Funkcialaj Ekvacioj, 40 (1997) 353-370
Asymptotic Behavior in Time of the Solutions of a Coupled System
of KdV Equations
By
E. BISOGNIN †, V. BISOGNIN † and G. Perla MENZALA ††
(† Federal University of Santa Maria and †† National Laboratory of Scientific Computation, Brazil)
§1.
Introduction
Our aim in this paper is to study the asymptotic behavior in time of the
solutions of a coupled system of Korteweg-de Vries equations. More precisely,
we consider global regular solutions of the “conservative” system:
(1.1)
$¥left¥{¥begin{array}{l}u_{t}+u_{xxx}+a_{3}v_{XXX}+u^{p}u_{X}+a_{1}v^{p}v_{x}+a_{2}(u^{p}v)_{¥chi}=0¥¥b_{¥mathrm{l}}v_{t}+v_{xxx}+b_{2}a_{3}u_{xxx}+v^{p}v_{x}+b_{2}a_{2}u^{p}u_{x}+b_{2}a_{1}(uv^{p})_{x}=0,¥end{array}¥right.$
where , , , ,
are real constants with $b_{1}>0$ and $b_{2}>0$ . System (1.1)
is considered for $-¥infty<x<+¥infty$ and
. The power is an integer bigger
than or equal to one. System (1.1) has the structure of a pair of Korteweg-de
Vries equations coupled through both dispersive and nonlinear effects. In case
$p=1$ system (1.1) was derived by Gear and Grimshaw in 1984 ([8]) as a model
to describe the strong interaction of weakly nonlinear, long waves. Mathematical results on system (1.1) were given by Bona, Ponce, Saut and Tom
([6]). They proved that problem (1.1) is globally well-posed in $H^{s}(R)¥times H^{s}(R)$
for any $s¥geq 1$ provided $|a_{3}|<1/¥sqrt{b_{2}}$ . Their result was recently improved by
J. Marshall Ash, J. Cohen and G. Wang ([14]). They proved that system
(1.1) (with $p=1$ ), is globally well-posed in $L^{2}(R)¥times L^{2}(R)$ provided that
. We shall also consider the “dissipative” system
$a_{1}$
$a_{2}$
$a_{3}$
$b_{1}$
$b_{2}$
$t$
$¥geq 0$
$p$
$|a_{3}|¥neq 1/¥sqrt{b_{2}}$
(1.2)
$¥left¥{¥begin{array}{l}u_{t}+u_{XXX}+a_{3}v_{xxx}+u^{p}u_{X}+a_{1}v^{p}v_{x}+a_{2}(u^{p}v)_{¥chi}-¥epsilon u_{XX}=0¥¥b_{1}v_{t}+v_{XXX}+b_{2}a_{3}u_{XXX}+v^{p}v_{x}+b_{2}a_{2}u^{p}u_{X}+b_{2}a_{1}(uv^{p})_{x}-¥epsilon v_{xx}=0,¥end{array}¥right.$
where
and the constants $a_{j},b_{k},j=1,2,3$ , $k=1,2$ and
are as
above. The extra terms
which appear in (1.2) correspond to
,
the presence of dissipative effects arising in the description of the phenomenon. Observe that if we consider (1.1) or (1.2) with $a_{1}=a_{2}=a_{3}=0$ then we
obtain the scalar Korteweg-de Vries equation. In that case, existence of global
solutions as well as the asymptotic behavior in time have been intensively
studied by several authors in recent year ([2], [3], [5], [12], [15] and the references
$¥epsilon>0$
$p$
$-¥epsilon u_{XX}$
therein).
$-¥epsilon v_{XX}$
E. BISOGNIN, V. BISOGNIN and G. Perla MENZALA
354
In order to study the asymptotic behavior of the solutions of system (1.1)
we need an existence result providing global solutions for large values of . We
obtain this result for small data using commutator estimates involving fractional
derivatives. Using the method of stationary phase we analyse the linear part of
system (1.1) and then using the integral version of our problem we arise to the
following result: There exists a constant $C>0$ such that
$¥mathrm{p}$
$||u(¥cdot, t)||_{W^{1,¥infty}}+||v(¥cdot, t)||_{W^{1,¥infty}}¥leq C(1+t)^{-1/3}$
,
, provided that the initial data at $t=0$ satisfy suitable assumptions:
¥
. Conceming system (1.2) the
they are sufficiently small, $p>4$ and
discussion is somehow easier due to the presence of dissipation. Certainly when
large) and
we consider “stronger” interaction (that is system (1.1) with
, similar results as above can be obtained for
and
dissipative terms
will be stronger due to the behavior of the
decay
small data. However, the
linear part. We also consider system (1.2) with large initial data. In this case
.
we only show asymptotic stability of the solutions as
$L^{q}(R)$
we shall denote the space (class
We shall use standard notation: By
power is integrable, with the norm
of) functions in $R$ whose
as
$ t¥rightarrow+¥infty$
$|a_{3}| sqrt{b_{2}}<1$
$p$
$-¥epsilon v_{XX}$
$-¥epsilon u_{xx}$
$f$
$||f||_{L^{q}}^{q}=$
$q^{th}$
$¥int_{-¥infty}^{+¥infty}|f(x)|^{q}dx$
,
$ 1¥leq q<¥infty$
By
.
essentially bounded functions in
we denote the space of measurable
with the norm
$L^{¥infty}(R)$
$R$
$||f||_{L^{¥infty}}=¥mathrm{e}¥mathrm{s}¥mathrm{s}¥sup_{-¥infty<x<+¥infty}|f(x)|$
For each
space
, the Sobolev space of order
with respect to the norm
$s¥in R$
$S(R)$
$s$
.
as the completion of the Schwartz
$||u||_{H^{s}}^{2}=¥int_{-¥infty}^{+¥infty}(1+|y|^{2})^{s}|¥hat{u}|^{2}dy$
The Fourier transform of
$f$
$¥rightarrow+¥infty$
.
is defined as
$¥hat{f}(y)=(2¥pi)^{-1/2}¥int_{-¥infty}^{+¥infty}e^{-ixy}f(x)dx$
and the inverse Fourier transform of
$f$
,
is given by
$¥check{f}(x)=(2¥pi)^{-1/2}¥int_{-¥infty}^{+¥infty}e^{ixy}¥hat{f}(y)dy$
.
be a real number, we denote by $¥mathrm{Y}_{m}=H^{m}(R)¥times H^{m}(R)$ with the norm
, we
matrix, $M$
. Let $M$ be an
$M$
$||M||=
¥
sum_{i,j=1}^{n}|m_{ij}|$
$||M||$
.
by
given
which is
the norm of the matrix
denote by
If $X$ is a Banach space we denote by $C(0, T;X)$ the space of continuous
functions $u:[0, T]¥rightarrow X$ . Various positive constants will be denote by $C$ .
Let
$m$
$||(u, v)||_{¥mathrm{Y}_{m}}^{2}=||u||_{H^{m}}^{2}+||v||_{H^{m}}^{2}$
$n¥times n$
$=[m_{ij}]_{n¥times n}$
Asymptotic Behavior in Time
They may vary line to line.
of the
Solutions
of KdV
355
Equations
We denote by
and
$J^{s}=(I-¥frac{¥partial^{2}}{¥partial_{X^{2}}})^{s/2}$
$D^{s}=(-¥frac{¥partial^{2}}{¥partial_{X^{2}}})^{s/2}$
,
the Bessel and Riesz potentials of order $-s$ respectively. Define $W^{s,p}=J^{-s}L^{p}$
. When $p=2$ we have the
whose norm will be denoted by .
$A$
$B$
$H^{2}(R)$
and
two operators, then the com. Let
classical Sobolev space
$[J^{s};
f]g=J^{s}(fg)-fJ^{s}g$ in which
$-BA$
. Thus,
mutator is given by $[A;B]=AB$
$f$ is regarded as a multiplication operator.
$||$
§2.
$||_{W^{s,p}}=||J^{s}||_{L^{p}}$
The Cauchy problem (local existence) and some technical lemmas
In this section we consider system (1.1) with initial data $u(x, 0)=u_{0}(x)$ ,
$v(x,0)=v_{0}(x)$ .
The following local existence theorem follows as in the paper
of J. Bona et. al. ([6]).
Theorem 2.1. Let $m¥geq 1$ and $(u_{0}, v_{0})¥in H^{m}(R)¥times H^{m}(R)$ . Consider system
(1.1) together with initial conditions $u(x,0)=u_{0}(x)_{¥mathit{3}}v(x,0)=v_{0}(x)$ . Let
an integer and
(real) constants $(j =1,2, 3; k=1,2)$ as indicated in the
introduction. Then, there exists $T_{0}=T_{0}(||u_{0}, v_{0})||_{¥mathrm{Y}_{m}},p)>0$ and a unique solution $(u(x, t), v(x, t))¥in X_{m}(T_{0})¥times X_{m}(T_{0})$ , of system (1.1) with initial data
where $X_{m}(T_{0})=C(0, T_{0};H^{m}(R))¥cap C^{1}(0, T_{0};H^{m-3}(R))$ . Moreover, the pair
in the sense that the mapping $(u_{0}, v_{0})¥mapsto(u, v)$ is
depends continuously on
into the space $X_{m}(T_{0})¥times X_{m}(T_{0})$ .
continuous from
$p¥geq 1_{¥mathit{3}}$
$a_{j_{¥mathit{3}}}b_{k}$
$p$
$(u_{0}, v_{0})_{¥mathit{3}}$
$(¥mathrm{w}, v)$
$(u_{0}, v_{0})$
$¥mathrm{Y}_{m}$
Proof.
The initial value problem associated with system (1.1) can be
written as
(2. 1)
$W_{t}+AW_{¥chi¥chi¥chi}=F(W)$
$W(x, 0)=W_{0}(x)$
where
$W=¥left(¥begin{array}{l}u¥¥v¥end{array}¥right)$
,
$A$
$=¥left¥{¥begin{array}{ll}1 & a_{3}¥¥a_{3}b_{2}/b_{¥mathrm{l}} & 1/b_{1}¥end{array}¥right¥}$
and
$F(W)=¥left¥{¥begin{array}{l}-u^{p}u_{X}-a_{¥mathrm{l}}v^{p}v_{X}-a_{2}(u^{p}v)_{x}¥¥-¥frac{1}{b_{1}}v^{p}v_{x}-b_{2}a_{2}/b_{1}u^{p}u_{X}-b_{2}a_{1}/b_{¥mathrm{l}}(uv^{p})_{X}¥end{array}¥right¥}$
.
356
E. BISOGNIN, V. BISOGNIN and G. Perla MENZALA
We first consider the associated linear system
(2.2)
$W_{t}+AW_{xxx}=¥mathit{0}$
,
$W(x, 0)=W_{0}(x)$ .
Using the Fourier transform we can (formally) write the solution of (2.2) as
(2.3)
$¥hat{W}(y, t)=¥exp(iy^{3}At)¥hat{W}_{0}(y)$
where
$a_{3}¥neq 0$
Here
The eigenvalues of
$¥hat{W}_{0}=(¥hat{u}_{0},¥hat{v}_{0})^{¥tau}$
.
.
$¥tau$
$A$
denotes transpose.
are given by
First let us assume that
.
(2.4)
$¥lambda_{1}=¥frac{1}{2}(1+¥frac{1}{b_{1}}+¥sqrt{(1-¥frac{1}{b_{1}})^{2}+¥frac{4b_{2}a_{3}^{2}}{b_{1}}})$
(2.5)
’
$¥lambda_{2}=¥frac{1}{2}(1+¥frac{1}{b_{1}}-¥sqrt{(1-¥frac{1}{b_{1}})^{2}+¥frac{4b_{2}a_{3}^{2}}{b_{1}}})$
which are distinct since $b_{1}>0,b_{2}>0$ and
matrix $¥exp$ ( At) to obtain
$a_{3}¥neq 0$
.
We can easily calculate the
$iy^{3}$
$¥exp(iy^{3}At)=¥left¥{¥begin{array}{ll}¥mathrm{e}¥mathrm{x}¥mathrm{p}(i¥lambda_{1}ty^{3})-c_{1}c_{2}¥mathrm{e}¥mathrm{x}¥mathrm{p}(i¥lambda_{2}ty^{3}) & -c_{¥mathrm{l}}¥mathrm{e}¥mathrm{x}¥mathrm{p}(i¥lambda_{1}ty^{3})+c_{1}¥mathrm{e}¥mathrm{x}¥mathrm{p}(i¥lambda_{2}ty^{3})¥¥c_{2}¥mathrm{e}¥mathrm{x}¥mathrm{p}(i¥lambda_{1}ty^{3})-c_{2}¥mathrm{e}¥mathrm{x}¥mathrm{p}(i¥lambda_{2}ty^{3}) & -c_{1}c_{2}¥mathrm{e}¥mathrm{x}¥mathrm{p}(i¥lambda_{1}ty^{3})+¥mathrm{e}¥mathrm{x}¥mathrm{p}(i¥lambda_{2}ty^{3})¥end{array}¥right¥}$
where $c_{1}=a_{3}/(¥lambda_{2}-1)$ and $c_{2}=$
. We note that
is well defined
and
. In fact, if
or
we would have that
which
$b_{1},b_{2}>0$
$c_{1}c_{2}=(
¥
lambda_{1}-1)/(
¥
lambda_{2}-1)
¥neq 1$
it is impossible because
and
. Also
because
. From (2.3) we get a formula for
and by inversion of
the Fourier transform we obtain the solution-pair of the initial value problem
$(¥lambda_{1}-1)/a_{3}$
$c_{2}¥neq 0$
$¥lambda_{2}=1$
$c_{1}$
$¥lambda_{1}=1$
$b_{2}a_{3}^{2}/b_{1}=0$
$a_{3}¥neq 0$
$¥hat{W}(y, t)$
$¥lambda_{1}¥neq¥lambda_{2}$
(2.2):
(2.6)
$¥tilde{u}(x, t)=(2¥pi)^{1/2}(1-c_{1}c_{2})^{-1}¥int_{-¥infty}^{+¥infty}¥{¥exp(i¥lambda_{1}ty^{3}+ixy)$
?
$c_{1}c_{2}¥exp(i¥lambda_{2}ty^{3}+ixy)¥}¥hat{u}_{0}(y)dy$
$+(2¥pi)^{-1/2}(1-c_{1}c_{2})^{-1}¥int_{-¥infty}^{+¥infty}¥{-c_{1}¥exp(i¥lambda_{1}ty^{3}+ixy)$
$+c_{1}¥exp(i¥lambda_{2}ty^{3}+ixy)¥}¥hat{v}_{0}(y)dy$
and
(2.7)
$¥tilde{v}(x, t)=(2¥pi)^{-1/2}(1-c_{1}c_{2})^{-1}¥int_{-¥infty}^{+¥infty}c_{2}¥exp(i¥lambda_{1}ty^{3}+ixy)$
?
$c_{2}¥exp(i¥lambda_{2}ty^{3}+ixy)¥}¥hat{u}_{0}(y)dy$
Asymptotic Behavior in Time
of the Solutions of KdV
Equations
357
$+(2¥pi)^{-1/2}(1-c_{1}c_{2})^{-1}¥int_{-¥infty}^{+¥infty}¥{-c_{1}c_{2}¥exp(i¥lambda_{1}ty^{3}+ixy)$
$+¥exp(i¥lambda_{2}ty^{3}+ixy)¥}¥hat{v}_{0}(y)dy$
.
We introduce the notation
$U(t)(u_{0}, v_{0})(x)=(¥tilde{u},¥tilde{v})(x, t)$
.
Using Duhamel’s formula, we know that problem (2.1) can be written as
(2.8)
$(u, v)(¥cdot, t)=U(t)(u_{0}, v_{0})(¥cdot)-¥int_{0}^{t}U(t-s)(f_{1},f_{2})(¥cdot,s)ds$
where
$f_{1}=u^{p}u_{x}+a_{1}v^{p}v_{x}+a_{2}(u^{p}v)_{x}$
and
$f_{2}=¥frac{1}{b_{1}}¥{v^{p}v_{X}+b_{2}a_{2}u^{p}u_{X}+b_{2}a_{1}(uv^{p})_{x}¥}$
.
If $a_{3}=0$ then the linear system (2.2) is decoupled, therefore representation (2.8)
it is evident with $U(t)$ being the group of operators generated by
$A$
$=¥left¥{¥begin{array}{ll}1 & 0¥¥0 & 1/b_{¥mathrm{l}}¥end{array}¥right¥}$
.
Now, the proof of local existence for (2.1) it follows from
representation (2.8) with the same sequence of ideas as in [6], [9], [11] or [13].
The following Lemmas will be useful in the next section.
Lemma 2.2.
Let
$f¥in S(R)_{¥mathit{3}}$
then
$||D^{s}f||_{L^{p}}¥leq C||D^{s_{0}}f||_{L^{p_{0}}}^{a}||D^{s_{1}}f||_{L^{p_{1}}}^{1-a}$
where
$p_{0}$
,
,
$p_{1}¥in(1, ¥infty)_{¥mathit{3}}s_{0},s_{1}¥in R$
$(0¥leq a¥leq 1)$
.
if
Moreover,
$s=as_{0}+(1-a)s_{1}$
$s_{0_{¥mathit{3}}}s_{1}¥in[0,$
$¥infty$
and $1/p=a/p0+(1-a)/p_{1}$ ,
) then
$||J^{s}f||_{L^{p}}¥leq C||J^{s0}f||_{L^{p}0}^{a}||J^{s_{1}}f||_{L^{p_{1}}}^{1-a}$
Proof. The
Lemma 2.3.
,
.
proof follows from Theorem 6.4.5 of [1].
Let
$f$
, $g¥in S(R)$ ,
$ 1<p<¥infty$
and
$s>0_{¥mathit{3}}$
then
$||J^{s}(fg)||_{L^{p}}¥leq C(||f||_{L^{p_{1}}}||J^{s}g||_{L^{p_{2}}}+||J^{s}f||_{L^{p_{3}}}||g||_{L^{p_{4}}})$
and
$||[J^{s};f]g||_{L^{p}}¥leq C(||f_{x}||_{L^{p_{1}}}||J^{s-1}g||_{L^{p_{2}}}+||J^{s}f||_{L^{p_{3}}}||g||_{L^{p_{4}}})$
where
$p_{2}$
,
$p_{3}¥in(1, ¥infty)$
and
$p_{1}$
,
$p_{4}$
are such that
$1/p_{1}+1/p_{2}=1/p=1/p_{3}+1/p_{4}$ .
E. BISOGNIN, V. BISOGNIN and G. Perla MENZALA
358
Proof. See [7], Appendix.
be the local solution of system (1.1) with initial
data $(u_{0}, v_{0})¥in H^{m}(R)¥times H^{m}(R)$ obtained in Theorem 2.1. Assume $m¥geq 1$ , ,
with
are real constants with $b_{1}>0$ and $b_{2}>0$ and $p¥geq 2$. Then for any
$1¥leq k¥leq m$ ,
Theorem 2.4.
Let
$(u, v)$
$a_{j}$
$b_{k}$
$k$
(2.9)
$¥sup_{t¥in[0,T_{0}]}||(u(¥cdot, t), v(¥cdot, t))||_{¥mathrm{Y}_{k}}¥leq C||(u_{0}, v_{0})||_{¥mathrm{Y}_{k}}¥exp(C¥int_{0}^{T_{0}}¥psi(s)ds)$
where
(2. 10)
$¥psi(s)=||u||_{L^{¥infty}}^{p-1}||u_{x}||_{L^{¥infty}}+||u||_{L^{¥infty}}^{p-2}||u_{X}||_{L^{¥infty}}||v||_{L^{¥infty}}+||u||_{L^{¥infty}}^{p-1}||v_{x}||_{L^{¥infty}}$
$+||v||_{L^{¥infty}}^{p-2}||||v_{X}||_{L^{¥infty}}||u||_{L^{¥infty}}+||v||_{L^{¥infty}}^{p-1}||u_{x}||_{L^{¥infty}}+||v||_{L^{¥infty}}^{p-1}||v_{x}||_{L^{¥infty}}$
.
The local well-posedness result mentioned in Theorem 2.1 implies
that the (formal) calculations below can be justified provided $1¥leq k¥leq m$ by
regularizing the initial data (see [4]), making the calculations for the associated
smooth functions and then passing to the limit. Since this procedure is
standard we do not repeat it here. We will prove (2.9) using results on
to
commutators mentioned in Lemmas 2.2 and 2.3. Applying the operator
the first equation of system (1.1) we obtain
Proof.
$J^{k}$
(2. 11)
$(J^{k}u)_{t}+¥frac{¥partial^{3}}{¥partial x^{3}}(J^{k}u)+a_{3}¥frac{¥partial^{3}}{¥partial_{X^{3}}}(J^{k}v)+[J^{k};u^{p}]u_{x}+u^{p}J^{k}u_{X}+a_{1}[J^{k};v^{p}]v_{X}$
$+a_{1}v^{p}J^{k}v_{x}+a_{2}[J^{k}; u^{p}]v_{X}+a_{2}u^{p}J^{k}v_{X}+a_{2}p[J^{k}; u^{p-1}v]u_{X}$
$+a_{2}pu^{p-1}vJ^{k}u_{x}=0$
Similarly, applying the operator
obtain
(2. 12)
$J^{k}$
to the second equation of system (1.1), we
$b_{1}(J^{k}v)_{t}+¥frac{¥partial^{3}}{¥partial x^{3}}(J^{k}v)+b_{2}a_{3}¥frac{¥partial^{3}}{¥partial x^{3}}(J^{k}u)+[J^{k}; v^{p}]v_{x}+v^{p}J^{k}v_{X}$
$+b_{2}a_{2}[J^{k}; u^{p}]u_{x}+b_{2}a_{2}u^{p}J^{k}u_{X}+b_{2}a_{1}p[J^{k}; v^{p-1}u]v_{X}$
$+b_{2}a_{1}pv^{p-1}uJ^{k}v_{x}+a_{1}b_{2}[J^{k}; v^{p}]u_{x}+b_{2}a_{1}v^{p}J^{k}u_{X}=0$
Now, we multiply equation (2. 11) by
$J^{k}u$
and equation (2. 12) by
.
$(1/b_{2})J^{k}v$
.
Inte-
Asymptotic Behavior in Time
of
the Solutions
of KdV
359
Equations
grating in the whole space and adding the results we obtain
(2. 13)
$¥frac{1}{2}¥frac{d}{dt}¥{||J^{k}u||_{L^{2}}^{2}+¥frac{b_{1}}{b_{2}}||J^{k}v||_{L^{2}}^{2}¥}=-¥int_{-¥infty}^{+¥infty}[J^{k}; u^{p}]u_{x}J^{k}udx$
?
?
$a_{1}¥int_{-¥infty}^{+¥infty}[J^{k}; v^{p}]v_{x}J^{k}udx-a_{2}p¥int_{-¥infty}^{+¥infty}[J^{k}; u^{p-1}v]u_{X}J^{k}udx$
$a_{2}¥int_{-¥infty}^{+¥infty}[J^{k}; u^{p}]v_{x}J^{k}udx-a_{2}¥int_{-¥infty}^{+¥infty}[J^{k}; u^{p}]u_{x}J^{k}vdx$
$-¥frac{1}{b_{2}}¥int_{-¥infty}^{+¥infty}[J^{k}; v^{p}]v_{x}J^{k}vdx-a_{1}p¥int_{-¥infty}^{+¥infty}[J^{k}; v^{p-1}u]v_{x}J^{k}vdx$
?
?
$a_{1}¥int_{-¥infty}^{+¥infty}[J^{k}, v^{p}]u_{X}J^{k}vdx-¥int_{-¥infty}^{+¥infty}u^{p}J^{k}u_{X}J^{k}udx$
$a_{2}p¥int_{-¥infty}^{+¥infty}u^{p-1}vJ^{k}u_{X}J^{k}udx$
$-¥frac{1}{b_{2}}¥int_{-¥infty}^{+¥infty}v^{p}J^{k}v_{x}J^{k}vdx-a_{1}p¥int_{-¥infty}^{+¥infty}v^{p-1}uJ^{k}v_{x}J^{k}vdx$
$+a_{2}p¥int_{-¥infty}^{+¥infty}u^{p-1}u_{x}J^{k}vJ^{k}udx+a_{1}p¥int_{-¥infty}^{+¥infty}v^{p-1}v_{x}J^{k}vJ^{k}udx$
.
Next, using Holder’s inequality, integration by parts and Lemmas 2.2 and
2.3 we estimate each term on the right hand side of (2.13):
a)
$|¥int_{-¥infty}^{+¥infty}[J^{k}; u^{p}]u_{x}J^{k}udx|¥leq||[J^{k}; u^{p}]u_{x}||_{L^{2}}||J^{k}u||_{L^{2}}$
$¥leq C(||u||_{L^{¥infty}}^{p-1}||u_{x}||_{L^{¥infty}}||J^{k}u||_{L^{2}}+||J^{k}u^{p}||_{L^{2}}||u_{X}||_{L^{¥infty}})||J^{k}u||_{L^{2}}$
$¥leq$
$¥mathrm{c}¥mathrm{o}¥mathrm{n}¥mathrm{s}¥mathrm{t}¥mathrm{a}¥mathrm{n}¥mathrm{t}(||u||_{L^{¥infty}}^{p-1}||u_{x}||_{L^{¥infty}}||J^{k}u||_{L^{2}}^{2})$
.
Similarly, we obtain
b)
c)
$|¥int_{-¥infty}^{+¥infty}[J^{k}; v^{p}]v_{x}J^{k}udx|¥leq C||v||_{L^{¥infty}}^{p-1}||v_{x}||_{L^{¥infty}}||J^{k}v||_{L^{2}}||J^{k}u||_{L^{2}}$
,
$|¥int_{-¥infty}^{+¥infty}[J^{k}; u^{p}]v_{x}J^{k}udx|¥leq C||u||_{L^{¥infty}}^{p-1}||v_{x}||_{L^{¥infty}}||J^{k}u||_{L^{2}}^{2}$
d)
$|¥int_{-¥infty}^{+¥infty}[J^{k}; v^{p}]v_{x}J^{k}vdx|¥leq C||v||_{L^{¥infty}}^{p-1}||v_{x}||_{L^{¥infty}}||J^{k}v||_{L^{2}}^{2}$
,
e)
$|¥int_{-¥infty}^{+¥infty}[J^{k}; v^{p}]u_{x}J^{k}vdx|¥leq C||v||_{L^{¥infty}}^{p-1}||u_{x}||_{L^{¥infty}}||J^{k}v||_{L^{2}}^{2}$
,
,
60
E. BISOGNIN, V. BISOGNIN and G. Perla MENZALA
,
1)
$|¥int_{-¥infty}^{+¥infty}[J^{k};u^{p}]u_{x}J^{k}vdx|¥leq C||u||_{L^{¥infty}}^{p-1}||u_{x}||_{L^{¥infty}}||J^{k}u||_{L^{2}}||J^{k}v||_{L^{2}}$
$r)$
$|¥int_{-¥infty}^{+¥infty}[J^{k}; u^{p-1}v]u_{x}J^{k}udx|¥leq||[J^{k}; u^{p-1}v]u_{X}||_{L^{2}}||J^{k}u||_{L^{2}}$
$¥leq C((||u||_{L^{¥infty}}^{p-1}||v_{X}||_{L^{¥infty}}+||u||_{L^{¥infty}}^{p-2}||v||_{L^{¥infty}}||u_{x}||_{L^{¥infty}})||J^{k}u||_{L^{2}}^{2}$
$+||u||_{L^{¥infty}}^{p-1}||J^{k}v||_{L^{2}}||u_{X}||_{L^{¥infty}}||J^{k}u||_{L^{2}})$
1)
,
$|¥int_{-¥infty}^{+¥infty}[J^{k}; v^{p-1}u]v_{x}J^{k}vdx|¥leq C((||v||_{L^{¥infty}}^{p-1}||u_{X}||_{L^{¥infty}}$
$+||v||_{L^{¥infty}}^{p-2}||u||_{L^{¥infty}}||v_{x}||_{L^{¥infty}})||J^{k}v||_{L^{2}}^{2}+||v||_{L^{¥infty}}^{p-1}||J^{k}u||_{L^{2}}||J^{k}v||_{L^{2}}||v_{x}||_{L^{¥infty}})$
$)$
$|¥int_{-¥infty}^{+¥infty}||u^{p}J^{k}u_{X}J^{k}udx|=¥frac{1}{2}|¥int_{-¥infty}^{+¥infty}u^{p}¥frac{¥partial}{¥partial x}(J^{k}u)^{2}dx|$
$¥leq¥frac{p}{2}||u||_{L^{¥infty}}^{p-1}||u_{X}||_{L^{¥infty}}||J^{k}u||_{L^{2}}^{2}$
$)$
k)
$|¥int_{-¥infty}^{+¥infty}||v^{p}J^{k}v_{X}J^{k}vdx|¥leq¥frac{p}{2}||v||_{L^{¥infty}}^{p-1}||v_{x}||_{L^{¥infty}}||J^{k}v||_{L^{2}}^{2}$
,
,
$|¥int_{-¥infty}^{+¥infty}||u^{p-1}vJ^{k}u_{x}J^{k}udx|=¥frac{1}{2}|¥int_{-¥infty}^{+¥infty}||u^{p-1}v¥frac{¥partial}{¥partial x}(J^{k}u)^{2}dx|$
$=¥frac{1}{2}|¥int_{-¥infty}^{+¥infty}((p-1)u^{p-2}u_{X}v+u^{p-1}v_{x})(J^{k}u)^{2}dx|$
,
$¥leq¥frac{p-1}{2}||u||_{L^{¥infty}}^{p-2}||u_{x}||_{L^{¥infty}}||v||_{L^{¥infty}}||J^{k}u||_{L^{2}}^{2}+||u||_{L_{¥infty}}^{p-1}||v_{x}||_{L^{¥infty}}||J^{k}u||_{L^{2}}^{2}$
1)
$|¥int_{-¥infty}^{+¥infty}v^{p-1}uJ^{k}v_{x}J^{k}vdx|=|¥frac{1}{2}¥int_{-¥infty}^{+¥infty}v^{p-1}u¥frac{¥partial}{¥partial x}(J^{k}u)^{2}dx|$
$¥leq¥frac{p-1}{2}||v||_{L^{¥infty}}^{p-2}||v_{x}||_{L^{¥infty}}||u||_{L^{¥infty}}||J^{k}v||_{L^{2}}^{2}+||v||_{L_{¥infty}}^{p-1}||u_{X}||_{L^{¥infty}}||J^{k}v||_{L^{2}}^{2}$
m)
n)
$|¥int_{-¥infty}^{+¥infty}||v^{p-1}v_{x}J^{k}vJ^{k}udx|¥leq||v||_{L^{¥infty}}^{p-1}||v_{X}||_{L^{¥infty}}||J^{k}v||_{L^{2}}||J^{k}u||_{L^{2}}$
$|¥int_{-¥infty}^{+¥infty}||u^{p-1}u_{x}J^{k}vJ^{k}udx|¥leq||u||_{L^{¥infty}}^{p-1}||u_{x}||_{L^{¥infty}}||J^{k}v||_{L^{2}}||J^{k}u||_{L^{2}}$
From (2.13) and all the above estimates we obtain
,
.
,
,
Asymptotic Behavior in Time
(2. 14)
of the Solutions of KdV
361
Equations
$¥frac{1}{2}¥frac{d}{dt}¥{||J^{k}u||_{L^{2}}^{2}+¥frac{b_{1}}{b_{2}}||J^{k}v||_{L^{2}}^{2}¥}$
$¥leq C¥{||u||_{L^{¥infty}}^{p-1}||u_{X}||_{L^{¥infty}}||J^{k}u||_{L^{2}}^{2}+||v||_{L^{¥infty}}^{p-1}||v_{x}||_{L^{¥infty}}||J^{k}v||_{L^{2}}^{2}$
$+||v||_{L^{¥infty}}^{p-1}||v_{X}||_{L^{¥infty}}||J^{k}u||_{L^{2}}||J^{k}v||_{L^{2}}+||u||_{L^{¥infty}}^{p-1}||u_{X}||_{L^{¥infty}}||J^{k}u||_{L^{2}}||J^{k}v||_{L^{2}}$
$+||u||_{L^{¥infty}}^{p-1}||v_{x}||_{L^{¥infty}}||J^{k}u||_{L^{2}}^{2}+||v||_{L^{¥infty}}^{p-1}||u_{X}||_{L^{¥infty}}||J^{k}v||_{L^{2}}^{2}$
$+||u||_{L^{¥infty}}^{p-2}||v||_{L^{¥infty}}||u_{x}||_{L^{¥infty}}||J^{k}u||_{L^{2}}^{2}+||v||_{L^{¥infty}}^{p-2}||u||_{L^{¥infty}}||v_{x}||_{L^{¥infty}}||J^{k}v||_{L^{2}}^{2}$
$+||v||_{L^{¥infty}}^{p-1}||u_{X}||_{L^{¥infty}}||J^{k}v||_{L^{2}}^{2}+||u||_{L^{¥infty}}^{p-1}||v_{X}||_{L^{¥infty}}||J^{k}u||_{L^{2}}^{2}¥}$
$¥leq C¥psi(t)¥{||J^{k}u||_{L^{2}}^{2}+||J^{k}v||_{L^{2}}^{2}¥}$
where is given by (2. 10). Integration of (2. 14) from zero to
Gronwall’s inequality, we conclude the proof of Theorem 2.4.
$t¥leq T_{0}$
$¥psi$
and using
Global solution and asymptotic behavior
§3.
In order to extend the local solution of system (1.1) obtained in the above
section it is sufficient to show that the function
given by (2.10) remains
$s
¥
geq
0$
bounded for all
. This will be the case as long as the initial data is chosen
sufficiently small. In what follows we shall use estimates obtained by Kenig,
Ponce and Vega in [13] conceming the solutions of the linear proof of the scalar
$¥psi(s)$
$¥mathrm{K}¥mathrm{d}¥mathrm{V}$
equation.
Lemma 3.1. ([13]). Let
$j=1,2$ and $i=¥sqrt{-1}$, then
$f¥in¥Psi(R)$ ,
$V^{j}(t)f(x)=¥int_{-¥infty}^{+¥infty}¥exp(ixy+i¥lambda_{j}ty^{3})¥hat{f}(y)dy$
$||V^{j}(t)f(¥cdot)||_{L^{¥infty}(R)}¥leq Ct^{-1/3}||f||_{L^{1}(R)}$
Proof.
$a$
See [13], Lemma 2.1 with
$a$
$=0$
,
.
or [12], Theorem 6.2 with
$¥beta=0$
,
$=1$ .
Theorem 3.2. Assume that $|a_{3}|¥sqrt{b}<1$ , $b_{1}>0$, $b_{2}>0$ and $p>4$. Let us
consider system (1.1) with initial data $(u_{0}, v_{0})¥in W^{1,1}(R)¥cap H^{s}(R)¥times W^{1,1}(R)¥cap$
$H^{s}(R)$ with $s¥geq 2$ .
There exists $¥delta>0$, such that if
$||(u_{0}, v_{0})||_{W^{1,1}¥times W^{1,1}}+||(u_{0}, v_{0})||_{¥mathrm{Y}_{s}}<¥delta$
then, system (1.1) with initial data
at $t=0$, has a unique global solution
$(u, v)$ in the class $C(0, ¥infty; H^{s}(R))¥times C(0, ¥infty; H^{s}(R))$ satisfying
$(u_{0}, v_{0})$
$¥sup_{t>0}(1+t)^{1/3}¥{||u(¥cdot, t)||_{W^{1,¥infty}}+||v(¥cdot, t)||_{W^{1,¥infty}}¥}<+¥infty$
.
E. BISOGNIN, V. BISOGNIN and G. Perla MENZALA
362
be the eigenvalues of
formulas (2.6) and (2.7) say that the solution
Proof.
Let
$¥lambda_{1}$
and
$¥lambda_{2}$
$A$
given by (2.4) and (2.5), then
of (2.1) is given by
$(¥tilde{u},¥tilde{v})$
(3.1)
$¥tilde{u}(x, t)=(2¥pi)^{-1/2}(1-c_{1}c_{2})^{-1}¥{V^{1}(t)u0-c_{1}c_{2}V^{2}(t)u_{0}+c_{2}V^{2}(t)v_{0}-c_{1}c_{2}V^{1}(t)v_{0}¥}$
,
,
$¥tilde{v}(x, t)=(2¥pi)^{-1/2}(1-c_{1}c_{2})^{-1}¥{c_{2}V^{1}(t)u_{0}-c_{2}V^{2}(t)u_{0}+V^{2}(t)v_{0}-c_{1}c_{2}V^{1}(t)v_{0}¥}$
was defined in Lemma 3.1. Observe that a density argument says
where
that Lemma 3.1 holds for $f¥in W^{1,1}(R)¥cap H^{s}(R)$ . Hence from (3.1) we deduce
$V^{j}(t)$
(3.2)
$||¥tilde{u}(¥cdot, t)||_{L^{¥infty}}¥leq Ct^{-1/3}(||u_{0}||_{L^{1}}+||v_{0}||_{L^{1}})$
,
$||¥tilde{v}(¥cdot, t)||_{L^{¥infty}}¥leq Ct^{-1/3}(||u_{0}||_{L^{1}}+||v_{0}||_{L^{1}})$
.
Now, we consider the derivative (with respect to
is the solution of
Clearly
$x$
) of system (2.2).
$(¥tilde{u}_{x},¥tilde{v}_{x})$
(3.3)
$¥left¥{¥begin{array}{l}u_{xt}+u_{¥mathrm{x}xxx}+a_{3}v_{xxxx}=0,¥¥v_{xt}+¥frac{1}{b_{1}}v_{xxxx}+¥frac{a_{3}b_{2}}{b_{1}}u_{XXXX}=0.¥end{array}¥right.$
Similar cal-
with initial conditions $¥tilde{u}_{x}(x, 0)=du_{0}(x)/dx,¥tilde{v}_{X}(x, 0)=dv_{0}(x)/dx$ .
culations to the ones we obtained in (3.2) are valid for
$(¥tilde{u}_{x},¥tilde{v}_{X})$
(3.4)
$¥left¥{¥begin{array}{l}||¥tilde{u}_{X}(¥cdot,t)||_{L^{¥infty}}¥leq C(||||||||_{L^{1}})t^{-¥mathrm{l}}/3,¥¥||¥tilde{v}_{x}(¥cdot,t)||_{L^{¥infty}}¥leq C(||||||||_{L^{1}})t^{-1}/3.¥end{array}¥right.$
Let us introduce the notation
$U(t)(u_{0}, v_{0})(x)=(¥tilde{u},¥tilde{v})(x, t)$
.
Thus (3.2) and (3.4) say, in particular, that
$||U(t)(u_{0}, v_{0})(x)||_{W^{1,¥infty}¥times W^{1,¥infty}}=||(¥tilde{u},¥tilde{v})(x, t)||_{W^{1,¥infty}¥times W^{1,¥infty}}$
(3.5)
$¥leq C(||u_{0}||_{W^{1,1}}+||v_{0}||_{W^{1,1}})t^{-1/3}$
Using Duhamel’s formula, we know that the solution
system (1.1) can be written as
(2.6)
$(¥mathrm{w}, v)$
.
of the nonlinear
$(u, v)(¥cdot, t)=U(t)(u_{0}, v_{0})(¥cdot)-¥int_{0}^{t}U(t-s)(f_{1},f_{2})(¥cdot,s)ds$
,
Asymptotic Behavior in Time
of the
Solutions
of KdV
363
Equations
where
$f_{l}=u^{p}u_{x}+a_{l}v^{p}v_{X}+a_{¥mathit{2}}(u^{p}v)_{x}$
,
$f_{2}=¥frac{1}{b_{1}}¥{v^{p}v_{X}+b_{2}a_{2}u^{p}u_{x}+b_{2}a_{1}(uv^{p})_{x}¥}$
Using the imbedding
(3.7)
$H^{j}(R)¥llcorner¥rightarrow L^{¥infty}(R),j¥geq 1$
.
we can estimate
$f_{1}$
and
$f_{2}$
as follows
$||f_{1}||_{W^{1,1}}+||f_{2}||_{W^{1,1}}¥leq C¥{||u||_{L^{¥infty}}^{p-1}||u||_{H^{2}}^{2}+||v||_{L^{¥infty}}^{p-1}||v||_{H^{2}}^{2}$
$+||u||_{L^{¥infty}}^{p-1}||u||_{H^{2}}||v||_{H^{2}}+||u||_{L^{¥infty}}^{p-2}||v||_{L^{¥infty}}||u||_{H^{2}}^{2}$
$+||v||_{L^{¥infty}}^{p-1}||v||_{H^{2}}||u||_{H^{2}}+||u||_{L^{¥infty}}^{p-2}||u||_{L^{¥infty}}||v||_{H^{2}}^{2}¥}<+¥infty$
in the interval of existence.
Let [0, ) the maximum interval of existence of the solution-pair $(u,v)$ of
system (1.1). Inequality (3.7) shows that for any $0¥leq t<T^{*}$ , the solution $(u, v)$
. Using estimates (3.5) we deduce that
belongs to
$T^{*}$
$H^{s}¥cap W^{1,¥infty}¥times H^{s}¥cap W^{1,¥infty}$
(3.8)
$||u(¥cdot, t)||_{W^{1,¥infty}}+||v(¥cdot, t)||_{W^{1,¥infty}}$
$¥leq C¥delta t^{-1/3}+C¥int_{0}^{t}(t-s)^{-1/3}¥{||f_{1}||_{W^{1,1}}+||f_{2}||_{W^{1,1}}¥}ds$
Next, let us define the function
(3.9)
$m(¥cdot)$
.
given by
$m(T)=¥sup_{0¥leq t¥leq T}(1+t)^{1/3}¥{||u(¥cdot, t)||_{W^{1,¥infty}}+||v(¥cdot, t)||_{W^{1,¥infty}}¥}$
is well defined, nonnegative continuous
where $0¥leq T<T^{*}$ . It follows that
and nondecreasing. From (3.7), the definition of $m(T)$ and Theorem 2.4 (with
$k=2)$ it follows that
$m(¥cdot)$
(3.10)
$||f_{1}(¥cdot,s)||_{W^{1,1}}+||f_{2}(¥cdot,s)||_{W^{1,1}}$
$¥leq Cm^{p-1}(T)(1+s)^{(1-p)/3}||(u_{0}, v_{0})||_{¥mathrm{Y}_{2}}¥exp(C¥int_{0}^{s}¥psi(¥tau)d¥tau)$
.
From (3.8) and (3.10) we deduce that
$||u(¥cdot, t)||_{W^{1¥infty}}¥}+||v(¥cdot, t)||_{W^{1,¥infty}}¥leq C¥delta(1+t)^{-1/3}$
$+C¥delta¥int_{0}^{t}(t-s)^{-1/3}(1+s)^{(1-p)/3}¥exp(c¥int_{0}^{s}¥psi(¥sigma)d¥sigma)m^{p-1}(T)ds$
$¥leq C¥delta(1+t)^{-1/3}+C¥delta m^{p-1}(T)¥exp(cm^{p}(T)¥int_{0}^{t}(1+¥sigma)^{-p/3}d¥sigma)$
.
$¥times¥int_{0}^{t}(t-s)^{-1/3}(1+s)^{(1-p)/3}ds$
364
E. BISOGNIN, V. BISOGNIN and G. Perla MENZALA
Now we use a lemma due to W. Strauss [18], p. 438, which says that if $p>4$
then
$¥int_{0}^{t}(t-s)^{-1/3}(1+s)^{(1-p)/3}ds¥leq C(1+t)^{-1/3}$
for some $C>0$ .
Consequently, we obtain that
$||u(¥cdot, t)||_{W^{1,¥infty}}+||v(¥cdot, t)||_{W^{1,¥infty}}¥leq C¥delta(1+t)^{-1/3}+C¥delta m^{p-1}(T)(1+t)^{-1/3}¥exp(Cm^{p}(T))$
or
$(1+t)^{1/3}¥{||u(¥cdot, t)||_{W^{1,¥infty}}+||v(¥cdot, t)||_{W^{1,¥infty}}¥}¥leq C¥delta+C¥delta m^{p-1}(T)¥exp(cm^{p}(T))$
which implies that
$m(T)¥leq C¥delta+C¥delta m^{p-1}(T)¥exp(Cm^{p}(T))$
where the constant does not depend on $T$ . Therefore as long as is sufficiently
small there exists a constant $K$ (which actually is the ffist positive root of
the function $f(x)=C¥delta+C¥delta x^{p-1}¥exp(cx^{p})-x)$ such that $m(T)<K$ for all
$0¥leq T<T^{*}$ .
This apriori estimate combined with Theorem 2.4 shows that the
solution pair $(u, v)$ exists globally and satisfies the desired decay.
$¥delta$
Remarks. The result of Theorem 3.2 could be extended in two directions:
a) We assumed that is an integer $>4$ just for simplicity. The method works
as well for any real number
) We may also consider more general nonby $a(u)$ and $a(v)$ respectively,
linearities in system (1.1) replacing
and
where
is a smooth function such that $|a(s)|¥leq C|s|^{p}$ for some positive
$C$
constant , any $s¥in R$ and $p>4$ (not necessarily an integer). Results of this
equation by C. Kenig, G. Ponce and L.
type were obtained for a single
Vega in [12] and [13].
$p$
$>4;¥mathrm{b}$
$u^{p}$
$v^{p}$
$a(¥cdot)$
$¥mathrm{k}¥mathrm{d}¥mathrm{V}$
§4.
The dissipative case
$(¥epsilon>0)$
In this section we shall study the asymptotic behavior in time of the
solutions of the “dissipative” system
(4. 1)
$¥left¥{¥begin{array}{l}u_{t}+u_{xxx}+a_{3}v_{xxx}+u^{p}u_{x}+a_{1}v^{p}v_{X}+a_{2}(u^{p}v)_{x}-¥epsilon u_{xx}=0,¥¥b_{1}v_{t}+v_{xxx}+b_{2}a_{3}u_{¥chi¥chi¥chi}+v^{p}v_{X}+b_{2}a_{2}u^{p}u_{x}+b_{2}a_{1}(uv^{p})_{x}-¥epsilon v_{xx}=0,¥end{array}¥right.$
and the coefficients , , ,
where
and
are
, $-¥infty<X<+¥infty$ ,
$b_{1}>0$
$b_{2}>0$
$a_{1}=a_{2}=a_{3}=0$
and
. In the particular case, where
real with
,
equation with dissipative term
.
system (4.1) reduces to the scalar
The asymptotic behavior in this situation was previously studied by P. Biller
$¥epsilon>0$
$t$
$¥geq 0$
$a_{1}$
$¥mathrm{k}¥mathrm{d}¥mathrm{V}$
$a_{2}$
$a_{3}$
$b_{1}$
$b_{2}$
$-¥epsilon u_{xx}$
Asymptotic Behavior in Time
of the Solutions of KdV Equations
365
([2]), J. Bona and L. Luo ([5]) and by V. Bisognin and G. Perla Menzala
([3]). Similar results as in Sections 2 and 3 could be obtained in this case,
however the decay will be stronger as
due to the behavior of the linear
part of (4.1). In fact, the linear part is
$ t¥rightarrow+¥infty$
(4.2)
$¥left¥{¥begin{array}{l}u_{t}+u_{XXX}+a_{3}v_{xxx}-¥epsilon u_{xx}=0,¥¥b_{1}v_{t}+v_{XXX}+b_{2}a_{3}u_{xxx}-¥epsilon v_{xx}=0,¥end{array}¥right.$
which can be written as
(4.3)
$b_{l}$
$U_{t}+AU_{xxx}+¥epsilon LU=¥mathit{0}$
,
where
$U=¥left(¥begin{array}{l}u¥¥v¥end{array}¥right)$
,
$A$
$=¥left¥{¥begin{array}{ll}b_{¥mathrm{l}} & b_{1}a_{3}¥¥b_{2}a_{3} & 1¥end{array}¥right¥}$
and
$L=[_{0}^{-b_{1^{¥frac{¥partial^{2}}{¥partial x^{2}}}}}$
$¥frac{0¥partial^{2}}{¥partial x^{2}}]$
.
If $a_{3}=0$ then system (4.2) decouples, therefore in this case the asymptotic
behavior was already studied by several authors (see [2], [3] or [5]). If ¥
then $¥det(A-¥lambda I)=¥lambda^{2}-(1+b_{1})¥lambda-b_{1}b_{2}a_{3}^{2}+b_{1}$ therefore $A$ is diagonalizable
¥
because $b_{1}>0$ , $b_{2}>0$ . If
the eigenvalues of $A$ are distinct.
Consequently, there exists a nonsingular matrix $P$ such that $P^{-1}AP=D$ is
diagonal. Let us consider $V=P^{-1}U$ . Substitution in (4.3) implies that $V$
satisfies
$a_{3} neq 0$
$a_{3} neq 0$
(4.4)
$b_{l}$
$V_{t}+DV_{XXX}+¥epsilon L_{l}V=¥mathit{0}$
,
where $L_{1}=P^{-1}LP$ . Observe that (4.4) also decouples system (4.2). It is
enough to obtain the asymptotic behavior of $V$ in order to get the same result
for and . The Fundamental solution of (4.4) is the
matrix $G$ given by
$u$
(4.5)
where
$2¥times 2$
$v$
$G(x, t)=(2¥pi)^{-1/2}¥int_{-¥infty}^{+¥infty}¥exp[ixyI+(ib_{1}^{-1}y^{3}D-b_{1}^{-1}¥epsilon y^{2}¥tilde{L})t]dy$
$¥tilde{L}=P^{-1}$
$¥left¥{¥begin{array}{ll}b_{1} & 0¥¥0 & 1¥end{array}¥right¥}$
$P$
and I denotes the identity matrix
(4.5) and using the change of variables
$z=b_{1}^{-1}¥epsilon y^{2}$
$¥left¥{¥begin{array}{ll}1 & 0¥¥0 & 1¥end{array}¥right¥}$
.
From
we obtain
$||G(x, t)||¥leq C¥int_{-¥infty}^{+¥infty}¥exp(-b_{1}^{-1}¥epsilon y^{2}t)dy=Ct^{-1/2}¥int_{-¥infty}^{+¥infty}¥exp(-z)z^{-1/2}dz=Ct^{-1/2}¥Gamma(¥frac{1}{2})$
where 1
denotes the Gamma function and
$G(x, t)$ .
We conclude that
$(¥cdot)$
is the norm of the matrix
In a similar manner we can
$||G(x, t)||$
$||G(¥cdot, t)||_{L^{¥infty}}¥leq Ct^{-1/2}$
.
E. BISOGNIN, V. BISOGNIN and G. Perla MENZALA
366
show that
$||G(¥cdot, t)||_{L^{2}}¥leq Ct^{-1/4}$
.
Using interpolation we deduce that
$||G(¥cdot, t)||_{L^{q}}¥leq Ct^{(1-q)/(2q)}$
for any
$ 2¥leq q¥leq+¥infty$
.
Now we can retum to system (4.2),
real, $b_{1}>0$,
and
Lemma 4.1. Consider system (4.2) with
$b_{2}>0$ .
Assume the initial data $u(x,0)=u_{0}(x)$ , $v(x,0)=v_{0}(x)$ is given with ,
$v_{0}¥in H^{m}(R)¥cap L^{1}(R)$, $(m ¥geq 2)$ , then
$¥epsilon>0$
$a_{3}$
$u_{0}$
(4.6)
$||u(¥cdot, t)||_{L^{q}}+||v(¥cdot, t)||_{L^{q}}¥leq Ct^{(1-q)/(2q)}$
for any
Proof.
$ 2¥leq q¥leq+¥infty$
data
$V_{0}$
Let
.
$V_{0}=P^{-1}U_{0}=P^{-1}(u_{0}, v_{0})$
is given by
$V(¥cdot, t)=G(¥cdot, t)*V_{0}$
.
.
The solution of (4.4) with initial
Consequently,
$||V(¥cdot, t)||_{L^{q}}¥leq||G(¥cdot, t)||_{L^{q}}||V_{0}||_{L^{1}}¥leq Ct^{(1-q)/(2q)}||V_{0}||_{L^{1}}$
which implies that (since $U=PV$ )
$||U(¥cdot, t)||_{L^{q}}=||PV(¥cdot, t)||_{L^{q}}¥leq||P||||V(¥cdot, t)||_{L^{q}}¥leq C||P||t^{(1-q)/q}||V_{0}||_{L^{1}}$
.
Now, with the help of Lemma 4.1 we can proceed as we did in Section 3 to
obtain the following result:
Let us consider system
$H^{m}(R)¥cap L^{1}(R)¥times H^{m}(R)¥cap L^{1}(R)$ and $m¥geq 2$.
are real constants with $b_{1}>0$ and $b_{2}>0$,
exists $¥delta>0$ such that if
Theorem 4.2.
$b_{2}$
(4.1) with initial data
$(u_{0}, v_{0})¥in$
and
We assume that , , ,
and $p¥geq 4$. There
,
$a_{2}$
$a_{1}$
$b_{2}a_{3}^{2}<1$
$a_{3}$
$b_{1}$
$¥epsilon>0$
$||(u_{0}, v_{0})||_{W^{1,1}¥times W^{1,1}}+||(u_{0}, v_{0})||_{¥mathrm{Y}_{m}}<¥delta$
,
at $t=0$ has a unique global {strong)
then system (4.1) with initial data
$(u, v)$ in the class $C(0, ¥infty; H^{m})¥times C(0, ¥infty; H^{m})$ satisfying
$Ct^{(1-q)/(2q)}$
.
for any $ 2¥leq q¥leq+¥infty$ , as
$(u_{0}, v_{0})$
$||u(¥cdot, t)||_{L^{q}}+||v(¥cdot, t)||_{L^{q}}¥leq$
$ t¥rightarrow+¥infty$
We will not give details of the proof here because the main steps are just as
in the proof of Theorem 3.2.
Remarks. The same remarks as we mentioned at the end of Theorem 3.2
and
apply here. Also, since in this case we gain the extra information that
belong to $L^{2}(R)$ it seems plausible that the final result of Theorem 4.2 should
remain valid for $p>3$ .
$u_{X}$
$v_{X}$
We will show now the asymptotic stability of the solutions
with large initial data:
$¥dot{¥mathrm{o}}$
$¥mathrm{f}$
system (1.2)
Asymptotic Behavior in Time
of
the Solutions
of KdV
Theorem 4.3. Let $(u, v)$ be the global solution of system
and
real numbers with , , $b_{2}>0$, $p=1$ and
,
the initial data belongs to $H^{m}(R)¥times H^{m}(R)$ , $m¥geq 2$ then
$b_{2}$
$b_{1}$
$a_{3}$
$¥epsilon$
a)
(1.2) with ,
$a_{3}^{2}b_{2}<1$
$b_{1}$
367
Equations
, ,
Assume that
$¥epsilon$
.
$a_{2}$
$a_{1}$
$¥lim_{t¥rightarrow+¥infty}(||u_{x}(¥cdot, t)||_{L^{2}}+||v_{x}(¥cdot, t)||_{L^{2}})=0$
and
b)
$¥lim_{t¥rightarrow+¥infty}(||u_{x}(¥cdot, t)||_{L^{¥infty}}+||(¥cdot, t)||_{L^{¥infty}})=0$
.
Let us multiply the first equation in (1.2) by
and the second
equation by . Integration in the whole space, some integrations by parts and
addition give us
Proof.
$b_{2}u$
$v$
(4.7)
$¥frac{d}{dt}¥{b_{2}||u(¥cdot, t)||_{L^{2}}^{2}+b_{1}||v(¥cdot, t)||_{L^{2}}^{2}¥}+2¥epsilon¥{b_{2}||u_{x}(¥cdot, t)||_{L^{2}}^{2}+||v_{x}(¥cdot, t)||_{L^{2}}^{2}¥}=0$
.
Integration in time of (4.7) gives us
$b_{2}||u(¥cdot, t)||_{L^{2}}^{2}+b_{1}||v(¥cdot, t)||_{L^{2}}^{2}+2¥epsilon¥int_{0}^{t}(b_{2}||u_{X}(¥cdot, t)||_{L^{2}}^{2}+||v_{X}(¥cdot, t)||_{L^{2}}^{2})ds$
$=b_{2}||u_{0}||_{L^{2}}^{2}+b_{1}||v_{0}||_{L^{2}}^{2}=$
for any
$t¥geq 0$
.
constant
In particular,
(4.8)
$¥int_{0}^{+¥infty}(b_{2}||u_{x}||_{L^{2}}^{2}+||v_{X}||_{L^{2}}^{2})ds<+¥infty$
and
(4.9)
$¥left¥{¥begin{array}{l}||u(¥cdot,t)||_{L^{2}}¥leq||u_{0}||_{L^{2}}+¥sqrt{¥frac{b_{1}}{b_{2}}}||v_{0}||_{L^{2}},¥¥||v(¥cdot,t)||_{L^{2}}¥leq||v_{0}||_{L^{2}}+¥sqrt{¥frac{b_{2}}{b_{1}}}||u_{0}||_{L^{2}}.¥end{array}¥right.$
Next, we multiply the first equation in (1.2) by
and the second
equation by
. Integration in the whole space and addition give us the
identity
$-b_{2}u_{XX}$
$-v_{xx}$
(4. 10)
$¥frac{1}{2}¥frac{d}{dt}¥{b_{2}||u_{x}||_{L^{2}}^{2}+b_{1}||v_{X}||_{L^{2}}^{2}¥}+¥epsilon b_{2}||u_{xx}||_{L^{2}}^{2}+¥epsilon||v_{¥mathrm{x}x}||_{L^{2}}^{2}$
$=b_{2}¥int_{-¥infty}^{+¥infty}u_{xx}uu_{x}dx+b_{2}a_{1}¥int_{-¥infty}^{+¥infty}u_{xx}vv_{x}dx+b_{2}a_{2}¥int_{-¥infty}^{+¥infty}u_{xx}(uv)_{x}dx$
$+¥int_{-¥infty}^{+¥infty}v_{XX}vv_{x}dx+b_{2}a_{2}¥int_{-¥infty}^{+¥infty}v_{xx}uu_{x}dx+b_{2}a_{1}¥int_{-¥infty}^{+¥infty}v_{xx}(uv)_{X}dx$
.
368
E. BISOGNIN, V. BISOGNIN and G. Perla MENZALA
Using Holder’s inequality and the imbedding
$H^{s}(R)¥leftrightarrow L^{¥infty}(R)$
, we obtain from
(4. 10)
(4. 11)
$¥frac{1}{2}¥frac{d}{dt}¥{b_{2}||u_{x}||_{L^{2}}^{2}+b_{1}||v_{x}||_{L^{2}}^{2}¥}+¥epsilon b_{2}||u_{xx}||_{L^{2}}^{2}+¥epsilon||v_{XX}||_{L^{2}}^{2}$
$¥leq b_{2}||u||_{L^{¥infty}}||u_{x}||_{L^{2}}||u_{xx}||_{L^{2}}+b_{2}|a_{1}|||v||_{L^{¥infty}}||v_{x}||_{L^{2}}||u_{xx}||_{L^{2}}$
$+b_{2}|a_{2}|||u||_{L^{¥infty}}||v_{x}||_{L^{2}}||u_{xx}||_{L^{2}}+b_{2}|a_{2}|||v||_{L^{¥infty}}||u_{X}||_{L^{2}}||u_{XX}||_{L^{2}}$
$+||v||_{L^{¥infty}}||v_{xx}||_{L^{2}}||v_{X}||_{L^{2}}+b_{2}|a_{2}|||u||_{L^{¥infty}}||v_{xx}||_{L^{2}}||u_{X}||_{L^{2}}$
$+b_{2}|a_{1}|||v||_{L^{¥infty}}||u_{x}||_{L^{2}}||v_{XX}||_{L^{2}}+b_{2}|a_{1}|||u||_{L^{¥infty}}||v_{x}||_{L^{2}}||v_{xx}||_{L^{2}}$
.
Now, we use Gagliardo-Niremberg’s inequality: If $f¥in H^{1}(R)$ then
and if $f¥in H^{2}(R)$ then
(see
[1] . Together with estimates (4.9), the right hand side of (4.11) is bounded by
$||f||_{L^{¥infty}}¥leq C||f_{X}||_{L^{2}}^{1/2}||f||_{L^{2}}^{1/2}$
$||f_{x}||_{L^{2}}¥leq C||f_{xx}||_{L^{2}}^{1/2}||f||_{L^{2}}^{1/2}$
$)$
(4. 12)
$C||u_{xx}||_{L^{2}}^{7/4}+C||v_{XX}||_{L^{2}}^{3/4}||u_{xx}||_{L^{2}}+C||u_{XX}||_{L^{2}}^{5/4}||v_{xx}||_{L^{2}}^{1/2}$
$+C||u_{xx}||_{L^{2}}^{3/2}||v_{xx}||_{L^{2}}^{1/4}+C||v_{xx}||_{L^{2}}^{7/4}+C||u_{xx}||_{L^{2}}^{3/4}||v_{xx}||_{L^{2}}$
.
$+C||v_{XX}||_{L^{2}}^{5/4}||u_{XX}||_{L^{2}}^{1/2}+C||u_{XX}||_{L^{2}}^{1/4}||v_{xx}||_{L^{2}}^{3/2}$
In each term in (4. 12) we shall use Young’s inequality: “Let , $y¥geq 0$ , $¥beta>0$ and
$0¥leq a<2$ .
Given $¥delta>0$ then there exists $C(¥delta)>0$ such that
,
where $¥gamma=(2¥beta)/(2-¥alpha)‘‘$ . Thus, from (4.11) and (4.12) we deduce that
$x$
$x^{a}y^{¥beta}¥leq¥delta x^{2}+c(¥delta)y^{¥gamma}$
$¥frac{1}{2}¥frac{d}{dt}¥{b_{2}||u_{x}||_{L^{2}}^{2}+b_{1}||v_{x}||_{L^{2}}^{2}¥}+¥epsilon b_{2}||u_{xx}||_{L^{2}}^{2}+¥epsilon||v_{XX}||_{L^{2}}^{2}$
$¥leq C+C¥delta^{2}||u_{xx}||_{L^{2}}^{2}+C¥delta^{2}||v_{xx}||_{L^{2}}^{2}$
Choosing
$¥delta>0$
such that
$C¥delta^{2}<¥min¥{¥epsilon b_{2},¥epsilon¥}$
.
then we deduce that
$¥frac{1}{2}¥frac{d}{dt}¥{b_{2}||u_{x}||_{L^{2}}^{2}+b_{1}||v_{X}||_{L^{2}}^{2}¥}+(¥epsilon b_{2}-c¥delta^{2})||u_{xx}||_{L^{2}}^{2}+(¥epsilon-C¥delta^{2})||v_{XX}||_{L^{2}}^{2}¥leq$
which proves that the function
derivative which together with (4.8) implies that
$H(t)=b_{2}||u_{x}||_{L^{2}}^{2}+b_{1}||v_{x}||_{L^{2}}^{2}$
(4. 13)
$¥lim_{t¥rightarrow+¥infty}(||u_{X}(¥cdot, t)||_{L^{2}}^{2}+||v_{X}(¥cdot, t)||_{L^{2}}^{2})=0$
.
Again, using Gagliardo-Niremberg’s inequality we know that
$||u(¥cdot, t)||_{L^{¥infty}}¥leq C||u_{X}(¥cdot, t)||_{L^{2}}^{1/2}||u(¥cdot, t)||_{L^{2}}^{1/2}$
.
constant,
has bounded
Asymptotic Behavior in Time
of
the Solutions
of KdV
Equations
369
Using (4.9) and (4.13) this implies that
$¥lim_{t¥rightarrow+¥infty}||u(¥cdot, t)||_{L^{¥infty}}=0$
.
Similarly, we show that
$¥lim_{t¥rightarrow+¥infty}||v(¥cdot, t)||_{L^{¥infty}}=0$
,
which proves Theorem 4.3.
Acknowledgements. We would like to express our sincere thanks to the
anonymous referee of this Joumal for his (hers) comments on our original
manuscript which led to this final version.
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$¥square u$
nuna adreso:
E. Bisognin, V. Bisognin
Department of Mathematics
Federal University of Santa Maria
Santa Maria, 97119-900, RS
Brazil
G.P. Menzala
National Laboratory of Scientific Computation
Rua Lauro Muller 455
Botafogo, 22290-160, RJ
Brazil
(Ricevita la 10-an de majo, 1996)