TAIWANESE JOURNAL OF MATHEMATICS
Vol. 8, No. 1, pp. 135-154, March 2004
This paper is available online at http://www.math.nthu.edu.tw/tjm/
ON SOME NONLINEAR DISSIPATIVE EQUATIONS WITH
SUB-CRITICAL NONLINEARITIES
Nakao Hayashi, Naoko Ito, Elena I. Kaikina and Pavel I. Naumkin
Abstract. We study the Cauchy problem for the nonlinear dissipative equations
(
½
¾
{
@t u + ® (¡¢) 2 u + ¯ juj u + ° juj u = 0; x 2 R n ; t > 0;
(1)
u(0; x) = u 0 (x); x 2 Rn ;
where ®; ¯ ; ° 2 C; Re ® > 0, ½ > 0, { > ¾ > 0: We are interested in the
critical case, ¾ = n½ and sub critical cases 0 < ¾ < ½n .¯R We assume
¯ that the
initial data u 0 are sufficiently small in a suiatble norm, ¯ u 0 (x) dx¯ > 0 and
Re¯ ±(®; ½; ¾ ) > 0, where
Z
¾
±(®; ½; ¾) = jG (x)j G (x) dx
½
and G (x) = F ¡1 e ¡®j»j : In the sub critical case we assume that ¾ is close to
½
: Then we prove global existence in time of solutions to the Cauchy problem
n
(1) satisfying the time decay estimate
8
1
1
< C (1 + t)¡ ¾ (log (2 + t))¡ ¾ if ¾ = ½ ;
n
ku (t)k L1 ·
¡ ½¢
1
:
¡¾
C (1 + t)
if ¾ 2 0; n :
1. INTRODUCTION
We study the Cauchy problem for the nonlinear dissipative equations with fractional power of the negative Laplacian and complex coefficients
(
½
@t u + ® (¡¢) 2 u + ¯ juj¾ u + ° juj{ u = 0; x 2 Rn; t > 0;
(2)
u (0; x) = u0 (x) ; x 2 Rn;
Received April 15, 2003.
Communicated by S. B. Hsu.
2000 Mathematics Subject Classification: 35Q35.
Key words and phrases: Nonlinear dissipative equations, Sub-critical nonlinearities.
135
136
Nakao Hayashi, Naoko Ito, Elena I. Kaikina and Pavel I. Naumkin
where ®; ¯; ° 2 C; Re ® > 0, ½ > 0, { > ¾ > 0: Furthermore we assume that
Re ¯±(®; ½; ¾) > 0 and initial data are sufficiently small in L1 \ L1;a where Lp,
1 · p · 1; is the usual Lebesgue space, the weighted Lebesgue space L1;b is
defined by
°
°
n
o
°
°
L1;b = Á 2 L1; kÁk L1;b = ° hxib Á° 1 < 1 ; b ¸ 0
L
and
±(®; ½; ¾) =
Z
³
´
½
jG (x)j¾ G (x) dx; G (x) = F ¡1 e¡®j»j :
We restrict our attention to the critical and subcritical cases 0 < ¾ · ½n and prove
global existence in time for small solutions to the Cauchy problem (2). In the super
critical case ¾ > n½ the problem is easier, and it was studied in [15] under the
restrictions ¾ > ½ > 14 , n = 1: Equation (2) with ½ = 2; ¾ = n2 is known as the
complex Landau-Ginzburg equation. Local existence in time for the solutions to the
Cauchy problem (2) with ½ = 2 was studied by many authors (see, e.g. [7],[8]) and
references cited therein). In the case ½ 6= 2; local existence in time can be easily
shown by the contraction mapping principle in L2 framework. Nonlinear dissipative
equations with a fractional power of the negative Laplacian in the principal part
were studied extensively (see, e.g., [1],[2],[20],[23] and references cited therein).
Blow-up in finite time of positive solutions to the Cauchy problem
(3)
½
@t u + (¡¢) 2 u ¡ u1+¾ = 0; u(0; x) = u0 (x) > 0
was proved in [5],[24] for the case 0 < ¾ < n2 ; ½ = 2; in [10],[18] for the case
¾ = n2 ; ½ = 2, and in paper [22] for the case 0 < ½ · 2; 0 < ¾ · n½ . Their
proofs of blow-up results are based on the positivity of linear evolution operator
½
F ¡1 e¡j»j ; associated with equation (3), for 0 < ½ · 2 (see book [25]), and do not
½
work for the case ½ > 2; since F ¡1e¡j»j is not necessarily positive. Large time
behavior of positive solutions was studied extensively for a particular case of (2)
written as
@tu ¡ ¢u + u1+¾ = 0;
with any ¾ > 0 (see [16]) for the super critical case ¾ > 2¡n , [6]¢ for the critical case
¾ = n2 and [3],[4],[9],[17] for the sub critical case ¾ 2 0; 2n ). Global existence
in time for small solutions to (3) in the super critical case ¾ > n2 ; ½ = 2 was shown
in [5]. Large time asymptotic behavior of small solutions to (2) with ½ = 2; ° = 0
was investigated in details in [12],[13],[14] under the condition that
¯
¯¡ n
³
´¡ n2
¯
2
2
2
2¯ 2
Re¯ (2 + ¾) j®j + ¾®
= j¯j ¯(2 + ¾) j®j + ¾® ¯
Ã
!
n
sin (2arg ®)
£ cos arg ¯ ¡ arctan
> 0:
2
1 + ¾2 + cos (2arg ®)
On Some Nonlinear Dissipative Equations with Sub-Critical Nonlinearities
137
The critical case ¾ = 2n ; n = 1 was studied in [12], when the initial data are small
in H1;0 \ H0;1, where Hm;s is the weighted Sobolev space
©
ª
Hm;s = Á 2 L2; kÁkHm;s = khirim hxis ÁkL2 < 1 ;
p
m; s ¸ 0; hxi = 1 + x2: In [13] the critical case ¾ = 2n ; was studied for any
space dimensions n ¸ 1, when the initial data are small in L 1 \ L1;a; a 2 (0; 1) ;
In [14] it was considered the sub critical case 1 < ¾ < n2 ; when 2 ¡ n¾ is small,
n ¸ 1; the initial data are small in L1 \ L1;a ; a 2 (0; 1) :
As far as we know the global existence in time for solutions to the Cauchy
problem (2) with ½ 6= 2 in critical and sub critical cases is not known. Our global
result stated below is applicable, in particular, to the problem
(4)
½
@t u + (¡¢) 2 u + ¸u1+¾ = ¹u1+{; u(0; x) = u0 (x) > 0;
with 0 < ¾ < { · ½n ; ¸; ¹ > 0: As we mentioned above when ¸ = 0; ¹ > 0;
0 < ½ · 2 the solutions of (4) blow up in finite time, and when ¸ > 0; ¹ = 0;
0 < ½ < 1 the solution exists globally in time. Our results stated below show that
the dissipation term u1+¾ in equation (4) is stronger than the blow-up term u1+{ :
Note that the problem of asymptotic behavior of solutions to (4) is still open for the
sub critical case 0 < { < ¾ · ½n even if ½ = 2:
n R
Let FÁ or Á^ be the Fourier transform of Á defined by Á^ (») = (2¼)¡ 2 e¡ix» Á
n R
(x) dx and F ¡1Á (x) = (2¼)¡ 2 eix» Á (x) dx is the inverse Fourier transform of
Á: By C (I; B) we denote the space of continuous functions from a time interval I
to the Banach space B.
Now we state the results of this paper. Denote g (t) = 1 + jµj¾ ´ log (1 + t) ;
´ = ½Re¯± (®; ½; ¾) ;
¾ (t) = g (t) if ¾ = n½ ; and
¾ (t) = 1 +
Theorem 1.1.
n¾jµj¾ ´ 1¡ n¾
½
½¡n¾ t
¡
¢
if ¾ 2 0; n½ :
Assume that Re® > 0, 0 < ¾ < { ·
½
n
and
u0 2 L1 \ L1;a; a 2 (0; min (1; ½)) :
Furthermore we suppose that
n
Re¯±(®; ½; ¾) > 0; j^
u0(0)j = µ (2¼)¡ 2 > 0:
Then there exists a positive " such that if ku0kL1 + ku0k L1;a · "; j^
u0(0)j ¸ C"
and the value ¾ is close to n½ ; so that n½ ¡ ¾ · C"¾ : then the Cauchy problem (2)
138
Nakao Hayashi, Naoko Ito, Elena I. Kaikina and Pavel I. Naumkin
¡
¢
has a unique global solution u (t; x) 2 C [0; 1); L1 \ L1;a satisfying the time
decay estimate
°
°
³ 1 ´ 1
°
n
°
¾
iÃ(t)
°u (t) ¡ µt¡ ½ G t¡ ½ (¢) ¡
°
¾ (t) e
°
° 1
L
8
1
1
¡
¡
½
1+¾
¾
¾
<
C"
(1 + t) g (t) if ¾ = n ;
³
´
·
: C"1+¾ (1 + t)¡ ¾1 if ¾ 2 ½ ¡ C"¾ ; ½ ;
n
n
and à (t) satisfies
¯
¯
1
¯
¯
¾ R t ¡¾
¡ ¾½ n
¯Ã (t) ¡ arg u
¯
b
(0)
+
jµj
é
Â
(¿)
(1
+
¿)
d¿
0
¾
¯
0
¯
8
Zt
1
½
¡1¡
>
>
¾ ¾ (¿) log ¾ (¿) (1 + ¿)¡1 d¿ if ¾ = ;
< C"1+¾
n
Z 0t
·
³½
¾
>
½´
¡1
>
1+2¾
: C"
¾ ¾ (¿) (1 + ¿)¡ ½ n d¿ if ¾ 2
¡ C"¾;
;
n
n
0
é = Im¯± (®; ½; ¾) :
For the convenience of the reader, we state the global existence result for the
super-critical case { > ¾ > n½ . In this case we do not need the positivity of
Re¯±(®; ½; ¾).
Theorem 1.2. Assume that Re® > 0, { > ¾ > n½ and u0 2 L 1 \ L1 : Then
there exists a positive " such that if ku0kL1 +ku
¡ 0kL1 · "; then the
¢ Cauchy problem
(2) has a unique global solution u(t; x) 2 C [0; 1); L1 \ L1 satisfying the time
decay estimate
¡n
ku (t)k L1 · C" (1 + t) ½ :
Remark 1.1. Our proof of the Theorem 1.1 depends on the positivity of the value
Re¯±(®; ½; ¾), which we need to derive better time decay properties of solutions
(see Lemma 2.3 below). If ½ = 2 then we can calculate explicitly the value of
± (®; ½; ¾) (see also [13,][14])
Ã
!
Z
2¾ 1
2
x
1
x
1
±(®; 2; ¾) =
e¡ 8 ( ® + ® )¡ 4® dx
n
n
n
n
(4¼t) 2 ¾+ 2 j®j 2 ¾ ® 2
0
1
n
=
22
(4¼)
n¾
2
B
@³
n
j®jn¡ 2 ¾
2
(2 + ¾) j®j + ¾®2
C
´n A:
2
On Some Nonlinear Dissipative Equations with Sub-Critical Nonlinearities
139
³
´
½
We do not need the positivity of the kernel G (x) = F ¡1 e¡®j»j of the Green
operator which was essentially used previously in proving the blow-up results. The
condition that the value ¾ should be close to n½ ; so that n½ ¡ ¾ · C"¾ is rather
technical and is caused by the application of the contraction mapping principle for
proving global existence of solutions.
Remark 1.2. We consider only the case of the linear operator of the form
½
®(¡¢) 2 in equation (2) for simplicity. Our method is also applicable to a more
general case of pseudodifferential operators.
We organize the rest of our paper as follows. We prove preliminary lemmas in
Section 2. In Lemma 2.1 we obtain estimates of the Green operator in the Lebesgue
spaces Lp, 1 · p · 1 and L1;a: Then in Lemma 2.2 we estimate the Green operator
in our basic norm
³
´
n
¡a
kÁkX = sup (1 + t) ½ kÁ (t)kL1 + kÁ (t)kL1 + (1 + t) ½ kÁ (t)kL1;a ;
t>0
where a 2 (0; 1) : Large time behavior of the mean value of the nonlinearity ¯ juj¾ u
in equation (2) is evaluated in Lemma 2.3. Section 3 is devoted to the proof of
Theorem 1.1. Theorem 1.2 is proved in Section 4.
2. PRELIMINARIES
We write the solution of the linear Cauchy problem
(
½
@tu + ® (¡¢) 2 u = f (t; x) ; x 2 Rn; t > 0;
u(0; x) = u0(x); x 2 Rn ;
with Re® > 0; ½ > 0; by virtue of the Duhamel formula
Z t
u (t) = G (t) u0 +
G (t ¡ ¿) f (¿) d¿;
0
where the Green operator G (t) is given by
(5)
n
½
^ = t¡ ½
G (t) Á = F ¡1e¡®j»j tÁ(»)
Z
³ 1
´
¡
G t ½ (x ¡ y) Á(y)dy
³
´
½
with a kernel G (x) = F ¡1 e¡®j»j :
We first collect some preliminary estimates of the Green operator G (t) in the
norms kÁk Lp and kÁk L1;a ; where kÁk L1;a = kh¢ia ÁkL1 ; a 2 (0; 1) ; 1 · p · 1:
140
Nakao Hayashi, Naoko Ito, Elena I. Kaikina and Pavel I. Naumkin
Lemma 2.1. Suppose that the function Á 2 L1 \ L1;a ; where a 2 (0; 1) :
Then the estimates
kG (t) ÁkLp · C kÁk Lp
and
°
³
³ 1 ´´°
° !
°
¡n
¡
°j¢j G (t) Á ¡ #t ½ G t ½ (¢) °
Lp
· Ct
n
½
³
1
p ¡1
´
+ !¡a
½
kÁkL1;a
are valid for all t > 0; where
1 · p · 1; 0 · ! · a; # =
Z
Á (x) dx:
³
´
½
Proof. Note that the kernel G (x) = F ¡1 e¡®j»j in representation (5) is a
smooth function G (x) 2 C1 (Rn) and decays at infinity so that
¯
¯
¯
¯
sup hxin+¹+k ¯@kxj G (x)¯ · C;
(6)
x2Rn
for all 1 · j · n; k = 0; 1; where 0 < ¹ < min (1; ½) (since ½ > 0 is a fractional
number). Indeed we have
¯
¯ ¯
³
´¯
°
°
½ ¯
½°
¯ k
¯ ¯
°
¯@xj G (x)¯ = ¯F ¡1 »jk e¡®j»j ¯ · C °»jke¡®j»j ° 1 · C:
L
and
¯
¯ ¯
³
³
´´¯
¯ ¹ n+k k
¯ ¯ ¡1
¯
¹ n+k
k ¡®j»j½
¯jxl j xl @xj G (x)¯ = ¯F
j@»l j @»l
»j e
¯
°
³
´°
½ °
°
· C °j@»l j¹ @n+k
»jk e¡®j»j ° 1
»l
L»
¯ ¯¹
for k = 0; 1; where the fractional derivative ¯@»j ¯ for ¹ 2 (0; 1) is defined by
Z ³ ³
³
´
´
´
¹
¡1
¹b
j@»l j Á (») = Fl
jxl j Á (x) = C
Á »e(y) ¡ Á (») jyj¡1¡¹ dy;
R
where » = (»³1; :::; »n) ; ´»e(y) = (»1; :::; »l ¡ y; :::; »n) (see [21]). We have with
½
Á (») = @»n+k
»jk e¡®j»j
l
Z ³ ³
´
´
j@»l j Á (») = C
Á »e(y) ¡ Á (») jyj¡1¡¹ dy = I1 + I2;
¹
R
where
I1 =
Z
j»j
2 ·jyj
³ ³
´
´
Á »e (y) ¡ Á (») jyj¡1¡¹ dy;
On Some Nonlinear Dissipative Equations with Sub-Critical Nonlinearities
I2 =
Since
Z
jyj· j»j
2
141
³ ³
´
´
Á »e(y) ¡ Á (») jyj¡1¡¹ dy:
¯
³
´¯
¯
½¡n ¡Cj»j½
k ¡®j»j½ ¯
jÁ (»)j = ¯@»n+k
»
e
e
¯ · C j»j
j
l
for all » 2 Rn; 1 · j; l · n; k = 0; 1; we estimate the first summand I1 by the
Young’s inequality
³¯ ³
´¯
´
R
R
¯ e
¯
kI1kL1 · C jyjdy
Á
»
(y)
+
jÁ
(»)j
d»
¯
1+¹ j»j·2jyj ¯
»
µ¯
¶
¯
R dy R
e j½
¯ e ¯½¡n ¡C j»(y)
½¡n ¡Cj»j½
· C jyj1+¹ j»j·2jyj ¯» (y)¯
e
+ j»j
e
d»
R
R
· C jyj·1 jyj½¡1¡¹ dy + C jyj>1 jyj¡1¡¹ dy · C
¤
since ¹ < ½: In the case jyj · j»j
2 we have with » = (»1; :::; » l ¡ ¸y; :::; »n ) ;
¸ 2 (0; 1)
¯ ³
´
¯
¯ e
¯
¯Á » (y) ¡ Á (»)¯ · C jyj j@»l Á (»¤ )j
¤ j½
· C jyj j»¤j½¡n¡1 e¡Cj»
for all » 2 Rn , jyj ·
Whence we get
j»j
2;
½
· C jyj j»j½¡n¡1 e¡Cj»j
since j»¤j · j»j + jyj ·
3
2
j»j and j»¤ j ¸ j»j ¡ jyj ¸ 12 j»j :
°R
°
½
°
°
kI2kL1 · C ° jyj· j»j j»j½¡n¡1 e¡Cj»j jyj¡¹ dy° 1
»
°
°2 °
° L»
° ½¡n¡¹ ¡Cj»j½ ° ° ¹¡1 R
°
¡¹
· C °j»j
e
dy°
· C:
° 1 °j»j
j»j jyj
jyj·
L»
2
L1
»
Therefore estimate (6) is true. By virtue of (6) we find
° ³ 1 ´°
n °
°
°
°
t ¡ ½ °G t¡ ½ (¢) ° 1 = kG (¢)kL1 · C ° hxi¡n¡¹°L1 · C;
x
L
hence by Young’s inequality we obtain the first estimate of the lemma
° n ³ 1 ´°
°
°
kG (t) ÁkLp · C °t¡ ½ G t ¡ ½ (¢) ° 1 kÁkLp · C kÁkLp
L
for all t > 0; where 1 · p · 1:
We write
³
³ 1
´´
n
jxj! G (t) Á ¡ #t ¡ ½ G t¡ ½ (x)
Z
³ ³ 1
´
³ 1 ´´
¡ n½
=t
jxj! G t¡ ½ (x ¡ y) ¡ G t ¡ ½ x Á(y)dy
142
Nakao Hayashi, Naoko Ito, Elena I. Kaikina and Pavel I. Naumkin
for any ! 2 [0; a] ; a 2 (0; 1) :We have
¡
¢
jG (x ¡ y) ¡ G (x)j · jG (x ¡ y)j + jG(x)j · C hx ¡ yi¡n¡¹ + hxi¡n¡¹
for all x; y 2 Rn ; and applying the Lagrange finite differences Theorem, in view
of (6) we obtain
jG(x ¡ y) ¡ G (x)j · jyj jr xG(x¤ )j · C jyj hx¤i¡n¡¹¡1 · C jyj hxi¡n¡¹¡1
for all x; y 2 Rn; jyj ·
jxj
2;
therefore
jxj! jG (x ¡ y) ¡ G (x)j · C jxj! jyj a hxi¡n¡¹¡a · C jyja hxi¡n¡¹
for all x; y 2 Rn; jyj ·
jxj
2 :
And if jyj ¸
jxj
2
we get
jxj! jG (x ¡ y) ¡ G (x)j · C jyj! (jG (x ¡ y)j + jG (x)j)
¡
¢
· C jyja hx ¡ yi¡n¡¹ + hxi¡n¡¹ :
Thus we obtain
³ ³ 1
´
³ 1 ´´
jxj! G t ¡ ½ (x ¡ y) ¡ G t¡ ½ x
µD
E¡n¡¹ D 1 E¡n¡¹¶
!¡a
1
· Ct ½ jyja t ¡ ½ (x ¡ y)
+ t¡ ½ x
for all x; y 2 Rn: Whence
°
³
³ 1 ´´°
° !
¡n
¡
°
°j¢j G (t) Á ¡ #t ½ G t ½ (¢) ° p
L
° µD
°
E¡n¡¹ D 1 E¡n¡¹¶
°
°
n
!¡a
R
1
¡½+ ½ °
¡½
¡½
a
· Ct
t (x ¡ y)
+ t x
jyj Á(y)dy °
°
° p
L
°D
°
³
´
E
°
¡n¡¹
°
n
1
!¡a
¡ n½ + !¡a
¡ 1½
a
a
½ p ¡1 + ½
°
°
½
· Ct
kh¢i ÁkL1
° t (¢)
° p kh¢i ÁkL1 · Ct
L
for all t > 0; where 1 · p · 1; ! 2 [0; a] ; a 2 (0; 1) : Thus the second estimate
of the lemma is true. Lemma 2.1 is proved.
Remark 2.1. The first estimate of Lemma 2.1 was shown in [19] or [23]. Here
we give a simpler proof.
In the next lemma we estimate the Green operator in our basic norm
³
´
n
a
kÁkX = sup (1 + t) ½ kÁ (t)kL1 + kÁ (t)kL1 + (1 + t)¡ ½ kÁ (t)kL1;a ;
t>0
where a 2 (0; 1) : Define the function g (t)
g (t) = 1 + · log (1 + t)
On Some Nonlinear Dissipative Equations with Sub-Critical Nonlinearities
143
with some · > 0:
R
Lemma 2.2. Let the function f (t; x) have a zero mean value f(t; x)dx = 0.
Then the following inequality
°
°
° k Z t ¡k
°
°g (t)
° · C k(1 + t) f (t; x)k
g
(¿)
G2(t
¡
¿)f
(¿)
d¿
X
°
°
0
X
is valid for k = 0; 1; provided that the right-hand side is finite.
Proof. View of the estimate g¡1 (¿) · C and Lemma 2.1 we get
°Z t
°
°Z t
°
°
°
°
°
° g ¡k (¿) G(t ¡ ¿)f (¿) d¿ ° + ° g ¡k (¿) G(t ¡ ¿)f (¿) d¿ °
°
° 1 °
° 1;a
0
0
L
L
Z 4
¡1+ a½
¡k
· C"
(1 + ¿)
d¿ · C"g (t)
0
for all 0 · t · 4: We now consider t > 4. Via
the condition of the lemma for the
a
¡ 2½
function g (t) we have the estimate (1 + t)
· Cg¡1 (t) and
¡
¡
p ¢¢¡1
sup¿2[pt;t] g¡1 (¿) · C 1 + ·log 1 + t
¡
¢¡1
· C 1 + ·2 log (1 + t)
· Cg ¡1 (t) ;
hence by virtue of Lemma 2.1 we obtain
°R
°
° t ¡k
°
° 0 g (¿) G(t ¡ ¿)f (¿) d¿ °
·
+Cg
R pt
C 0
¡k
(t)
(t ¡ ¿)
R 2t
p (t ¡
t
+Cg ¡k (t)
Rt
n
½
³
¿)
n
³
n
½
³
´
Lp
¡ a½
a
´
1 ¡1
p
´
1
p ¡1 ¡1
a
¿ ½ ¡1 d¿ sup ¿>0 (1 + ¿)¡ ½ k(1 + ¿) f (¿)kL1;a
¡ a½
a ¡1
¿½
¡ a½
d¿ sup¿>0 (1 + ¿)
¡n
½
·
d¿ sup¿ >0 (1 + ¿)
³
1
p ¡1
´
k(1 + ¿) f (¿ )kL1;a
k(1 + ¿) f (¿)k Lp
µ n ³1 ´ a
³
´¶
n 1
¡1 ¡ 2½
¡1
· C t½ p
+ g ¡k (t) t ½ p
k(1 + t) f (t; x)k X
t
2
¿½
1 ¡1
p
n
Cg¡k (t) t ½
³
´
1 ¡1
p
k(1 + t) f (t; x)kX
for 1 · p · 1; and
°R
°
° t ¡k
°
° 0 g (¿) G(t ¡ ¿)f (¿) d¿ ° 1;a
L
R pt a ¡1
a
¡
· C 0 ¿ ½ d¿ sup¿>0 (1 + ¿) ½ kf (¿)k L1;a
R
a
a
+Cg ¡k (t) pt t ¿ ½ ¡1 d¿ sup¿>0 (1 + ¿)¡ ½ kf (¿)kL1;a
³ a
a´
· C" t 2½ + g¡k (t) t ½ k(1 + t) f (t; x)k X
a
· Cg¡k (t) t ½ k(1 + t) f (t; x)kX
144
Nakao Hayashi, Naoko Ito, Elena I. Kaikina and Pavel I. Naumkin
for all t > 4. Hence the result of the lemma follows. Lemma 2.2 is proved.
Next lemma will be used in the proof of the theorem to evaluate large time
behavior of the mean value of the nonlinearity in equation (2). We use the notation
´ = ½ Re¯± (®; ½; ¾) > 0,
Z
± (®; ½; ¾) = jG (x)j¾ G (x) dx
and
¾ (t) = g (t) if ¾ =
³ ½´
½
n¾ jµj¾ ´ 1¡ ¾½ n
and  ¾ (t) = 1 +
t
if ¾ 2 0;
;
n
½ ¡ n¾
n
where g (t) = 1 + jµj¾ ´ log (1 + t) :
1
1;a
Lamma 2.3. Assume
¯R , the norm
¯ kv0kL1 +kv0 kL1;a = " is
R that v0 2 L \L
¯
sufficiently small and v0 (x) dx = µ = u0 (x) dx¯ : Let function v (t; x) satisfy
the estimates
1
kvkL1 · C" (1 + t)¡ ¾ ; kvkL1 · C" and
kv (t) ¡ G (t) v0 kL1 · C"1+¾ g¡k (t) ;
where k = 1 in the critical case ¾ = ½n and k = 0 in the sub critical case
0 < ¾ < n½ :
Then the inequality
¯
¯
Z t
Z
¯
¯
¾
¯1 + ¾
¯
d¿Re
¯
jvj
v(¿;
x)dx
¡
Â
(t)
¾
¯
¯
µ 0
8
½
2¾
(7)
< C" log ¾ (t) if ¾ = ;
·
³ n´
: C"2¾ Â (t) if ¾ 2 0; ½
¾
n
is valid for all t > 0:
Proof. In view of the condition kvkL1 + kvk L1 · C" we get
(8)
¯ Z t
¯
Z
¯¾
¯
¾
¯
¯ · C"¾t;
d¿Re
¯
jvj
v(¿;
x)dx
¯µ
¯
0
whence estimate (7) follows for all 0 < t < 1:
We now consider the case t ¸ 1: By the second estimate of Lemma 2.1 we get
°
³
´°
°
¡n
¡1 °
¡a
°G (t) v0 ¡ µt ½ G xt ½ ° 1 · C"t ½ ;
L
145
On Some Nonlinear Dissipative Equations with Sub-Critical Nonlinearities
where µ =
R
v0 (x) dx: Hence we find
°
¯ ³
´¯
³
´°
° ½
¾ ¡ 1+¾
n¯
¡ 1½ ¯¾
¡ 1½ °
½
°jvj v ¡ jµj µt
¯G xt
¯ G xt
°
L1
³
· C kv (t) ¡ G (t) v0k L1
°
³
´° ´
°
¡ n½
¡ 1½ °
+ °G (t) v0 ¡ µt G xt
° 1
L
³
´
¡ n¾
£ kvk¾L1 + kGv0k¾L1 + jµj¾ t ½ kGk ¾L1
· C"1+2¾ t¡
n¾
½
g¡k (t) + C"1+¾ t¡
¾n+a
½
for all t ¸ 1; where k = 1 if ¾ = n½ and k = 0 if 0 < ¾ < n½ : Since
Z ¯ ³
Z
´¯
³
´
¡ n½
¯
¡ 1½ ¯¾
¡ 1½
t
¯G xt
¯ G xt
dx = jG(x)j¾ G (x) dx = ± (®; ½; ¾) ;
and Re¯± (®; ½; ¾) =
´
½
> 0 we get
¯ Z
¯
¯
¯
¾
¯Re ¯ jvj¾ v (t; x) dx ¡ jµj¾ µt¡ ½ n ´ ¯
¯
½¯
°
¯ ³
´ ¯¾ ³
´°
1+¾
1
° ½
¡½ ¯
¡ 1½ °
¾ ¡ ½ n¯
· C °jvj v ¡ jµj µt
¯G xt
¯ G xt
°
L1
· C"
n¾
1+2¾ ¡ ½ ¡k
t
g
(t) + C"
¾n+a
1+¾ ¡ ½
t
for all t ¸ 1; where 0 < ¾ · ½: Therefore
¯ Zt
¯
Z
¯½
¯
¾
¾
¯
d¿Re ¯ jvj v(¿; x)dx ¡ jµj ´ log t¯¯
¯µ
1
Zt
Z t
a
C"2¾ d¿
¾
·
+ C"
¿ ¡1¡ ¾ d¿
¾
1 ¿ (1 + jµj ´ log(1 + ¿))
1
· C"2¾ log (1 + jµj¾ ´ log (1 + t)) + C"¾
for all t ¸ 1: Thus in view of (8) we obtain (7) in the case ¾ = n½ : In the same
manner we have the inequality
¯ Zt
¯
Z
¯¾
¾ n¯
n¾´
1¡
¾
¾
¯
d¿Re ¯ jvj v(¿; x)dx ¡ jµj
t ½ ¯¯
¯µ
½ ¡ n¾
0
¾
· C"2¾ t1¡ ½ n + C"¾ t1¡
¾n+a
½
for all t ¸ 0; which in view of (8) implies (7) in the case 0 < ¾ < n½ . Lemma 2.3
is proved.
146
Nakao Hayashi, Naoko Ito, Elena I. Kaikina and Pavel I. Naumkin
3. P ROOF OF THEOREM 1.1
We make a change of the dependent variable u (t; x) = v (t; x) e¡'(t)+iÃ(t) as
in [11]. Then for the new function v (t; x) we get the following equation
@t v + ® (¡¢) 2 v + ¯e¡¾' jvj¾ v + °e¡{' jvj{ v ¡ ('0 ¡ iÃ0)v = 0:
½
We assume that
Z
¡
¢
¯e¡¾' jvj¾ v + °e¡{' jvj{ v ¡ ('0 ¡ iÃ0 )v dx = 0
R
then the mean value of new function v (t; x) satisfies a conservation law: dtd v (t; x) dx =
0; hence b
v (t; 0) = b
v0 (0) for all t > 0: We can choose '(0) = 0 and à (0) such
n
that ^v0 (0) = j^
u0 (0)j = µ (2¼)¡ 2 > 0. Thus we consider the Cauchy problem for
the new dependent variables (v (t; x) ; ' (t))
8
¡ ¾
R ¾
¢
½
>
2 v = ¡¯e¡¾' jvj ¡ 1
@
v
+
®
(¡¢)
jvj
vdx
v
>
t
µ
>
>
>
¡
R
¢
>
{
{
<
¡°e¡{' jvj ¡ 1µ jvj vdx v;
(9)
¡
R
R
¢
>
>
@t ' (t) = 1µ e¡¾' Re¯ jvj¾ vdx + e(¾¡{)'Re° jvj{ vdx ;
>
>
>
>
:
v (0; x) = v0 (x) ; '(0) = 0:
We denote h (t) = e¾'(t) and write (9) as
8
½
<
@t v + ®(¡¢) 2 v = F (v; h) ; v (0; x) = v0 (x) ;
³
´
(10)
: @ h = ¾ Re¯ R jvj¾ vdx + h1¡ {¾ Re° R jvj{ vdx ; h(0) = 1;
t
µ
where
F (v; h) = ¡¯h
¡1
µ
¶
µ
¶
Z
Z
1
1
¾
¾
{
{
¡{
¾
jvj ¡
jvj vdx v ¡ °h
jvj ¡
jvj vdx v:
µ
µ
We note that the mean value of the nonlinearity F\ (v; h) (t; 0) = 0 for all t > 0:
{
It is expected that the second summand °h¡ ¾ jvj{ v of the nonlinearity F (v; h)
decays in time more rapidly than the first one ¯h¡1 jvj¾ v. We now prove the existence of the solution (v (t; x) ; h(t)) for the Cauchy problem (10) by the successive
approximations (vm (t; x) ; hm (t)) ; m = 1; 2; :::; defined as follows
8
½
@t vm + ® (¡¢) 2 vm = F (vm¡1 ; hm¡1 ) ;
>
>
>
<
µ
¶
Z
Z
¾
1¡ {
¾
{
(11)
¾
@t hm =
Re¯ jvm¡1j vm¡1dx + hm¡1 Re° jvm¡1j vm¡1dx ;
>
>
µ
>
:
vm (0; x) = v0 (x) ; hm(0) = 1;
On Some Nonlinear Dissipative Equations with Sub-Critical Nonlinearities
147
for all m ¸ 2; where v1 = G (t) v0; h1 = ¾ (t),
¾ (t) = g (t) if ¾ =
³ ½´
½
n¾ jµj¾ ´ 1¡ ¾½ n
and ¾ (t) = 1 +
t
if ¾ 2 0;
;
n
½ ¡ n¾
n
where g (t) = 1 + jµj¾ ´ log(1 + t) : We now prove by induction the following
estimates
kv mkX · C"; kvm (t) ¡ G (t) v0 kL1 · C"1+¾ g¡k (t) ;
(
(12)
C"2¾ log ¾ (t) if ¾ = ½n ;
jhm (t) ¡ ¾ (t)j ·
¡
¢
C"2¾Â ¾ (t) if ¾ 2 ½n ¡ C"¾ ; ½n
for all m ¸ 1; where k = 1 in the critical case ¾ = ½n and k = 0 in the sub critical
case 0 < ¾ < ½n ; the norm k¢kX is defined as above by
³
´
n
a
kÁkX = sup (1 + t) ½ kÁ (t)kL1 + kÁ (t)kL1 + (1 + t)¡ ½ kÁ (t)kL1;a ;
t>0
where a 2 (0; min (1; ½)) : By virtue of Lemma 2.1 we have
n
kG (t) v 0kL1 · C" (1 + t)¡ ½ ; kG (t) v0kL1 · C",
° ³
³ 1 ´´°
° a
°
¡ n½
¡
°j¢j G (t) v0 ¡ µt G t ½ (¢) ° 1 · C"
L
and
° n
³ 1 ´°
° ¡½ a
¡
°
°t j¢j G t ½ (¢) °
L1
a
· Ct ½ :
Therefore estimates (12) are valid for m = 1: We assume that estimates (12) are
true with m replaced by m¡1: The integral equation associated with (11) is written
as
8
Z t
>
>
vm (t) = G (t) v0 +
G(t ¡ ¿)F (vm¡1 (¿) ; hm¡1 (¿)) d¿;
>
>
>
0Z
>
µ
Z
<
¾ t
hm (t) = 1 +
d¿ Re¯ jv m¡1 j¾ vm¡1 dx
>
µ 0Z
>
¶
>
>
>
1¡ {
{
>
¾
:
+hm¡1Re° jvm¡1 j vm¡1 dx :
Note that in the critical case ¾ =
½
n
we have
148
Nakao Hayashi, Naoko Ito, Elena I. Kaikina and Pavel I. Naumkin
kF (vm¡1 (t) ; hm¡1 (t))k L1
µ
¶
1
1+¾
¡1
· Chm¡1 (t) kvm¡1 (t)k L1 1 + kvm¡1 (t)kL1
µ
¡1¡ 1¾ ¡1
1+¾
· C"
(1 + t)
g (t) ;
(13)
kF (vm¡1 (t) ; hm¡1 (t))k L1
µ
¶
1
¾
2
¡1
· Chm¡1 (t) kvm¡1 (t)k L1 kvm¡1 (t)kL1 + kvm¡1 (t)k L1
µ
· C"1+¾ (1 + t)¡1 g ¡1 (t)
and
kF (vm¡1 (t) ; hm¡1 (t))kL1;a
·
¾
Ch¡1
m¡1 (t) kvm¡1 (t)k L1 kvm¡1 (t)k L1;a
· C"1+½ (1 + t)
¡1+ a½
g¡1 (t)
µ
¶
1
1 + kv m¡1 (t)kL1
µ
for all t > 0; provided
¡
¢that (vm¡1 (t) ; hm¡1 (t)) satisfies (12). Similarly in the sub
critical case ¾ 2 0; n½ we obtain
kF (vm¡1 (t) ; hm¡1 (t))kL1
¡
¢
1+¾
1
· Ch¡1
m¡1 (t) kvm¡1 (t)kL1 1 + µ kvm¡1 (t)kL1
³
´
¡ 1+¾ n
njµj¾ 1¡ ¾ n ¡1
· C"1+¾ (1 + t) ½
1 + ½¡n¾ t ½
¡
¢
¡1¡ n
½ ;
· C" n½ ¡ ¾ (1 + t)
(14)
kF (vm¡1 (t) ; hm¡1 (t))kL1
³
´
¾
2
1
· Ch¡1
(t)
kv
(t)k
kv
(t)k
+
kv
(t)k
1
1
1
m¡1
m¡1
m¡1
m¡1
L
L
L
µ
¡½
¢
¡1
· C" n ¡ ¾ (1 + t)
and
kF (vm¡1 (t) ; hm¡1 (t))kL1;a
·
¾
Ch¡1
m¡1 (t) kvm¡1 (t)k L1 kvm¡1 (t)k L1;a
· C"
³½
´
¡1+ ®½
¡ ¾ (1 + t)
µ
¶
1
1 + kv m¡1 (t)kL1
µ
n
for all t > 0: This yields the estimate
°
°
°
°
°(1 + t) gk (t) F (vm¡1 (t) ; hm¡1 (t))° · C"1+¾
X
On Some Nonlinear Dissipative Equations with Sub-Critical Nonlinearities
149
if we suppose that ½ ¡ n¾ · C"¾ : Since F (vm¡1 (¿) ; hm¡1 (¿)) have the zero
mean value we get via Lemma 2.2
°
°
° k Z t ¡k
°
° g (t)
° · C"1+¾
g
(¿)
G(t
¡
¿)F
(v
(¿)
;
h
(¿))
d¿
m¡1
m¡1
°
°
0
X
whence it follows that
kvm kX · C"; kvm (t) ¡ G (t) v0kL1 · C"1+¾ g¡k (t) :
By virtue of Lemma 2.3 we find that
(
jhm (t) ¡ ¾ (t)j ·
C"2¾ log ¾ (t) if ¾ = ½n ;
¡
¢
C"2¾ ¾ (t) if ¾ 2 n½ ¡ C"¾; n½
for all t > 0: Thus by induction we see that estimates (12) are valid for all m ¸ 1:
In the same way by induction we can prove that
1
kvm¡1 ¡ vm¡2kX ;
4
1
sup ¡1
kvm¡1 ¡ vm¡2kX
¾ (t) jhm (t) ¡ h m¡1 (t)j ·
4
t>0
1
+ sup ¡1
(t) jhm¡1 (t) ¡ hm¡2 (t)j
4 t>0 ¾
kvm ¡ vm¡1kX ·
for all m > 2: Therefore taking limits
lim vm (t; x) = v (t; x) and lim hm (t) = h (t)
m!1
m!1
we obtain a unique solution v (t; x) 2 X, h(t) = e¾'(t) 2 C (0; 1) satisfying
equations
8
R
v (t) = G (t) v0 + 0t G(t ¡ ¿)F (v (¿) ; h (¿)) d¿;
<
³
´
(15)
: h(t) = 1 + ¾ R t d¿ Re¯ R jvj¾ vdx + h1¡ {¾ Re° R jvj{ vdx ;
µ
and estimates
and
(16)
0
kv (t) ¡ G (t) v0 kL1 · C"1+¾ g¡k (t)
jh (t) ¡ ¾ (t)j ·
(
C"2¾ log ¾ (t) if ¾ = ½n ;
¡
¢
C"2¾Â¾ (t) if ¾ 2 ½n ¡ C"¾ ; ½n :
150
Nakao Hayashi, Naoko Ito, Elena I. Kaikina and Pavel I. Naumkin
We also have by applying (13) and (14) to (15)
8
< C"1+¾ (1 + t)¡ ¾1 g¡1 (t) if ¾ = ½ ;
n
(17)
kv (t) ¡ G (t) v0k L1 ·
¡½
¢
:
¡ n½
1+¾
¾
C"
(1 + t)
if ¾ 2 n ¡ C" ; n½ ;
and by the definition of à we see that
\ 0 (0) ¡ 1
à (t) = arg u
µ
Z
0
t
¡¾'
e
µ
Z
Z
¶
¾
(¾¡·)'
·
Im¯ jvj vdx + e
Im° jvj vdx :
By the time decay property of the solution v we have
¯
¯
Zt
¯
¯
·¡¾
¡ ¾½ n
¡ ¾1
\ 0 (0) + jµj¾ é
¯Ã (t) ¡ arg u
h (¿) (1 + ¿)
d¿ ¯¯ · C"·t¡ ½ n;
¯
0
where é = Im¯± (®; ½; ¾) : Hence by (16)
¯
¯
1
¯
¯
R
¡ ¾½ n
¾
¯Ã (t) ¡ arg u
\ 0 (0) + jµj¾ é t  ¡
¯
(¿)
(1
+
¿)
d¿
¾
¯
0
¯
1
R
¾
¡1¡
· C jµj¾ é 0t ¾ ¾ (¿) jh (¿) ¡ ¾ (¿)j (1 + ¿)¡ ½ n d¿
(18)
8 R ¡1¡ 1
< C t ¾ ¾ (¿) (log ¾ (¿)) (1 + ¿)¡1 d¿ if ¾ = ½ ;
0
n
·
1
R
¾
¡
¢
:
t ¡
¡ n
C 0 ¾ ¾ (¿) (1 + ¿) ½ d¿ if ¾ 2 ½n ¡ C"¾; ½n
1
for large t > 0. Then via formulas u (t; x) = e¡'(t)+iÃ(t) v (t; x) = h¡ ¾ (t) eiÃ(t) v (t; x)
we find the estimates
°
³ 1 ´
°
°
°
¡n
¡
°u(t) ¡ µt ½ G t ½ (¢) e¡'(t)+iÃ(t) ° 1
L
°
°
· °u(t) ¡ (G (t) v0 ) e¡'(t)+iÃ(t) ° L1 +
°³
³ 1 ´´
°
°
°
¡n
¡
(19)
° G (t) v0 ¡ µt ½ G t ½ (¢) e¡'(t)+iÃ(t) ° 1
L
8
1
< C"1+¾ (1 + t)¡ ¾ g ¡1 (t) if ¾ = ½ ;
n
·
¡
¢
1
:
¡
½
C"1+¾ (1 + t) ¾ if ¾ 2 n ¡ C"¾ ; ½n ;
where we have used the estimate
°³
°
³ 1 ´´
°
°
¡n
¡
° G (t) v0 ¡ µt ½ G t ½ (¢) e¡'(t)+iÃ(t) °
L1
1
a
· Ct¡ ¾ ¡ ½ kÁkL1;a
151
On Some Nonlinear Dissipative Equations with Sub-Critical Nonlinearities
and (17). We also have by (16)
°
°
³ 1 ´ ¡1
° ¡ n ³ ¡1 ´ ¡ 1
°
n
¡
¡
iÃ(t)
iÃ(t)
°µt ½ G t ½ (¢) h ¾ (t) e
°
¡ µt ½ G t ½ (¢) ¾ ¾ (t) e
°
°
·
n ¡1¡ 1
C"t¡ ½ ¾ ¾
L1
(t) jh(t) ¡ ¾ (t)j
whence via (19) it follows that
°
°
³ 1 ´ 1
°
°
n
¾
iÃ(t) °
°u (t) ¡ µt¡ ½ G t¡ ½ (¢) ¡
(t)
e
¾
°
° 1
8
L
1
1
(20)
< C"1+¾ (1 + t)¡ ¾ g ¡ ¾ (t) if ¾ = ½ ;
n
·
¡
¢
1
:
C"1+¾ (1 + t)¡ ¾ if ¾ 2 n½ ¡ C"¾; n½ :
This completes the proof of Theorem 1.1.
4. PROOF OF THEOREM 1.2
e with norm
We consider the function space X
³
´
n
½
kÁkX
e = sup (1 + t) kÁ (t)k L1 + kÁ (t)kL1
t>0
and define the mapping u = Mv as follows
Z t
(21)
u (t) = G (t) u0 +
G(t ¡ ¿)F (v (¿)) d¿;
0
e and kvk e · C": Using Lemma 2.1
where F (v (¿)) = ¯ jvj¾ v + ° jvj{ v; v 2 X
X
in (21) we obtain
R
ku(t)kL1 · kG (t) v0k L1 + 0t kG(t ¡ ¿)F (v (¿))kL1 d¿
n
(22)
· C (ku0kL1 + ku0kL1 ) (1 + t)¡ ½
Rt
¡n
+C 02 (t ¡ ¿) ½ (kv (¿)k¾L1 + kv (¿)k·L1 ) kv (¿)kL1 d¿
Rt
+C t (kv (¿)k¾L1 + kv (¿)k·L1 ) kv (¿)kL1 d¿
2
n
and
(23)
n
· C (ku0kL1 + ku0kL1 ) (1 + t)¡ ½ + C"1+¾ (1 + t)¡ ½
ku (t)k L1 · kG (t) v0kL1 +
·
Rt
0 kG(t ¡ ¿)F (v (¿))k L1 d¿
Rt
C ku0kL1 + C 0 (kv (¿)k¾L1 + kv (¿)k·L1 ) kv (¿)k L1 d¿
· C ku0kL1 + C"1+¾;
152
Nakao Hayashi, Naoko Ito, Elena I. Kaikina and Pavel I. Naumkin
where we have used the fact that ¾; · > n½ : Hence we have by (22) and (23)
(24)
1+¾
kukX
e · C (ku0kL1 + ku 0kL1 ) + C"
which implies that if the initial function is sufficiently small, then the mapping M
e to itself. In the same way as in the proof
defined by u = Mv is a mapping from X
of (23) we find that M is a contraction mapping. Hence we have a unique solution
e satisfying the integral equation
u2X
Z t
(25)
u (t) = G (t) u0 +
G(t ¡ ¿) (¯ juj¾ u(¿) + ° juj{ u (¿)) d¿:
0
Applying the same arguments as in the proof of (22), it follows that
(26)
n
ku(t) ¡ G (t) u0k · C"1+¾ (1 + t)¡ ½ :
The result of the theorem follows from (26) and the second estimate of the Lemma
2.1 with ! = 0; a > 0 and p = 1:
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154
Nakao Hayashi, Naoko Ito, Elena I. Kaikina and Pavel I. Naumkin
Nakao Hayashi,
Department of Mathematics
Graduate School of Science, Osaka University
Osaka Toyonaka 560-0043,
Japan
E-mail: nhayashi@math.wani.osaka-u.ac.jp
Naoko Ito
Department of Mathematics
Faculty of Science
Tokyo University of Science
26 Wakamiya-cho, Shinjuku-ku
Tokyo, 162-0827,
Japan
E-mail: j1102701@ed.kagu.tus.ac.jp
Elena I. Kaikina
Departamento de Ciencias Básicas
Instituto Tecnológico de Morelia
CP 58120, Morelia,
Michoacán
M éxico
Pavel I. Naumkin
Instituto de Matemáticas
UNAM, Campus Morelia
AP 61-3 (Xangari), Morelia
CP 58089, Michoacán,
M éxico
E-mail: pavelni@matmor.unam.mx