Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 27, pp. 1–18.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.27

PERSISTENCE PROPERTIES OF SOLUTIONS FOR

MULTI-COMPONENT NOVIKOV EQUATIONS

XIN LIU, XINGLONG WU

Abstract. In this article, we investigate the asymptotic behavior of the so-

lution for a multi-component Novikov equation in weighted Sobolev spaces.
We introduce a set of weighted functions, and prove that the strong solution

will retain the corresponding decay properties when the initial data U0(x) and

its derivative U0,x(x) decay logarithmically, algebraically, and exponentially
at infinity.

1. Introduction

In this article, we consider the initial value problem (IVP) for a multi-component
Novikov equation

∂tmk =

N∑
i=1

(−2mkvi∂xui −mkui∂xvi − uivi∂xmk −mivi∂xuk + ukmi∂xvi),

∂tnk =

N∑
i=1

(−2nkui∂xvi − nkvi∂xui − viui∂xnk − niui∂xvk + vkni∂xui),

(uk(t, x), vk(t, x))|t=0 = (uk,0(x), vk,0(x)),

(1.1)

where mk = uk − ∂2xuk, nk = vk − ∂2xvk, t > 0, x ∈ R, k = 1, 2, . . . , N . Equation
(1.1) was proposed by Li et al. [9] to derive its bi-Hamiltonian structure. Mi and
Guo proved the local well-posedness of the system in a range of the Besov spaces
using the Littlewood-Paley theory and transport equations [15]. Li and Wu et al.
[11] deduced blow-up criteria for (1.1) and global existence of two-component case
in Hs(R), s > 1/2. Moreover, they verified that the system possesses peakons and
periodic peakons.

For N = 1, equation (1.1) turns into the Geng-Xue (GX) equation

∂tm+ 3vm∂xu+ uv∂xm = 0, t > 0, x ∈ R,
∂tn+ 3un∂xv + uv∂xn = 0, t > 0, x ∈ R,
m = u− ∂2xu, n = v − ∂2xv, t > 0, x ∈ R,
u(0, x) = u0(x), v(0, x) = v0(x), x ∈ R,

(1.2)

2020 Mathematics Subject Classification. 35G25, 35L05.
Key words and phrases. Multi-component Novikov equation; asymptotic properties;

logarithmic decay; algebraical decay; exponential decay.
©2025. This work is licensed under a CC BY 4.0 license.

Submitted November 21, 2024. Published March 13, 2025.

1



2 X. LIU, X. WU EJDE-2025/27

which is a two-component CH-type equation constructed by Geng and Xue. They
also showed that (1.2) admits multi-peakons and infinitely many conserved quan-
tities [5]. In 2013, Li and Liu obtained a bi-Hamiltonian structure of this equation
[10]. Luo and Yin [14] established local well-posedness of the system in Besov spaces
Bs−1

l,r ×Bs
l,r with l, r ∈ [1,∞], s > max{1+ 1

l ,
3
2} by the Littlewood-Paley decomposi-

tion. Moreover, they introduced blow-up criteria for the system based on conserva-
tion laws. In 2018, Zhou and Li [25] investigated the persistence properties of strong
solutions for two-component Novikov equation in weighted Lp

ϕ
.
= Lp(R, ϕp(x) dx)

spaces for a large class of moderate weights.
If we take u = v, then (1.2) becomes the Novikov equation

∂tm+ u2∂xm+ 3um∂xu = 0, t > 0, x ∈ R,
m = (1− ∂2x)u, t > 0, x ∈ R,

u(0, x) = u0(x), x ∈ R.
(1.3)

This equation was discovered by Novikov [17]. Hone and Wang [7] showed that (1.3)
has a bi-Hamiltonian structure and infinitely many conserved quantities. Also, it
admits peakon solutions and conserves the H1-norm as well as CH equation. The
Cauchy problem of (1.3) has attracted a great deal of attention in [6, 16, 20, 21, 22].
Specifically, Ni and Zhou [16] showed (1.3) is locally well-posed in the Besov spaces

B
3/2
2,1 (R). Furthermore, they verified two results on the persistence properties of the

strong solution. Wu and Yin [21] proved (1.3) possesses a global strong solution
on the initial value u0 ∈ Hs(R) for s > 3/2. They also proved the existence
and uniqueness of global weak solutions to (1.3) with the initial data satisfying
certain sign conditions [20]. Furthermore, they established the local well-posedness
of (1.3) in Besov space Bs

p,r(R), p, r ∈ [1,∞], s > max{ 3
2 , 1 + 1

p} and proved the

equation is ill-posed in B
3/2
2,∞(R) [22]. The global existence and blow-up for the

weakly dissipative Novikov equation were considered in [24].
When N = 1, v = 1, equation (1.1) transforms into the Degasperis-Procesi (DP)

equation
mt + umx + 3uxm = 0, t > 0, x ∈ R,

m = u− uxx, t > 0, x ∈ R,
u(0, x) = u0(x), x ∈ R,

(1.4)

where mt = ∂tm(t, x), mx = ∂xm(t, x), ux = ∂xu(t, x). Equation (1.4) was pre-
sented by Degasperis et al. [2] through the construction of a Lax pair and considered
as a model for nonlinear shallow water dynamics [3, 13]. Constantin and Ivanov
[1] proved that (1.4) has a bi-Hamiltonian structure and an infinite many of con-
servation laws. Moreover, the authors presented the traveling wave solutions and
classified all weak traveling wave solutions for the DP equation in [8, 18].

However, the persistence properties of solutions to (1.1) have not been studied
yet. Inspired by the recent works [4, 19, 23], we study some new decay properties
of solutions to (1.1) with a set of weighted functions, which comes form [23]. The
focus of Theorems 3.3–3.10 is to estimate a class of norms like ∥IF (ui, uix)(s)∥Lp

and ∥IFx(ui, uix)(s)∥Lp , essentially to investigate Lemma 3.1. The decay of the
solution in this article covers and extents the results in [25].

The rest of this article is structured as follows. In Section 2, we recall the
local well-posedness result and introduce several lemmas which are needed for later
proofs. In Section 3, we establish persistence properties of strong solutions to (1.1)



EJDE-2025/27 MULTI-COMPONENT NOVIKOV EQUATIONS 3

provided the initial data U0 and ∂xU0 decay logarithmically, algebraically, and
exponentially at infinity.

Notation. As all function spaces considered are over R, for convenience, we sim-
plify our notation by omitting R when there is no ambiguity. we denote by ∗ spatial
convolution on R and use A⊤ stands for transpose of vector A. The notation

.
=

stands for the definition of functions. For 1 ≤ p ≤ ∞, we denote the norm of the
Banach space Lp(R) by ∥·∥Lp and the norm in the classical Sobolev spaces Hs,p(R)
by ∥ · ∥Hs,p , s ∈ R. In addition, for constant K ≥ 0, we denote

f(x) ∼ O(g(x)) as|x| → ∞, if lim
x→∞

|f(x)
g(x)

| ≤ K.

2. Preliminaries

In this section, we first recall the following local well-posedness result from [12],
and present three key weighted functions, which will be used in Section 3.

Lemma 2.1 ([12]). Let p1 ∈ (1,∞) and s > max{ 5
2 , 2 +

1
p1
}. Assume that

U0 = (u1,0, . . . , uN,0, v1,0, . . . , vN,0)
⊤ ∈ (Hs,p1)2N ,

there exists a time T > 0 and a unique solution U of (1.1) such that

U = (u1, . . . , uN , v1, . . . , vN )⊤ ∈ (L∞([0, T ];Hs,p1) ∩ Lip([0, T ];Hs−1,p1))2N .

Then the data-to-solution map of the initial value problem (1.1),

Φ : U0 = (u1,0, . . . , uN,0, v1,0, . . . , vN,0)
⊤ 7→ U = (u1, . . . , uN , v1, . . . , vN )⊤,

is continuous from (Hs,p1)2N into (L∞([0, T ];Hs,p1)∩Lip([0, T ];Hs−1,p1))2N , where
c > 0 is a constant depending on s, p1.

Lemma 2.2 ([23]). We define the weighted function

I(x) =

{
(ln(e2 + |x|))α, |x| ∈ [0,K],

(ln(e2 +K))α, |x| ∈ (K,∞),

where α ∈ [0,∞) and K ∈ R+. Let J(x) = (ln(e2+ |x|))α, then we have J(x+y) ≤
J(x)J(y). Furthermore, if there exists C0 ≥ 0 such that I(x + y) ≤ C0J(x)I(y),
then we say that the function I(x) is J(x)-moderate.

Lemma 2.3 ([23]). For θ ∈ [0,∞) and K ∈ R+, we define the algebraic weighted
function

Q(x) =

{
(1 + |x|)θ, |x| ∈ [0,K],

(1 +K)θ, |x| ∈ (K,∞).

Let P (x) = (1+|x|)θ, then P (x+y) ≤ P (x)P (y). Additionally, if there exists C0 ≥ 0
such that Q(x+ y) ≤ C0P (x)Q(y), then the function Q(x) is P (x)-moderate.

Lemma 2.4 ([23]). Define the weighted function ψ(x) = min
{
e|x|, K

}
, where

x ∈ R and K ∈ R+. Let ϕ(x) = e|x|. Then we have ϕ(x + y) ≤ ϕ(x)ϕ(y). If
there exists C0 ≥ 0 such that ψ(x + y) ≤ C0ϕ(x)ψ(y), then the function ψ(x) is
ϕ(x)-moderate. Moreover, ψ1/2(x) is ϕ1/2(x)-moderate.



4 X. LIU, X. WU EJDE-2025/27

3. Logarithmic, algebraical, and exponential decay of solutions

In this section, using ideas from [23], we prove that the solution of (3.2) keep
corresponding decay properties, provided the initial data decays logarithmically,
algebraically, and exponentially at infinity.

For the sake of brevity, we let mi = ui − ∂2xui, ni = vi − ∂2xvi. Then (1.1) can
be rephrased as follows

∂tui +

N∑
j=1

uixujvj + F (ui, uix) = 0, t > 0, x ∈ R,

∂tvi +

N∑
j=1

vixujvj +H(vi, vix) = 0, t > 0, x ∈ R,

ui(0, x) = ui,0(x), vi(0, x) = vi,0(x), x ∈ R,

(3.1)

where the nonlocal terms are

F (ui, uix) = Λ−2∂x

N∑
j=1

(uiujxvjx + uixujvjx)

+ Λ−2
N∑
j=1

(uixujvj + 2uiujxvj − uiujxvj,xx),

H(vi, vix) = Λ−2∂x

N∑
j=1

(viujxvjx + vixujxvj)

+ Λ−2
N∑
j=1

(vixujvj + 2viujvjx − viuj,xxvjx).

Note that (1 − ∂2x)
−1f = G ∗ f for all f ∈ Lp, where G(x) = 1

2e
−|x|, x ∈ R. Then

we can rewrite (3.1) as

∂tui +

N∑
j=1

uixujvj +Gx ∗ (f1 + f2) +G ∗ (f3 + f4 + f5) = 0,

∂tvi +

N∑
j=1

vixujvj +Gx ∗ (h1 + h2) +G ∗ (h3 + h4 + h5) = 0,

ui(0, x) = ui,0(x), vi(0, x) = vi,0(x),

(3.2)

where

f1 =

N∑
j=1

uiujxvjx, f2 =

N∑
j=1

uixujvjx, f3 =

N∑
j=1

uixujvj ,

f4 = 2

N∑
j=1

uiujxvj , f5 = −
N∑
j=1

uiujxvj,xx,

(3.3)



EJDE-2025/27 MULTI-COMPONENT NOVIKOV EQUATIONS 5

h1 =

N∑
j=1

viujxvjx, h2 =

N∑
j=1

vixujxvj , h3 =

N∑
j=1

vixujvj ,

h4 = 2

N∑
j=1

viujvjx, h5 = −
N∑
j=1

viuj,xxvjx.

(3.4)

First, we establish the logarithmic decay of the strong solution to (3.2), but
before proving the result, we need to show a useful lemma.

Lemma 3.1. Let I(x) and J(x) be the weighted functions defined in Lemma 2.2,
and fk, gk be given by (3.3) and (3.4). If I(x) is J(x)-moderate, then for p ≥ 1,
f, g ∈ Lp, we have

∥I(f ∗ g)∥Lp ≤ ∥Jf∥L1∥Ig∥Lp ,

where f = G, Gx, G(x) =
1
2e

−|x|, and g = fk, hk, k = 1, 2, . . . , 5.

Proof. If f = G and g = f1 =
∑N

j=1 uiujxvjx, by I(x + y) ≤ J(x)I(y) and the
Young inequality to yield

∥I(G ∗ f1)∥Lp = ∥
∫
R
I(x)G(x− y)f1(y)dy∥Lp

≤ ∥
∫
R
|J(x− y)I(y)G(x− y)f1(y)|dy∥Lp

= ∥(JG) ∗ (If1)∥Lp

≤ ∥JG∥L1∥If1∥Lp ,

which leads to ∥I(G∗f1)∥Lp ≤ ∥JG∥L1∥If1∥Lp . Similarly, one can easily check the
following inequalities

∥I(G ∗ fk)∥Lp ≤ ∥JG∥L1∥Ifk∥Lp , k = 1, 2, . . . , 4,

∥I(Gx ∗ fk)∥Lp ≤ ∥JGx∥L1∥Ifk∥Lp , k = 1, 2, . . . , 5,

∥I(G ∗ hk)∥Lp ≤ ∥JG∥L1∥Ihk∥Lp , k = 1, 2, . . . , 5,

∥I(Gx ∗ hk)∥Lp ≤ ∥JGx∥L1∥Ihk∥Lp , k = 1, 2, . . . , 5.

This completes the proof. □

We shall estimate ∥IF (ui, uix)(s)∥Lp , ∥IFx(ui, uix)(s)∥Lp , ∥IH(ui, uix)(s)∥Lp ,
and ∥IHx(ui, uix)(s)∥Lp in the following lemma, and will be referenced multiple
times.

Lemma 3.2. For p > 1, the following estimates hold:

∥IF (ui, uix)(s)∥Lp + ∥IFx(ui, uix)(s)∥Lp ≤ c(∥uiI∥Lp + ∥uixI∥Lp), (3.5)

∥IH(vi, vix)(s)∥Lp + ∥IHx(vi, vix)(s)∥Lp ≤ c(∥viI∥Lp + ∥vixI∥Lp), (3.6)

where c > 0 is a constant depending on α,N,M , for i = 1, 2, . . . , N .

Proof. We will only give a proof of (3.5), the other can be proved similarly. For
convenience, let

M = sup
t∈[0,T ]

{∥U(t, ·)∥Hs,p1 }.

The Sobolev embedding theorem leads to

∥ui(t)∥L∞ , ∥ui,x(t)∥L∞ , ∥ui,xx(t)∥L∞ , ∥vi(t)∥L∞ , ∥vi,x(t)∥L∞ , ∥vi,xx(t)∥L∞ ≤M.



6 X. LIU, X. WU EJDE-2025/27

Lemma 3.1 yields that

∥IF (ui, uix)(s)∥Lp

≤ ∥I (Gx ∗ (f1 + f2)) ∥Lp + ∥I(G ∗ (f3 + f4 + f5))∥Lp

≤ ∥I(Gx ∗ f1)∥Lp + ∥I(Gx ∗ f2)∥Lp + ∥I(G ∗ (f3 + f4 + f5))∥Lp

≤ ∥JGx∥L1(∥If1∥Lp + ∥If2∥Lp) + ∥JG∥L1(∥If3∥Lp + ∥If4∥Lp + ∥If5∥Lp).

(3.7)

For the first term of (3.7), in view of the definition of f1 in (3.3) and Hölder’s
inequality to arrive at

∥JGx∥L1∥If1∥Lp =

N∑
j=1

∥JGx∥L1∥uiujxvjxI∥Lp

≤
N∑
j=1

∥JGx∥L1∥ujxvjx∥L∞∥uiI∥Lp

≤ 2α−1(2 + 1/e2)NM2∥uiI∥Lp .

Using the same method, it follows that

∥IF (ui, uix)(s)∥Lp ≤ 2α−1(2 + 1/e2)NM2(∥uiI∥Lp + ∥uixI∥Lp)

≤ c(∥uiI∥Lp + ∥uixI∥Lp).

Since Gxx ∗ f = G ∗ f − f , G(x) = 1
2e

−|x|, x ∈ R, one obtains

Fx(ui, uix) = Gxx ∗ (f1 + f2) +Gx ∗ (f3 + f4 + f5)

= G ∗ (f1 + f2)− (f1 + f2) +Gx ∗ (f3 + f4 + f5).

Therefore, one can easily check that

∥IFx(ui, uix)(s)∥Lp

≤ ∥I (G ∗ (f1 + f2)) ∥Lp + ∥I(f1 + f2)∥Lp + ∥I (Gx ∗ (f3 + f4 + f5)) ∥Lp

≤ c(∥uiI∥Lp + ∥uixI∥Lp) +

N∑
j=1

∥I(uiujxvjx + uixujvjx)∥Lp

≤ c(∥uiI∥Lp + ∥uixI∥Lp),

where the constant c depends on α, N , M , and the second inequality comes from

∥JG∥L1 , ∥JGx∥L1 ≤ 2α−1(2 + 1/e2).

The proof is complete. □

Theorem 3.3. Assume the initial data U0 = (u0, v0)
⊤ ∈ (Hs,p1)2N , s > 2 +

1
p1
, p1 ∈ (1,∞), and T > 0. Then there exists a unique solution U(t, x) ∈

[C([0, T ];Hs,p1(R))]2N to (3.2) with the initial data U0. For p > 1, if the initial
data satisfies for some C > 0,

∥U0(x)I(x)∥Lp + ∥U0,x(x)I(x)∥Lp ≤ C,

then the solution satisfies

∥U(t, ·)I∥Lp + ∥Ux(t, ·)I∥Lp ≤ C,



EJDE-2025/27 MULTI-COMPONENT NOVIKOV EQUATIONS 7

uniformly in the interval [0, T ], where for α ∈ [0,∞) and K ∈ R+, the weighted
function I(x) is given by

I(x) =

{
(ln(e2 + |x|))α, |x| ∈ [0,K],

(ln(e2 +K))α, |x| ∈ (K,∞).

In particular, if the initial data U0 and U0,x decay logarithmically as

|U0(x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞,

|U0,x(x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞,

then the solution of (3.2) decays logarithmically as

|U(t, x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞,

|Ux(t, x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞,

uniformly in the interval [0, T ].

Proof. Multiplying the first equation in (3.1) by I, we obtain

(uiI)t +

N∑
j=1

uixujvjI + F (ui, uix)I = 0. (3.8)

Then by this equality and |uiI|p−2(uiI) with p > 1, and integrating the result on
R with respect to x-variable, it follows that

1

p

d

dt

∫
R
|uiI|pdx = −

N∑
j=1

∫
R
uixujvjI|uiI|p−2(uiI)dx

−
∫
R
F (ui, uix)I|uiI|p−2(uiI)dx.

(3.9)

Note that

1

p

d

dt

∫
R
|uiI|pdx =

1

p

d

dt
∥uiI∥pLp = ∥uiI∥p−1

Lp

d

dt
∥uiI∥Lp .

In view of uixI = (uiI)x − uiIx and 0 ≤ Ix ≤ βI, where β = α
2e2 > 0, we have∣∣ ∫

R
uixujvjI|uiI|p−2(uiI)dx

∣∣
≤

∫
R
ujvj [(uiI)x − uiIx]|uiI|p−2(uiI)dx

≤
∫
R
ujvj(uiI)x|uiI|p−2(uiI)dx+

∫
R
ujvjuiIx|uiI|p−2(uiI)dx

≤ 1

p

∫
R
ujvj(|uiI|p)xdx+ β

∫
R
ujvjuiI|uiI|p−2(uiI)dx

≤ 1

p

∫
R
(ujvj)x|uiI|pdx+ β∥ujvj∥L∞∥uiI∥pLp

≤ (2 + β)M2∥uiI∥pLp ,

whereM is a constant defined as Lemma 3.2. Using Holder’s inequality, one obtains∫
R
IF (ui, uix)|uiI|p−2(uiI)dx ≤ ∥IF (ui, uix)∥Lp∥uiI∥p−1

Lp .



8 X. LIU, X. WU EJDE-2025/27

Combining (3.8) with the above relations to yield

d

dt
∥uiI∥Lp ≤ (2 + β)NM2∥uiI∥Lp + ∥IF (ui, uix)∥Lp . (3.10)

Differentiating the first equation in (3.1) with respect to x, and multiplying it by
I, one has that

(uixI)t +

N∑
j=1

(ujvjui,xx + uixujxvj + uixujvjx)I + IFx(ui, uix) = 0. (3.11)

Multiplying (3.11) by |uixI|p−2(uixI) and integrating the result on R with respect
to x, one obtains

∥uixI∥p−1
Lp

d

dt
∥uixI∥Lp

= −
N∑
j=1

∫
R
(uixujvjx + uixujxvj)I|uixI|p−2(uixI) dx

−
N∑
j=1

∫
R
ujvjui,xxI|uixI|p−2(uixI) dx−

∫
R
IFx(ui, uix)|uixI|p−2(uixI) dx

.
= E1 + E2 + E3.

(3.12)
Applying Hölder’s inequality, it follows that

E1 ≤
N∑
j=1

(∥ujvjx∥L∞ + ∥ujxvj∥L∞)∥uixI∥pLp ≤ NM2∥uixI∥pLp .

From ui,xxI = (uixI)x − uixIx and 0 ≤ Ix ≤ βI, where β = γ
2e2 > 0, we have

|E2| = | −
N∑
j=1

∫
R
ujvj [(uixI)x − uixIx]|uixI|p−2(uixI) dx|

≤ |1
p

N∑
j=1

∫
R
ujvj(|uixI|p)xdx+

N∑
j=1

∫
R
ujvjuixIx|uixI|p−2(uixI) dx|

≤ 1

p

N∑
j=1

∫
R
(ujvj)x|uixI|pdx+ β

N∑
j=1

∥ujvj∥L∞∥uixI∥pLp

≤ (2 + β)NM2∥uixI∥pLp .

E3 is treated similarly to obtain∫
R
IFx(ui, uix)|uixI|p−2(uixI) dx ≤ ∥IFx(ui, uix)∥Lp∥uixI∥p−1

Lp .

Consequently, plugging the above inequalities into (3.12), we obtain

d

dt
∥uixI∥Lp ≤ (3 + β)NM2∥uixI∥Lp + ∥IFx(ui, uix)∥Lp . (3.13)

Taking into account (3.10) and (3.13) we have

d

dt
(∥uiI∥Lp + ∥uixI∥Lp) ≤ (3 + β)NM2 (∥uiI∥Lp + ∥uixI∥Lp)

+ ∥IF (ui, uix)∥Lp + ∥IFx(ui, uix)∥Lp ,
(3.14)



EJDE-2025/27 MULTI-COMPONENT NOVIKOV EQUATIONS 9

which along with Gronwall’s inequality leads to

∥uiI∥Lp + ∥uixI∥Lp

≤ e(3+β)NM2t (∥ui0I∥Lp + ∥ui0,xI∥Lp)

+ e(3+β)NM2t

∫ t

0

(
∥IF (ui, uix)(s)∥Lp + ∥IFx(ui, uix)(s)∥Lp

)
ds.

(3.15)

Analogously, one can easily deduce that

∥viI∥Lp + ∥vixI∥Lp

≤ e(3+β)NM2t (∥vi0I∥Lp + ∥vi0,xI∥Lp)

+ e(3+β)NM2t

∫ t

0

(
∥IH(vi, vix)(s)∥Lp + ∥IHx(vi, vix)(s)∥Lp

)
ds.

(3.16)

Let U = (u, v)⊤, where u = (u1, u2, . . . , uN )⊤, v = (v1, v2, . . . , vN )⊤ and we define

∥U(t)∥Lp
.
= ∥u(t)∥Lp + ∥v(t)∥Lp .

Combining (3.15) with (3.16) and applying Lemma 3.2, one has

∥UI∥Lp + ∥UxI∥Lp ≤ e(3+β)NM2t(∥U0I∥Lp + ∥U0,xI∥Lp)

+ e(3+β)NM2t

∫ t

0

(∥UI∥Lp + ∥UxI∥Lp)ds.

Assuming Z(t) = ∥U(t, ·)I∥Lp + ∥Ux(t, ·)I∥Lp , we deduce that

Z(t) ≤ e(3+β)NM2t
(
Z(0) + c

∫ t

0

Z(s)ds
)
. (3.17)

Using Gronwall’s inequality, one gets

Z(t) ≤ CZ(0) ≤ C (∥U0(x)I(x)∥Lp + ∥U0,x(x)I(x)∥Lp) , (3.18)

where C = C(c, β,M,N, T ) is a positive constant.
If for some C > 0, the data U0 and U0,x satisfy

∥U0(x)I(x)∥Lp + ∥U0,x(x)I(x)∥Lp ≤ C,

then for all t ∈ [0, T ], we can show that the solution satisfies

∥U(t, ·)I∥Lp + ∥Ux(t, ·)I∥Lp ≤ C.

Particularly, if the initial data U0 and U0,x decay logarithmically as

|U0(x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞,

|U0,x(x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞.

Taking the limit as p→ ∞ and K → ∞ in the above inequality, we have

|U(t, x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞,

|Ux(t, x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞,

uniformly in the interval [0, T ]. This completes the proof. □

Corollary 3.4. In fact, under the assumption of Theorem 3.3, if U0, U0,x and
U0,xx satisfy

∥U0(x)I(x)∥Lp + ∥U0,x(x)I(x)∥Lp + ∥U0,xx(x)I(x)∥Lp ≤ C,



10 X. LIU, X. WU EJDE-2025/27

hat for some C > 0, then the solution satisfies

∥U(t, ·)I∥Lp + ∥Ux(t, ·)I∥Lp + ∥Uxx(t, ·)I∥Lp ≤ C,

uniformly in the interval [0, T ]. In particular, if the initial data U0 satisfies

|U0(x)|, |U0,x(x)|, |U0,xx(x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞,

then the solution U(t, x) decays logarithmically as

|U(t, x)|, |Ux(t, x)|, |Uxx(t, x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞,

uniformly in the interval [0, T ].

Proof. Differentiating the first equation in (3.1) twice with respect to x, and mul-
tiplying it by I, applying the obtained result by |ui,xxI|p−2(ui,xxI), integration by
parts, it yields that∫

R
(ui,xxI)t|ui,xxI|p−2(ui,xxI) dx

= −
∫
R
IFxx(ui, uix, ui,xx)|ui,xxI|p−2(ui,xxI) dx

− 2

N∑
j=1

∫
R
ui,xx(ujxvj + ujvjx)I|ui,xxI|p−2(ui,xxI) dx

− 2

N∑
j=1

∫
R
uixujxvjxI|ui,xxI|p−2(ui,xxI) dx

−
N∑
j=1

∫
R
uix(uj,xxvj + ujvj,xx)I|ui,xxI|p−2(ui,xxI) dx

−
N∑
j=1

∫
R
ui,xxxujvjI|ui,xxI|p−2(ui,xxI) dx.

(3.19)

By using Hölder’s inequality, one can easily check that∫
R
(ui,xxI)t|ui,xxI|p−2(ui,xxI) dx = ∥ui,xxI∥p−1

Lp

d

dt
∥ui,xxI∥Lp ,∫

R
IFxx(ui, uix, ui,xx)|ui,xxI|p−2(ui,xxI) dx ≤ ∥IFxx(ui, uix, ui,xx)∥Lp∥ui,xxI∥p−1

Lp ,

2

∫
R
ui,xx(ujxvj + ujvjx)I|ui,xxI|p−2(ui,xxI) dx

≤ 2(∥ujxvj∥L∞ + ∥ujvjx∥L∞)∥ui,xxI∥pLp

≤ 4M2∥ui,xxI∥pLp ,

2

∫
R
uixujxvjxI|ui,xxI|p−2(ui,xxI) dx ≤ 2∥ujxvjx∥L∞∥uixI∥Lp∥ui,xxI∥p−1

Lp

≤ 2M2∥uixI∥Lp∥ui,xxI∥p−1
Lp ,∫

R
uix(uj,xxvj + ujvj,xx)I|ui,xxI|p−2(ui,xxI) dx

≤ (∥uj,xxvj∥L∞ + ∥ujvj,xx∥L∞)∥uixI∥Lp∥ui,xxI∥p−1
Lp

≤ 2M2∥uixI∥Lp∥ui,xxI∥p−1
Lp .



EJDE-2025/27 MULTI-COMPONENT NOVIKOV EQUATIONS 11

In view of ui,xxxI = (ui,xxI)x − ui,xxIx and 0 ≤ Ix ≤ βI, where β = α
2e2 > 0, we

have ∫
R
ui,xxxujvjI|ui,xxI|p−2(ui,xxI) dx

=

∫
R
ujvj [(ui,xxI)x − ui,xxIx]|ui,xxI|p−2(ui,xxI) dx

=
1

p

∫
R
ujvj(|ui,xxI|p)xdx−

∫
R
ujvjui,xxIx|ui,xxI|p−2(ui,xxI) dx

≤ 1

p

∫
R
(ujxvj + ujvjx)|ui,xxI|pdx+ β

∫
R
|ujvj ||ui,xxI|pdx

≤
(2
p
+ β

)
M2∥ui,xxI∥pLp

≤ (2 + β)M2∥ui,xxI∥pLp .

Inserting the above relations into (3.19) yields

d

dt
∥ui,xxI∥Lp ≤ 4NM2∥uixI∥Lp + (6 + β)NM2∥ui,xxI∥Lp

+ ∥IFxx(ui, uix, ui,xx)∥Lp .
(3.20)

Combining (3.14) with (3.20), it follows that

d

dt
(∥uiI∥Lp + ∥uixI∥Lp + ∥ui,xxI∥Lp)

≤ (6 + β)NM2(∥uiI∥Lp + ∥uixI∥Lp + ∥ui,xxI∥Lp)

+ ∥IF (ui, uix)∥Lp + ∥IFx(ui, uix)∥Lp + ∥IFxx(ui, uix, ui,xx)∥Lp .

(3.21)

Using Gronwall’s inequality, we obtain

∥uiI∥Lp + ∥uixI∥Lp + ∥ui,xxI∥Lp

≤ e(6+β)NM2t (∥ui0I∥Lp + ∥ui0,xI∥Lp + ∥ui0,xxI∥Lp)

+ e(6+β)NM2t

∫ t

0

(
∥IF (ui, uix)(s)∥Lp + ∥IFx(ui, uix)(s)∥Lp

+ ∥IFxx(ui, uix, ui,xx)(s)∥Lp

)
ds.

Similarly, one can easily check that

∥viI∥Lp + ∥vixI∥Lp + ∥vi,xxI∥Lp

≤ e(6+β)NM2t (∥vi0I∥Lp + ∥vi0,xI∥Lp + ∥vi0,xxI∥Lp)

+ e(6+β)NM2t

∫ t

0

(
∥IH(vi, vix)(s)∥Lp

+ ∥IHx(vi, vix)(s)∥Lp + ∥IHxx(vi, vix, vi,xx)(s)∥Lp

)
ds.

From Gxx ∗ f = G ∗ f − f and G(x) = 1
2e

−|x| for x ∈ R, we have

Fxx(ui, uix, ui,xx) = Gx ∗ (f1 + f2)− (f1x + f2x) +Gxx ∗ (f3 + f4 + f5)

= Gx ∗ (f1 + f2) +G ∗ (f3 + f4 + f5)

− (f1x + f2x + f3 + f4 + f5) .



12 X. LIU, X. WU EJDE-2025/27

Now, we just need to estimate ∥IFxx(ui, uix, ui,xx)(s)∥Lp , by Lemma 3.1 to obtain

∥IFxx(ui, uix, ui,xx)(s)∥Lp

≤ ∥I (Gx ∗ (f1 + f2)) ∥Lp + ∥I (G ∗ (f3 + f4 + f5)) ∥Lp

+ ∥I(f1x + f2x + f3 + f4 + f5)∥Lp

≤ ∥JGx∥L1(∥If1∥Lp + ∥If2∥Lp) + ∥JG∥L1(∥If3∥Lp + ∥If4∥Lp + ∥If5∥Lp)

+ ∥I(f1x + f2x + f3 + f4 + f5)∥Lp

≤ c(∥uiI∥Lp + ∥uixI∥Lp + ∥ui,xxI∥Lp),

where the constant c depends on α, N, M , the last inequality comes from

∥JG∥L1 , ∥JGx∥L1 ≤ 2α−1(2 + 1/e2).

Combining the above estimates, we obtain

∥UI∥Lp + ∥UxI∥Lp + ∥UxxI∥Lp

≤ e(6+β)NM2t(∥U0I∥Lp + ∥U0,xI∥Lp + ∥U0,xxI∥Lp)

+ e(6+β)NM2t

∫ t

0

(∥UI∥Lp + ∥UxI∥Lp + ∥UxxI∥Lp)ds.

Setting Y (t) = ∥U(t, ·)I∥Lp + ∥Ux(t, ·)I∥Lp + ∥Uxx(t, ·)I∥Lp , we obtain

Y (t) ≤ e(6+β)NM2t
(
Y (0) + c

∫ t

0

Y (s)ds
)
. (3.22)

Applying Gronwall’s inequality, there exists a constant C(c, α,M,N, T ) such that
for all t ∈ [0, T ],

Y (t) ≤ CY (0)

≤ C (∥U0(x)I(x)∥Lp + ∥U0,x(x)I(x)∥Lp + ∥U0,xxI∥Lp) .
(3.23)

If U0, U0,x and U0,xx satisfy

∥U0(x)I(x)∥Lp + ∥U0,x(x)I(x)∥Lp + ∥U0,xx(x)I(x)∥Lp ≤ C

for some C > 0, then for all t ∈ [0, T ], the solution satisfies

∥U(t, ·)I∥Lp + ∥Ux(t, ·)I∥Lp + ∥Uxx(t, ·)I∥Lp ≤ C.

Particularly, if the initial data U0 decays logarithmically as

|U0(x)|, |U0,x(x)|, |U0,xx(x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞.

Taking K → ∞ and p → ∞ in (3.23), the solution U(t, x) decays logarithmically
as

|U(t, x)|, |Ux(t, x)|, |Uxx(t, x)| ∼ O
(
(ln(e2 + |x|))−α

)
, as |x| → ∞,

uniformly in the interval [0, T ]. So, the proof is complete. □

Secondly, we will show the algebraic decay of the solution to (3.2). As before,
we first state the following lemma.

Lemma 3.5. Let Q(x) and P (x) be the weighted functions defined in Lemma 2.3;
and let fk, gk be given by (3.3) and (3.4). If Q(x) is P (x)-moderate, then for p ≥ 1,
f, g ∈ Lp, one has

∥Q(f ∗ g)∥Lp ≤ ∥Pf∥L1∥Qg∥Lp ,

where f = G,Gx, G(x) =
1
2e

−|x|, and g = fk, hk, k = 1, 2, . . . , 5.



EJDE-2025/27 MULTI-COMPONENT NOVIKOV EQUATIONS 13

The proof of the above lemma is similar to the proof of Lemma 3.1 we omit its
proof.

Theorem 3.6. Suppose U0 = (u0, v0)
⊤ ∈ (Hs,p1)2N , s > 2 + 1

p1
, p1 ∈ (1,∞).

Then there exist T > 0 and a unique solution U(t, x) ∈ [C([0, T ];Hs,p1(R))]2N to
(3.2). Furthermore, if U0 and U0,x satisfy

∥U0(x)Q(x)∥Lp + ∥U0,x(x)Q(x)∥Lp ≤ C,

for p > 1 and some C > 0, then the solution satisfies

∥U(t, ·)Q∥Lp + ∥Ux(t, ·)Q∥Lp ≤ C,

uniformly in the interval [0, T ], where for θ ∈ [0,∞) and K ∈ R+, the weighted
function Q(x) is defined by

Q(x) =

{
(1 + |x|)θ, |x| ∈ [0,K],

(1 +K)θ, |x| ∈ (K,∞).

In particular, if the initial data U0 and U0,x decay algebraically as

|U0(x)| ∼ O
(
(1 + |x|)−θ

)
, as |x| → ∞,

|U0,x(x)| ∼ O
(
(1 + |x|)−θ

)
, as |x| → ∞,

then the solution U(t, x) decays algebraically as

|U(t, x)| ∼ O
(
(1 + |x|)−θ

)
, as |x| → ∞,

|Ux(t, x)| ∼ O
(
(1 + |x|)−θ

)
, as |x| → ∞,

uniformly in the interval [0, T ].

The proof of the above theorem is similar to that of Theorem 3.3. In view of
Lemma 2.3 and Lemmas 3.2 and 3.5, we obtain the required statement.

As the proof of the following corollary is very similar to Corollary 3.4 just with
a slight modification, thus we only show the result here.

Corollary 3.7. Under the conditions of Theorem 3.6, if U0, U0,x, and U0,xx satisfy,
for some C > 0,

∥U0(x)Q(x)∥Lp + ∥U0,x(x)Q(x)∥Lp + ∥U0,xx(x)Q(x)∥Lp ≤ C,

then the solution satisfies

∥U(t, ·)Q∥Lp + ∥Ux(t, ·)Q∥Lp + ∥Uxx(t, ·)Q∥Lp ≤ C,

uniformly in the interval [0, T ]. In particular, if the initial data U0 satisfy

|U0(x)|, |U0,x(x)|, |U0,xx(x)| ∼ O
(
(1 + |x|)−θ

)
, as |x| → ∞,

then the solution U(t, x) decays algebraically as

|U(t, x)|, |Ux(t, x)|, |Uxx(t, x)| ∼ O
(
(1 + |x|)−θ

)
, as |x| → ∞,

uniformly in the interval [0, T ].

To derive the exponential decay of the strong solution to (3.2), we first give the
following lemma.



14 X. LIU, X. WU EJDE-2025/27

Lemma 3.8. Let ψ(x) and ϕ(x) be the weighted functions defined in Lemma 2.4.
If the function ψ1/2(x) is ϕ1/2(x)-moderate, then for p ≥ 1, f, g ∈ Lp, one has

∥ψ1/2(f ∗ g)∥Lp ≤ C0∥ϕ1/2f∥L1∥ψ1/2g∥Lp ,

where C0 ≥ 0, f = G,Gx and g = fk, hk, k = 1, 2, . . . , 5 are given by (3.3) and
(3.4).

The proof of the above lemma is analogous to the proof of Lemma 3.1, we omit
it. We now shall prove the exponential decay of the solution to (3.2).

Theorem 3.9. Suppose that U0 = (u0, v0)
⊤ ∈ (Hs,p1)2N , s > 2 + 1

p1
, p1 ∈ (1,∞).

Then there exist T > 0 and a unique solution U(t, x) ∈ [C([0, T ];Hs,p1(R))]2N to
(3.2). Additionally, if the initial data U0 and U0,x admit p > 1 and some C > 0
such that

∥U0(x)ψ(x)∥Lp + ∥U0,x(x)ψ(x)∥Lp ≤ C,

then the solution satisfies

∥U(t, ·)ψ∥Lp + ∥Ux(t, ·)ψ∥Lp ≤ C,

uniformly in the interval [0, T ], where for K > 0, the weighted function ψ(x) =
min{e|x|, K}.

The proof of the above theorem is similar to the proof of Theorem 3.3; in view
of Lemmas 2.4, 3.2, and Lemma 3.8, the desired result follows.

Theorem 3.10. Suppose U0 = (u0, v0)
⊤ ∈ (Hs,p1)2N , s > 2 + 1

p1
, p1 ∈ (1,∞).

There exist T > 0 and a unique solution U(t, x) ∈ [C([0, T ];Hs,p1(R))]2N to (3.2).
Additionally, for p > 1 and some C > 0, if the initial data U0 and U0,x satisfy

∥U0(x)ψ(x)∥L∞ + ∥U0,x(x)ψ(x)∥L∞ ≤ C,

then the solution satisfies

∥U(t, ·)ψ∥L∞ + ∥Ux(t, ·)ψ∥L∞ ≤ C,

uniformly in the interval [0, T ]. In particular, if the initial data U0 and U0,x satisfy

|U0(x)| ∼ O
(
e−|x|), as |x| → ∞,

|U0,x(x)| ∼ O
(
e−|x|), as |x| → ∞,

then we the solution U(t, x) decays exponentially as

|U(t, x)| ∼ O
(
e−|x|), as |x| → ∞,

|Ux(t, x)| ∼ O
(
e−|x|), as |x| → ∞,

uniformly in the interval [0, T ].

Proof. By Lemma 2.4, the function ψ is ϕ-moderate, and ψ1/2 is ϕ1/2-moderate.
Multiplying the first equation in (3.1) by ψ1/2, by the method of estimate (3.14),
one gets

d

dt

(
∥uiψ1/2∥Lp + ∥uixψ1/2∥Lp

)
≤ (3 + β)NM2

(
∥uiψ1/2∥Lp + ∥uixψ1/2∥Lp

)
+ ∥ψ1/2F (ui, uix)∥Lp + ∥ψ1/2Fx(ui, uix)∥Lp .

(3.24)



EJDE-2025/27 MULTI-COMPONENT NOVIKOV EQUATIONS 15

Next, we need to consider ∥ψ1/2F (ui, uix)(τ)∥Lp and ∥ψ1/2Fx(ui, uix)(τ)∥Lp . Note
that

F (ui, uix) = Gx ∗ (f1 + f2) +G ∗ (f3 + f4 + f5),

Fx(ui, uix) = G ∗ (f1 + f2)− (f1 + f2) +Gx ∗ (f3 + f4 + f5).

In view of |G|, |Gx|, |Gxx| ≤ 1
2e

−|x|, and ψ1/2 is ϕ1/2-moderate, one can easily check
that

∥ψ1/2F (ui, uix)(s)∥Lp

≤
(
∥ψ1/2 (Gx ∗ (f1 + f2)) ∥Lp + ∥ψ1/2(G ∗ (f3 + f4 + f5))∥Lp

)
≤ ∥ψ1/2(Gx ∗ f1)∥Lp + ∥ψ1/2(Gx ∗ f2)∥Lp + ∥ψ1/2(G ∗ (f3 + f4 + f5))∥Lp

≤ ∥ϕ1/2Gx∥L1

(
∥ψ1/2f1∥Lp + ∥ψ1/2f2∥Lp

)
+ ∥ϕ1/2G∥L1

(
∥ψ1/2f3∥Lp + ∥ψ1/2f4∥Lp + ∥ψ1/2f5∥Lp

)
≤ c

(
∥uiψ1/2∥Lp + ∥uixψ1/2∥Lp

)
,

∥ψ1/2Fx(ui, uix)(s)∥Lp

≤ ∥ψ1/2 (G ∗ (f1 + f2)) ∥Lp + ∥ψ1/2(f1 + f2)∥Lp + ∥ψ1/2 (Gx ∗ (f3 + f4 + f5)) ∥Lp

≤ ∥ϕ1/2G∥L1

(
∥uiψ1/2∥Lp + ∥uixψ1/2∥Lp

)
+

N∑
j=1

∥ψ1/2(uiujxvjx + uixujvjx)∥Lp

+ ∥ϕ1/2Gx∥L1

(
∥uiψ1/2∥Lp + ∥uixψ1/2∥Lp

)
≤ c

(
∥uiψ1/2∥Lp + ∥uixψ1/2∥Lp

)
,

where in the last inequality we have used the estimates ∥ϕ1/2G∥L1 , ∥ϕ1/2Gx∥L1 ≤ 2,
and the constant c depends on M,N .

Inserting the above estimates into (3.24), it follows that

d

dt

(
∥uiψ1/2∥Lp + ∥uixψ1/2∥Lp

)
≤ c

(
∥uiψ1/2∥Lp + ∥uixψ1/2∥Lp

)
. (3.25)

By Gronwall’s inequality, we derive that

∥uiψ1/2∥Lp + ∥uixψ1/2∥Lp ≤ cect
(
∥ui0ψ1/2∥Lp + ∥ui0xψ1/2∥Lp

)
. (3.26)

Note that ψ is ϕ-moderate, it yields that

∥ψF (ui, uix)(s)∥L∞

≤ ∥ψ (Gx ∗ (f1 + f2)) ∥L∞ + ∥ψ(G ∗ (f3 + f4 + f5))∥L∞

≤ ∥ψ(Gx ∗ f1)∥L∞ + ∥ψ(Gx ∗ f2)∥L∞ + ∥ψ(G ∗ (f3 + f4 + f5))∥L∞

≤ ∥ϕGx∥L∞(∥ψf1∥L1 + ∥ψf2∥L1) + ∥ϕG∥L∞(∥ψf3∥L1 + ∥ψf4∥L1 + ∥ψf5∥L1)

≤ c

N∑
j=1

(
∥uiψ1/2∥L2∥ujvjψ1/2∥L2 + ∥uixψ1/2∥L2∥ujvjxψ1/2∥L2

+ ∥uixψ1/2∥L2∥ujvjxψ1/2∥L2

)



16 X. LIU, X. WU EJDE-2025/27

+ c

N∑
j=1

(
∥uiψ1/2∥L2∥ujxvjψ1/2∥L2 + ∥uiψ1/2∥L2∥ujxvj,xxψ1/2∥L2

)
≤ cect, (3.27)

∥ψFx(ui, uix)(s)∥L∞

≤ ∥ψ (G ∗ (f1 + f2)) ∥L∞ + ∥ψ(f1 + f2)∥L∞

+ ∥ψ (Gx ∗ (f3 + f4 + f5)) ∥L∞

≤ ∥ψ(G ∗ f1)∥L∞ + ∥ψ(G ∗ f2)∥L∞ + ∥ψ(f1 + f2)∥Lp

+ ∥ψ(Gx ∗ (f3 + f4 + f5))∥L∞

≤ ∥ϕG∥L∞(∥ψf1∥L1 + ∥ψf2∥L1) + ∥ϕGx∥L∞(∥ψf3∥L1

+ ∥ψf4∥L1 + ∥ψf5∥L1) + c(∥uiψ∥Lp + ∥uixψ∥Lp)

≤ c(∥uiψ∥Lp + ∥uixψ∥Lp) + cect,

(3.28)

where we have applied (3.26) with p = 2 in the fourth inequality. By replacing ψ1/2

with ψ in (3.24), it follows that

d

dt
(∥uiψ∥Lp + ∥uixψ∥Lp) ≤ (3 + β)NM2 (∥uiψ∥Lp + ∥uixψ∥Lp)

+ ∥ψF (ui, uix)∥Lp + ∥ψFx(ui, uix)∥Lp .
(3.29)

Taking (3.27), (3.28) into (3.29), and letting p→ ∞, we obtain

d

dt
(∥uiψ∥L∞ + ∥uixψ∥L∞) ≤ c (∥uiψ∥L∞ + ∥uixψ∥L∞) + cect. (3.30)

Similarly, it implies that

d

dt
(∥viψ∥L∞ + ∥vixψ∥L∞) ≤ c (∥viψ∥L∞ + ∥vixψ∥L∞) + cect. (3.31)

By combining (3.30) with (3.31), it follows that

d

dt
(∥Uψ∥L∞ + ∥Uxψ∥L∞) ≤ c(∥Uψ∥L∞ + ∥Uxψ∥L∞) + cect.

By Gronwall’s inequality, one obtains

∥U(x)ψ∥L∞ + ∥Ux(x)ψ∥L∞ ≤ C. (3.32)

Particularly, if the initial data U0 and U0,x decay exponentially as

|U0(x)| ∼ O
(
e−|x|), as |x| → ∞,

|U0,x(x)| ∼ O
(
e−|x|), as |x| → ∞.

Taking the limit as K → ∞ in inequality (3.32) to derive the result of theorem. □

Corollary 3.11. Suppose U0 = (u0, v0)
⊤ ∈ (Hs,p1)2N , s > 2 + 1

p1
, p1 ∈ (1,∞).

Then there exist T > 0 and a unique solution U(t, x) ∈ [C([0, T ];Hs,p1(R))]2N to
(3.2). For p > 1, if U0, U0,x and U0,xx satisfying

∥U0(x)ψ(x)∥Lp + ∥U0,x(x)ψ(x)∥Lp + ∥U0,xx(x)ψ(x)∥Lp ≤ C,

for some C > 0, then the solution satisfies

∥U(t, ·)ψ∥Lp + ∥Ux(t, ·)ψ∥Lp + ∥Uxx(t, ·)ψ∥Lp ≤ C,

uniformly in the interval [0, T ], where the weighted function ψ(x) = min
{
e|x|, K

}
.



EJDE-2025/27 MULTI-COMPONENT NOVIKOV EQUATIONS 17

In addition, for p ∈ (1,∞), if the initial data U0 satisfy

|U0(x)|, |U0,x(x)|, |U0,xx(x)| ∼ O
(
e−|x|), as |x| → ∞,

then the solution U(t, x) decays exponentially as

|U(t, x)|, |Ux(t, x)|, |Uxx(t, x)| ∼ O
(
e−|x|), as |x| → ∞,

uniformly in the interval [0, T ].

Acknowledgments. This work was partially supported by the NSFC (Grant No.
11771442) and by the Fundamental Research Funds for the Central University
(WUT:2021III056JC). The authors want to thank Professor Boling Guo and Doctor
Lijun Du, Zhengyan Liu for their helpful discussions and constructive suggestions.

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Xin Liu
School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070,

China

Email address: liuxin316587@whut.edu.cn

Xinglong Wu (corresponding author)

School of Mathematics and Statistics, Guangdong University of Foreign Studies,

Guangzhou 510006, China
Email address: wxl8758669@aliyun.com


	1. Introduction
	Notation

	2. Preliminaries
	3. Logarithmic, algebraical, and exponential decay of solutions
	Acknowledgments

	References