L R L R W R

L R

E u u u u

u u

f x u x

             (15)

On the other hand, by virtue of condition 2˚





_{ }

   

1

, , n d

n

p L R

R

f t x u c a x u x

_

x (16)

and





₂

   

2

, , d

n

p R

f t x u c a x u x

_

x. (17)

Using the Holder inequality, from (16) we have





_{ }

 

1

1 1

1

, , n d d

n n q q pq q q _q L R R R

f t x u c a x x u x x

      _  _{ } 



 



   .

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

 





_{ }

 

_{ }

  3 1 1 , , , n q L R p q p L R l

f t u

c a u D u

           _ _ 





 (18)

where 1 1 2 n q l pq   _   

 , (see [10]). (19)

Analogously from (17) we have

 





 

2

_{ }

2 1  2

, , , n q p p L R l

f t u

c a u D u

              _ _ 



 (20) where 2 1 2 n q l pq   _   

 . (21)

From (12), (16) and (20) we have the following esti-mates

 













 

2 4 0 1 2 ₁ 4 0 2 2 1

, 1 ,

1 , .

, . , . d

n l l l

, .

p

t n l _p

l

p p

l

D u t c t E u u

с t u D u

u D u

    _    _            _           _ _     _ _  _ _       _ _    _







,(22)

 













 

_{ }

 

_{ }

4 4 0 1 1 0 2 2 1

, 1 ,

1 , .

, . , . d

n l n l , . p t p l p p l

u t c t E u u

с t u D u

u D u

  _    _                   _ _     _ _  _ _       _ _    _







 . (23)

(3)

 



 







 

_{ }





 

_{ }

4 4

1

0 1

1 2

2 1 2

1 2

1 1

1

t

n n

l l

p p

G t cd c t t

G G

  



  

d ;

  



 



   

    



_ 



  _





(24)

 













 

_{ }







 

_{ }

2 2

4 4

2

0 1

1 2

2 1 2

1 2

1 1

1

1 d

t

n l n l

l l

p p

G t cd c t t

G G

  



  

 _ 

 



   

    

 _ 



  _





(25)

where G t1

 

and G t2

 

are defined by

  



4

 

1 1

n l

G t  t u t, , (26)

  



2

 

4

2 1

n l l

l

G t t D u t

 



 



, , (27)

and

,

4 2 2

np p np

p

l l



       . (28)

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



1



1 1

.

4 2 2 2 2

n q

np p n q np

l l pq l lq

   _   _  

  , (29)



2



2 1

2 2 2

n q

np _p n q np

l l pq l lq

    _   _  

  . (30)

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

1

   .

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

 





1 2( ) , 0, max .

G t G t c t T (31)

Thus the a priori estimate (9) is satisfied, so T . 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

 

1



,



0, ,

l l

tt t

n

u u u f t x

t x R

    

 

,u ,

1

(32)

 

0, 0

   

, t 0,

 

,

n

u x u x u x u x

x R

 

 , (33)

where







2



1 , ,

1

p s

f t x u u

x 

 .

The weak solution of inequality (32) with initial data (33) where

 

0 . 1 , 1 . 1

l n n

u W R u L R

is called a function

 

, 1





n

u t x L R_R

which, and u t x

 

, satisfies the following inequality:

 

   

 

   

 





 

0 1 0

0 0

0,

0, d d

, , , 1 , d

, , , , d d ,

n n

n

R R

l l

tt t

R

x

u x u x x x u x x

t

u t x t x t x t x x t

f t x u t x t x x t

 

  







 _  _ 



 

 _     _





 

d

for any function

 

. 02,2



l

C R R

  _ n



, where

 

t x, 0,

 

t x, R Rn

    .

From Theorem 1 it follows that if n2l and

2 2 1 l s,

p

n



 

 _ 

 , (34)

then there exists  0 0 such that for any

   



u0 . ,u1 .



U0, problems (30) and (31) have a uni-

que solution

 







 







 





2 1

2

, 0, ;

0, ;

l n

n

u t x C W R

С L R

 



 .

Theorem 2. Let

2 2 1 p 1 l s

n



   , (35)

and

 

3

0 1 d

R

u x u x x

 

 



0. (36)

Then problems (32) and (33) havenonontrivial solutions.

5. Proof of Theorem 2

We assume that u t x

 

, is a global solution of (32) and (33). Let C R2



; 0,1

 



be such that

 

r 1,r 1,

 

r 0,r 2

     

and, choose

 

, 2 2 2

l l

t x

t x R

R

 _  _, 0

 

(see [8]).

(4)

 

   





   

 

0 1 2 0 0 0 0, d 1

, , d

1

0, d

, , ,

1 , d d .

n n n n R p s R R tt t R l _l

u x u x x x

u t x t x x t

x

u x x

t

u t x t x t x

t x x t

                   _      _



 



 

d (37)

The choose of 

 

. implies that

   

0 0, d n R x

u x x

t   0  



. (38)

Define  



 

t x, 



0, 



R tn, 2 x2l 2





. Again,

by the choice of 



t x, , it is easy to show that



2



1 0

1 d d

n p sp p p p t R

x   x t

 

 _



 

 

C  ,



2



2 0

1 d d

n p sp p p p tt R

x   x t

 

 _



 

 

C  ,



2



3 0

1 d d

n p sp p l p p R

x   x t

 

 _ _

  

 

C  ,

,n

Take scaled variables ,

then we have

2

, , 1,

l

i i

t  x y i _

 

   





 

3 1 2 0 1 2 0

1 2 3

0, d

1

, , d

1 , n n R p s R

u x u x x x

u t x t x x t

x                      



 

d (39)

where



 



1 2 2 2 1

d d ,

sp

p

p p

p l

c _ y _ y

1 2 4 2 1 1 s lp l n p p     

  , (43)

2 3

2 2

1 1

s lp _n

p p

    

  . (44)

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

 





 

0 1 2 0 d 1

, d d

1 n n R p s R

u x u x x

u t x x t C

x       .     



 

(45)

Taking into account condition (36), from (45) it fol-lows that



2



 

0

1

, d d

1

n

p s

R

u t x x t C

x



   

 

. (46)

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

 





   





 





4

2 2 2

4

2 2 2

2

0 1 4

2 0 1 2 2 2 2 d 1

, , d d

1

, d d

1

d d . 1 n l x p s R p p s t x p l tt t sp t x x

u x u x h x

u t x t x x t

x

u t x x t

x x t x                             _{ } _                     _   _     



 



 (47)

Letting   in (47), owing to (45), we get

c          _     _  _    



  (40)

 





 

0 1 2 0 1 d , 1 n n p s R R

u x u x x u t x x t

x



d d 0.

  

 

 





 

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

 

, 0

u t x  .



 



2

2 ₂

1

d d ,

sp

p

p p

p l

c _ y _ y c

         _     _     



 

6. Acknowledgments

 (41)









3

2

3 2

1

d d ,

sp

p

p _l p

p l

c _ y y c

         _     _  _    



 

This work was supported by the Science Development Foundation under the President of the Republic of Azer- baijan Grant No EIF-2011-1(3)-82/18-1.

 (42)

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(5)

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