Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 84, pp. 1–25.

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

DOI: 10.58997/ejde.2025.84

ASYMPTOTIC BEHAVIOR OF KIRCHHOFF TYPE PLATE EQUATIONS

WITH NONLOCAL WEAK DAMPING, ANTI-DAMPING AND

SUBCRITICAL NONLINEARITY

LING XU, YANNI WANG, BIANXIA YANG

Abstract. In this work we study the global well-posedness, dissipativity and existence of global
attractors for Kirchhoff type plate equations with nonlocal weak damping and anti-damping,

when the nonlinear term g(u) satisfies a subcritical growth condition. Firstly, we show the

global well-posedness of this system by the monotone operator theory with locally Lipschitz
perturbation. Secondly, we construct a refined Gronwall’s inequality and then apply the barrier

method to prove the dissipativity for this system. Lastly, the asymptotic smoothness by taking

advantage of the energy reconstruction method, we deduce the existence of a global attractor
for this system.

1. Introduction

This paper discusses the existence of global attractors for nonlinear Kirchhoff type plate equa-
tion with nonlocal weak damping and anti-damping,

utt + k∥ut∥put +∆2u−m(∥∇u∥2)∆u+ g(u) = h(x) +

∫
Ω

K(x, y)ut(y)dy, x ∈ Ω, t ≥ 0,

u(x, t) = ∆u(x, t) = 0, x ∈ ∂Ω, t ≥ 0,

u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω,

(1.1)

where Ω ⊂ Rn is an open bounded domain with the smooth boundary ∂Ω, k∥ut∥put is a nonlo-
cal weak damping term, k, p are positive constants, h(x) ∈ L2(Ω) is the external forcing term,∫
Ω
K(x, y)ut(y)dy is the anti-damping term and K ∈ L2(Ω × Ω), and the assumptions on m(·)

and g(·) will be given in Section 2.
In 1950, Woinowsky-Krieger [20] firstly constructed the mathematical model of a class of ex-

tensible beams with transverse deflection of u(x, t) in the one-dimensional case

utt +
EI

ρ
uxxxx −

(H
ρ

+
EA

2ρl

∫ l

0

|ux|2dx
)
uxx = 0,

where H = EA∆/l is the axial force of the beam, l is the length, A is the cross-sectional area
of the beam, and ρ is the density of the beam, E is the Young’s modulus, I is the second-order
moment on the cross section of the beam. If H > 0, it represents the tension of the beam at rest.
Especially, he also proposed a class of scalable beam models

utt +∆2u− (α+ β∥∇u∥2)∆u = f.

The Berger equation with Kirchhoff type term was studied in [1]:

utt +∆2u− (Q+

∫
Ω

|∇u|2dx)∆u = p(u, ut, x),

2020 Mathematics Subject Classification. 35B40, 35B41, 35Q35.

Key words and phrases. Plate equation; nonlocal weak damping; anti-damping.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted February 7, 2025. Published August 11, 2025.

1



2 L. XU, Y. WANG, B. YANG EJDE-2025/84

where Q denotes the plane force acting on the plate, p is the transverse load, and the degree of the
load depends on the velocity ut and the displacement u. In 2012, Ma [13] studied the long-term
behavior of an extensible beam model with nonlinear boundary dissipation

utt + uxxxx −M(

∫ L

0

|ux|2dx)uxx = h,

where 0 < x < L, t > 0, M ≥ 0 is a non-decreasing function of C1. Subsequently, the asymptotic
behavior of the plate equations with weak damping δut, the damping term (−∆)θut(0 < θ ≤ 1),
and nonlinear damping g(ut) have been extensively studied, one can refer to [2, 8, 4, 11, 12, 16,
18, 21, 24, 25] and references therein. There exists a wealth of papers that focus on the long-time
dynamical behavior of hyperbolic equations with non-local damping and a nonlinear source term.

Ma and Narciso [14] discussed the existence of bounded absorbing sets and global attractors
for nonlinear beam equations with nonlinear damping

utt +∆2u−M(

∫
Ω

|∇u|2dx)∆u+ f(u) + g(ut) = h,

where Ω is a bounded domain of RN , M is a nonnegative real function, and h ∈ L2(Ω). Recently,
Zhao et al[26] studied the existence of global attractors for wave equations with nonlocal weak
damping and anti-damping

utt −∆u+ k∥ut∥put + f(u) =

∫
Ω

K(x, y)ut(y)dy + h(x),

where k and p are positive numbers, K ∈ L2(Ω× Ω), h ∈ L2(Ω). f ∈ C1(R) and the polynomial
growth index q satisfies the growth condition of the subcritical index: 0 < q < 2

n−2 if n ≥ 3 and

0 < q < ∞ if n ≤ 2. In [27], the author studied the existence of global attractors for the beam
equation with nonlocal weak damping in a bounded smooth domain

utt +∆2u−m(∥∇u∥2)∆u+ ∥ut∥put + f(u) = h, (1.2)

when f satisfies subcritical growth, where p > 0, m(·) are nonlocal coefficients, h ∈ L2(Ω) is the ex-
ternal forcing term. In [23], the existence of a compact minimal forward attractor is demonstrated
for non-autonomous strongly damped wave equations with asymptotically vanishing damping

utt −∆ut −∆u+ ϕ(x, t)ut + f(u) = g(x, t).

This work introduces innovative integral conditions for external forces, offering fresh insights into
degenerate damping problems.

Based on the above series of works, our main goal in this paper is to investigate the well-
posedness and the long-time dynamics for Kirchhoff type plate equation (1.1) with nonlocal weak
damping and anti-damping when the nonlinear term g(u) satisfies the subcritical growth condition.
In our opinion, the main difficulties and innovations are presented in the following:

(i) The nonlocal coefficient k∥ut∥p reflects the effect of kinetic energy on damping in physics.
Different from many other works in the literature, we cannot use the standard Fatou-Galerkin
method to prove the well-posedness. The reason lies in that when estimating the energy bounded-
ness, we can only obtain the boundedness of ut in the L2(Ω)-norm, then we can only get the weak
convergence of ut in the L2(Ω)-norm, this is insufficient to ensure that the nonlocal coefficients
∥ut∥p converge to the same limit. Besides, when the velocity ut is very small, the nonlocal damping
is weaker than the linear damping, and it is more difficult to obtain the asymptotic smoothness by
utilizing the decomposition of semigroup or contractive functions method than in the case of linear
damping ut. In particular, we don’t impose any restriction on the growth index p of the coefficient
in the nonlocal coefficient k∥ut∥p, which creates special obstacles to prove the dissipation of the
system and the existence of the global attractor.

(ii) The term
∫
Ω
K(x, y)ut(y)dy is an anti-damping because it may provide energy. The presence

of anti-damping term leads to energy along the orbit is not gradually weakened, and the effect of
energy supplement brought by the anti-damping term needs to be overcome by the damping, which
makes it invalid to prove the dissipation by constructing the commonly used Gronwall inequality.



EJDE-2025/84 KIRCHHOFF TYPE PLATE EQUATIONS 3

(iii) when k = 1, K(x, y) ≡ 0, the equation (1.1) degenerates into the equation (1.2), so we
proceed to further extend the results associated with it.

To overcome these problems, we first prove the global well-posedness of the solution is es-
tablished by the monotone operator theory with locally Lipschitz perturbation. Secondly, by
constructing a refined Gronwall’s inequality and then using the barrier method to prove the dissi-
pation of the system. Afterwards, the asymptotic compactness of Kirchhoff type plate equations
is obtained by taking advantage of the energy reconstruction method given by Chueshov and
Lasiecka [6]. Finally, the existence of a global attractor is obtained when f is of subcritical growth
condition.

The layout of this paper is as follows. In Section 2, we provide the concepts and hypothesis used
in this paper. The global well-posedness result of problem (1.1) is established in Section 3. The
existence of bounded absorbing sets of problem (1.1) is discussed in Section 4. In Section 5, we
prove the asymptotic smoothness of the dynamical system. In Section 6, we obtain the existence
of the global attractor for this system in the natural energy space H2(Ω) ∩H1

0 (Ω)× L2(Ω).
Throughout this paper, we use the symbol C to represent a normal number, and the symbol C

in the same line may also represent different normal numbers. Simultaneously, C(·) still represents
a normal number, and its value depends on the amount in parentheses.

2. Preliminaries

Let H = L2(Ω), D(A1/2) = H2(Ω) ∩H1
0 (Ω), and denote the corresponding inner products and

norms by

(u, v) =

∫
Ω

u(x)v(x)dx, ∥u∥ =
(∫

Ω

|u(x)|2dx
)1/2

, ∀u, v ∈ H,

((u, v)) =

∫
Ω

∆u(x)∆v(x)dx, ∥∆u∥ =
(∫

Ω

|∆u|2dx
)1/2

, ∀u, v ∈ D(A1/2).

In general, for for s ∈ R, Hs = D(A s
2 ) is a Hilbert space with the inner product and the norm

(u, v)Hs = (As/4u,As/4v) =

∫
Ω

As/4u · As/4vdx,

∥u∥2Hs = (u, u)Hs = ∥As/4u∥2.

Unusually, D(A) = {u ∈ H : Au ∈ H} = {u ∈ H4 : u,∆u ∈ H1
0}, D(A0) = H, D(A1/4) = H1

0 (Ω),
where A = ∆2,A1/2 = −∆. Then, the norm of the space W = D(A1/2)×H is defined as

∥(u, v)∥2W = ∥∆u∥2 + ∥v∥2.

Finally, by the Poincaré inequality, we obtain

∥∆u∥2 ≥ λ1∥u∥2, ∥∆u∥2 ≥ λ
1/2
1 ∥∇u∥2, ∀u ∈ D(A1/2), (2.1)

where λ1 > 0 is the first eigenvalue of A.
Now, we introduce assumptions on the functions m(·) and f(·) as follows:
(A1) The Kirchhoff coefficient m ∈ C1(R+) and satisfies

m(s) ≥ 0, m(s)s ≥ 1

2
M(s)− θs, (2.2)

where 0 ≤ θ ≤ 1
2λ

1/2
1 , M(s) =

∫ s

0
m(τ)dτ .

(A2) The nonlinear function g ∈ C1(R), without loss of generality, g(0) = 0 and satisfies

|g′(s)| ≤ C(1 + |s|q), (2.3)

where 1 ≤ q <∞ if n ≤ 4 and 1 ≤ q < 4
n−4 if n > 4.

lim inf
|s|→∞

g′(s) > −λ1, (2.4)



4 L. XU, Y. WANG, B. YANG EJDE-2025/84

where λ1 > 0 is the first eigenvalue of the bi-harmonic operator ∆2 with boundary condi-
tion (1.1)2. Note G(s) =

∫ s

0
g(τ)dτ , there is∫

Ω

G(s)dx ≥ −λ+ λ1
2

∥u∥2 − C, (2.5)

for some λ > λ1.
(A3) θ and λ satisfy

1− λ

λ1
− 2θ√

λ1
> 0. (2.6)

We assume that (X, ∥ · ∥) is a real Banach space, and X∗ is its dual space. The following gives
some relevant conclusions used in proving well-posedness (see [26, 27, 17, 9, 5, 22, 6, 15, 3, 10]).

Definition 2.1. A mapping A : X → X∗, it is said to be

(i) strongly and weakly continuous, if xn → x in X implies Axn ⇀ Ax in X∗;
(ii) quasi-weakly continuous, if t 7→ (A(x+ ty), z) is continuous on [0, 1], for all x, y, z ∈ X;
(iii) bounded, if A maps any bounded set in X to a bounded set in X∗;

(iv) mandatory, if lim∥x∥→∞
(Ax,x)
∥x∥ = +∞.

Definition 2.2. Let A : X → X∗ be a mapping. If x ̸= y implies (Ax− Ay, x− y) ≥ (>)0, and
for all x, y ∈ X, then A is said to be monotonic (strictly monotonic).

Corollary 2.3 ([17]). Let X be a reflexive Banach space, A : X → X∗ is quasi-weakly continuous,
monotone and bounded, then A must be strongly and weakly continuous.

Corollary 2.4 ([9]). Let X be reflexive Banach space, A : X → X∗ is quasi-weakly continuous,
monotone and coercive, then A must be a surjection.

In the following, we introduce some conclusions of accretive operators on a Hilbert space.
Assume that H is a Hilbert space, and A is a binary relation on H, that A is a subset of H×H.

The domain of A is D(A) := {x : [x, y] ∈ A}, the range of A is R(A) := {y : [x, y] ∈ A}, and the
inverse of is A−1 := {[y, x] : [x, y] ∈ A}. Here, due to the binary relations on space H, the linear
operation is defined as follows

λA = {[x, λy] : [x, y] ∈ A}, ∀λ ∈ R,
A+B = {[x, y + z] : [x, y] ∈ A, [x, z] ∈ B},

then

D(λA) = D(A), λ ̸= 0,

D(A+B) = D(A) ∩D(B).

Definition 2.5. Let H be Hilbert space, A is said to be

(i) accretive, if (w1 − w2, x1 − x2)H ≥ 0, for all [x1, w1], [x2, w2] ∈ A;
(ii) maximal accretive, if A is accretive and there is no accretive binary relation on H that

really contains A;
(iii) m-proliferative, if A is proliferative and satisfies R(I +A) = H.

Lemma 2.6 ([17]). If A,B are operators on H, A is m-accretive and B is accretive and Lipschitz,
then A+B is m-accretive.

Lemma 2.7 (Gronwall inequality [19]). Let y(t) be a nonnegative absolutely continuous function
on [0, t]. If y(t) satisfies the inequality

y′(t) + γy(t) ≤ h(t),

where h ≥ 0, γ ≥ 0, then

y(t) ≤ y(0)e−γt +

∫ t

0

e−γ(t−s)h(s)ds.

In particular, if h(t) = C, then

y(t) ≤ y(0)e−γt + Cγ−1.



EJDE-2025/84 KIRCHHOFF TYPE PLATE EQUATIONS 5

Corollary 2.8 ([5]). Suppose that A : D(A) ⊆ H → H is a m-accretive operator, B : H → H is
locally Lipschitz continuous and 0 ∈ A0. The initial value problem

ut +Au+Bu ∋ f,

u = u0 ∈ H,
(2.7)

(i) has a unique strong solution u on the interval [0, tmax), if tmax ≤ +∞, u0 ∈ D(A) and
f ∈W 1,1(0, t;H) for all t > 0;

(ii) has a unique generalized solution u ∈ C([0, tmax);H), if u0 ∈ D(A) and f ∈ L1(0, t;H)
for all t > 0.

In both cases we have limt→tmax
∥u(t)∥H = ∞ provided tmax <∞.

Now we give some theorems and concepts on the existence of global attractors in autonomous
dynamical systems.

Definition 2.9. Let {S(t)}t≥0 be a continuous semigroup on space X. If there exists a bounded
set B0 ⊂ X such that S(t)B ⊂ B0(∀t ≥ tB , tB ≥ 0) for any bounded subset B ⊂ X, then B0 is a
bounded absorbing set or {S(t)}t≥0 is called bounded dissipative.

Definition 2.10. Let {S(t)}t≥0 be a continuous semigroup on a complete metric space X. A ⊂ X
is called a global attractor of {S(t)}t≥0, if

(i) (compactness) A is a compact set;
(ii) (invariance) S(t)A = A,∀t ≥ 0;
(iii) (attractivity) dist(S(t)B,A) → 0 as t→ ∞, for each bounded set B ⊂ X, where dist(A,B)

denotes the Hausdorff semi-distance define as

dist(A,B) = sup
x∈A

inf
y∈B

dist(x, y).

Theorem 2.11 ([22]). Let un : K → X(n = 1, 2, 3, · · · ) be a measurable function sequence (K
is a finite real number interval). If limn→∞ un(t) = u(t), a.e. t ∈ K and there exists a Lebesgue
integrable function g : K → R such that ∥un(t)∥ ≤ g(t) for all n ≥ 1, a.e. t ∈ K, then the
function u is Bochner integrable on K and limn→∞

∫
K un(t)dt =

∫
K u(t)dt. Furthermore, there is

limn→∞
∫
K ∥un(t)− u(t)∥dt = 0.

Theorem 2.12 (Arzelà-Ascoli theorem [6]). Suppose X is a Banach space. A set F ⊂ C(a, b;X)
is relatively compact if and only if

(i) F (t) := {f(t) : f ∈ F} is relatively compact in X for each t ∈ [a, b];
(ii) F is equicontinuous; that is, for all ε > 0, there exists δ > 0 such that

∥f(t)− f(s)∥X ≤ ε, ∀f ∈ F, ∀t, s ∈ [a, b] and |t− s| ≤ δ.

Theorem 2.13 ([27]). Let (X,S(t)) be a dynamical system on the complete metric space (X, d).
Assume that for any bounded positive invariant set B ⊂ X , for each ε > 0, there exists T > 0, a
continuous non-decreasing function q: R+ → R+ and a pseudometric ρTB,ε on the C(0, T ;X) such
that

(i) q(0) = 0 and q(s) < s for s > 0;
(ii) the pseudometric ρTB,ε is precompact (with respect to X) in the following sense: any se-

quence {xn} ⊂ B has a subsequence {xnk
} such that the sequence {yn} ⊂ C(0, T,X) of

elements yk = Sτxnk
is Cauchy with respect to ρTB,ε;

(iii) the following inequality holds

d(ST y1, ST y2) ≤ q
(
(1 + ε)d(y1, y2) + ρTB,ε({Sτy1}, {Sτy2})

)
,

for every y1, y2 ∈ B, where Sτyi ⊂ C(0, T,X), yi(τ) = Sτyi.

Then (X,S(t)) is an asymptotically smooth dynamical system.

Theorem 2.14 ([3, 10]). Let {S(t)}t≥0 be a continuous semigroup on a complete metric space X.
{S(t)}t≥0 has a global attractor A in X if and only if



6 L. XU, Y. WANG, B. YANG EJDE-2025/84

(i) {S(t)}t≥0 has a bounded absorbing set in X, and the positive orbit of the bounded set is
ultimately bounded;

(ii) {S(t)}t≥0 is asymptotically smooth on X.

3. Well-posedness

In this section, we discuss the well-posedness of solution for problem (1.1). Firstly, we give the
definition of solution.

Definition 3.1 ([7]). A function u ∈ C([0, T ];D(A1/2)) ∩ C1([0, T ];L2(Ω)) possessing the prop-
erties u(0) = u0 and ut(0) = u1 is said to be a strong solution to (1.1) on the interval [0, T ],
if

• u ∈W 1,1(a, b;D(A1/2)) and ut ∈W 1,1(a, b;L2(Ω)) for any 0 < a < b < T ;
• k∥ut∥put +∆2u ∈ [L2(Ω)]′ for almost all t ∈ [0, T ];
• the problem (1.1) is satisfied in [L2(Ω)]′ for almost all t ∈ [0, T ].

A generalized solution to (1.1) on the interval [0, T ], if there exists a sequence of strong solution
{uj} to (1.1) with initial value (uj0, uj1) instead of (u0, u1) such that

lim
j→∞

max
t∈[0,T ]

{|∂tu(t)− ∂tuj(t)|+ |A1/2(u(t)− uj(t))|} = 0.

And a weak solution to (1.1) on the interval [0, T ], if∫
Ω

ut(t, x)ϕ(x)dx

=

∫
Ω

u1ϕ(x)dx+

∫ t

0

[ ∫
Ω

h(x)ϕ(x)dx+

∫
Ω×Ω

K(x, y)ut(τ, y)ϕ(x)dxdy

−
∫
Ω

∆u(τ, x)∆ϕ(x)dx− k∥ut(τ)∥p
∫
Ω

ut(τ, x)ϕ(x)dx

−m(∥∇u∥2)
∫
Ω

∇u(τ, x)∇ϕ(x)dx−
∫
Ω

g(u(τ, x))ϕ(x)dx
]
dτ,

(3.1)

for every ϕ ∈ D(A1/2) and for almost all t ∈ [0, T ].

Now we give the well-posedness results for problem (1.1).

Theorem 3.2. Let T > 0 be arbitrary. Under conditions (A1) and (A2) the following statements
hold:

(i) for all (u0, u1) ∈ D(A1/2)×D(A1/2) such that k∥u1∥pu1+∆2u0 ∈ L2(Ω), then the problem
(1.1) has a unique strong solution u on [0, T ] which satisfies

(ut, utt) ∈ L∞(0, T ;D(A1/2)× L2(Ω)), ut ∈ Cr([0, T ];D(A1/2)),

utt ∈ Cr([0, T ];L
2(Ω)), k∥ut∥put +∆2u ∈ Cr([0, T ]; [L

2(Ω)]′),
(3.2)

where Cr is denoted the space of right continuous functions;
(ii) for each (u0, u1) ∈ D(A1/2) × L2(Ω), there exists a unique generalized solution, which is

also a weak solution to (1.1);
(iii) the generalized solution and weak solution satisfy the energy relation

X (u(t), ut(t)) + k

∫ t

0

∥ut(τ)∥p+2dτ

= X (u0, u1) +

∫ t

0

∫
Ω×Ω

K(x, y)ut(τ, y)ut(τ, x)dydxdτ,

(3.3)

where

X (u(t), ut(t)) =
1

2
∥ut(t)∥2 +

1

2
∥∆u(t)∥2 + 1

2
M(∥∇u∥2)

+

∫
Ω

G(u(t, x))dx−
∫
Ω

h(x)u(t, x)dx.



EJDE-2025/84 KIRCHHOFF TYPE PLATE EQUATIONS 7

Proof. This is done three steps. The first step is to prove the local well-posedness of the problem
(1.1). First of all, the equation (1.1) is written as a first-order equation. So A : D(A) ⊆ W → W
is introduced, where W = D(A1/2)× L2(Ω). Let U = (u, v)T , v = ut. We define

A =

(
0 −I
∆2 k∥v∥p

)
, (3.4)

and

D(A) = {(u, v)T ∈ D(A1/2)×D(A1/2) | k∥v∥pv +∆2u ∈ [L2(Ω)]′}.

Then the original problem (1.1) is equivalent to the problem

d

dt
U +AU = B(U), t > 0,

U(0) = U0 = (u0, u1)
T ,

(3.5)

where B : W → W is defined as

B(U) =

(
0∫

Ω
K(x, y)ut(y)dy + h(x)− g(u) +m(∥∇u∥2)∆u

)
,

for all U = (u, v)T ∈ W.
The following proves that the operator A is accretive. For each v1, v2 ∈ L2(Ω), we note that

(∥v1∥pv1 − ∥v2∥pv2, v1 − v2)

= ∥v1∥p(∥v1∥2 − (v1, v2)) + ∥v2∥p(∥v2∥2 − (v1, v2))

≥ ∥v1∥p[∥v1∥2 −
1

2
(∥v1∥2 + ∥v2∥2)] + ∥v2∥p[∥v2∥2 −

1

2
(∥v1∥2 + ∥v2∥2)]

=
1

2
(∥v1∥2 − ∥v2∥2)(∥v1∥p − ∥v2∥p) ≥ 0.

(3.6)

And for each U1 = (u1, v1)
T and each U2 = (u2, v2)

T ∈ D(A), we obtain

(AU1 −AU2, U1 − U2)W

=

((
v2 − v1

∆2u1 −∆2u2 + k∥v1∥pv1 − k∥v2∥pv2

)
,

(
u1 − u2
v1 − v2

))
W

= (∆(v2 − v1),∆(u1 − u2)) + (∆(u1 − u2),∆(v1 − v2))

+ (k∥v1∥pv1 − k∥v2∥pv2, v1 − v2)

= (k∥v1∥pv1 − k∥v2∥pv2, v1 − v2) ≥ 0.

(3.7)

Therefore, the operator A is accretive.
Next, we verify that the accretive operator A is maximal; that is, R(I + A) = W, namely the

following equation has a solution

(A+ I)U =

(
−v + u

∆2u+ k∥v∥pv + v

)
=

(
g0
g1

)
, (3.8)

for all (g0, g1)
T ∈ W such that U = (u, v)T ∈ D(A). Removing u from (3.8), we obtain

∆2v + k∥v∥pv + v = g1 −∆2g0 ∈ [D(A1/2)]′. (3.9)

For all u ∈ D(A1/2), we define E : D(A1/2) → [D(A1/2)]′ and denote as E(u) = ∆2u+k∥u∥pu+u.
Then, for any u1, u2 ∈ D(A1/2), there exists that a continuous function of the real variable λ is

(E(u1 + λu2), u2) = (∆2(u1 + λu2) + k∥u1 + λu2∥p(u1 + λu2) + u1 + λu2, u2)

= (∆(u1 + λu2),∆u2) + (1 + k∥u1 + λu2∥p)(u1 + λu2, u2).
(3.10)



8 L. XU, Y. WANG, B. YANG EJDE-2025/84

We infer from (3.9), for any u1, u2 ∈ D(A1/2) such that

(E(u1)− E(u2), u1 − u2) = (∆2u1 + k∥u1∥pu1 + u1 −∆2u2 − k∥u2∥pu2 − u2, u1 − u2)

= (∆2(u1 − u2) + k∥u1∥pu1 − k∥u2∥pu2 + u1 − u2, u1 − u2)

= ∥∆(u1 − u2)∥2 + k(∥u1∥pu1 − ∥u2∥pu2, u1 − u2) + ∥u1 − u2∥2 ≥ 0.

(3.11)
In addition, if ∥∆u∥ → ∞, then

(E(u), u)

∥∆u∥
=

∥∆u∥2 + k∥u∥p+2 + ∥u∥2

∥∆u∥
→ +∞. (3.12)

In summary, E is quasi-weakly continuous, monotone and coercive. From the Corollary 2.4, we
can obtain that E is surjective and R(I + A) = W holds. From (3.7) and R(I + A) = W, we
deduce that A is a m-accretive operator.

We prove that B(U) is locally Lipschitz. For any u1, u2 ∈ D(A1/2), we derived from (2.2)[ ∫
Ω

(
g(u1)− g(u2)

)2

dx
]1/2

=
{∫

Ω

[ ∫ 1

0

g′(u2 + ϑ(u1 − u2))(u1 − u2)dϑ
]2
dx

}1/2

≤
{∫

Ω

[ ∫ 1

0

C(1 + |u2 + ϑ(u1 − u2)|q)|u1 − u2|dϑ
]2
dx

}1/2

≤ C
{∫

Ω

[
(|u1|q + |u2|q + 1)|u1 − u2|

]2
dx

}1/2

≤ C
{∫

Ω

(|u1|2q + |u2|2q + 1)|u1 − u2|2dx
}1/2

≤ C
{(∫

Ω

|u1|2q|u1 − u2|2dx
)1/2

+
(∫

Ω

|u2|2q|u1 − u2|2dx
)1/2

+
(∫

Ω

|u1 − u2|2dx
)1/2}

.

(3.13)

When n > 4, we take r = n
(n−4)q and r = n

n−(n−4)q , then by q < 4
n−4 , it is clear that

1
r + 1

r = 1,

2qr < 2n
n−4 and 2r < 2n

n−4 . When n ≤ 4, we take r = r = 2. So, for any n ∈ N+, we obtain

D(A1/2) ↪→ L2qr(Ω) and D(A1/2) ↪→↪→ L2r(Ω). And for any U1 = (u1, v1)
T , U2 = (u2, v2)

T ∈ W,
there exists a positive constant r̂ such that ∥Ui∥W ≤ r̂, i = 1, 2. Thus[ ∫

Ω

(
g(u1)− g(u2)

)2

dx
]1/2

≤ C
{(∫

Ω

|u1|2qrdx
) 1

2r
(∫

Ω

|u1 − u2|2rdx
) 1

2r

+
(∫

Ω

|u2|2qrdx
) 1

2r
(∫

Ω

|u1 − u2|2rdx
) 1

2r

+
(∫

Ω

|u1 − u2|2dx
)1/2}

≤ C(∥∆u1∥q + ∥∆u2∥q + 1)∥∆(u1 − u2)∥
≤ L(r̂)∥∆(u1 − u2)∥.

(3.14)

Similarly, using the assumption (A1), the mean value theorem and Sobolev embedding theorem
(D(A1/2) ↪→ H1

0 (Ω)), we obtain

∥m(∥∇u1∥2)∆u1 −m(∥∇u2∥2)∆u2∥
= ∥m(∥∇u1∥2)∆u1 −m(∥∇u1∥2)∆u2 +m(∥∇u1∥2)∆u2 −m(∥∇u2∥2)∆u2∥
≤ ∥m(∥∇u1∥2)∆u1 −m(∥∇u1∥2)∆u2∥+ ∥m(∥∇u1∥2)∆u2 −m(∥∇u2∥2)∆u2∥
≤ C(r̂)∥∆(u1 − u2)∥+ C(r̂)∥∇(u1 − u2)∥
≤ L(r̂)∥∆(u1 − u2)∥.

(3.15)



EJDE-2025/84 KIRCHHOFF TYPE PLATE EQUATIONS 9

In addition, from Hölder’s inequality, we obtain∥∥∥∫
Ω

K(x, y)(v1(y)− v2(y))dy
∥∥∥ =

{∫
Ω

(∫
Ω

K(x, y)(v1(y)− v2(y))dy
)2

dx
}1/2

≤
{∫

Ω

[( ∫
Ω

K2(x, y)dy
)1/2

∥v1 − v2∥
]2
dx

}1/2

≤
[ ∫

Ω×Ω

K2(x, y)dxdy
]1/2

∥v1 − v2∥,

(3.16)

for all v1, v2 ∈ L2(Ω). We deduce from (3.14)-(3.16) that B(U) is locally Lipschitz continuous. So
far, we have proved that A is a m-accretive operator, B(U) is locally Lipschitz continuous, and

D(A) = W. Therefore, from Corollary 2.8, there exists tmax < +∞ such that the problem (1.1)
has a unique strong solution on the interval [0, tmax) and satisfies (3.2), for any (u0, u1) ∈ D(A).
Meanwhile, the problem (1.1) has a unique generalized solution (u, ut) ∈ C([0, tmax);W), for any
(u0, u1) ∈ W. Further, the strong solution and generalized solution have the following properties:
if tmax < +∞, then

lim
t→tmax

∥(u, ut)∥W = ∞. (3.17)

The second step is to prove the global well-posedness for problem (1.1). Let

X (u(t), ut(t)) =
1

2
∥ut(t)∥2 +

1

2
∥∆u(t)∥2 + 1

2
M(∥∇u∥2) +

∫
Ω

G(u(t, x))dx−
∫
Ω

h(x)u(t, x)dx

and

I(t) = 1

2
∥ut(t)∥2 +

1

2
∥∆u(t)∥2.

Using (2.5) and Poincaré inequality, we have∫
Ω

G(u)dx ≥ −λ+ λ1
4

∥u∥2 − C ≥ −λ+ λ1
4λ1

∥∆u∥2 − C. (3.18)

Combining Poincaré inequality, Young inequality and Hölder inequality, we deduce that∣∣ ∫
Ω

hudx
∣∣ ≤ ∥h∥∥u∥

≤ 4

λ1 − λ
∥h∥2 + 1

16
(λ1 − λ)∥u∥2

≤ 1

16

(
1− λ

λ1

)
∥∆u∥2 + C.

(3.19)

According to assumptions (A1), (3.18) and (3.19), we know that

X (u(t), ut(t)) ≥
1

2
∥ut(t)∥2 +

1

2
∥∆u(t)∥2 + 1

2
M(∥∇u∥2)

− λ+ λ1
4λ1

∥∆u∥2 − C − 1

16
(1− λ

λ1
)∥∆u∥2 − C

≥ νI(t)− C,

(3.20)

where 0 < ν < 1. By (2.3), we have

|G(s)| ≤ C(s2 + |s|q+2), ∀s ∈ R. (3.21)

Using the range of q in condition (2.3) and Sobolev embedding theorem D(A1/2) ↪→ Lq+2(Ω), we
have ∣∣ ∫

Ω

G(u)dx
∣∣ ≤ ∫

Ω

|G(u)|dx ≤
∫
Ω

C(u2 + |u|q+2)dx ≤ C(∥∆u∥2 + ∥∆u∥q+2). (3.22)



10 L. XU, Y. WANG, B. YANG EJDE-2025/84

Combining (3.19) and (3.22), we obtain

X (u(t), ut(t)) ≤
1

2
∥ut(t)∥2 +

1

2
∥∆u(t)∥2 + 1

2
M(∥∇u∥2)

+ C(∥∆u∥2 + ∥∆u∥q+2) +
1

16
(1− λ

λ1
)∥∆u∥2 + C

≤ C(∥ut∥2 + ∥∆u∥2 + ∥∆u∥q+2 + 1).

(3.23)

Multiplying (1.1) by ut and integrating on Ω yields

d

dt
X (u(t), ut(t)) = −k∥ut∥p+2 +

∫
Ω×Ω

K(x, y)ut(y)ut(x)dydx, (3.24)

where t ∈ [0, tmax). Using the similar calculation method in (3.16), we deduce that∥∥∥∫
Ω

K(x, y)ut(y)dy
∥∥∥ =

[ ∫
Ω

(∫
Ω

K(x, y)ut(y)dy
)2

dx
]1/2

≤
[ ∫

Ω

((∫
Ω

K2(x, y)dy
)1/2

∥ut(y)∥
)2

dx
]1/2

≤
(∫

Ω×Ω

K2(x, y)dxdy
)1/2

∥ut∥

= ∥K∥L2(Ω×Ω)∥ut∥.

(3.25)

Furthermore, ∣∣ ∫
Ω×Ω

K(x, y)ut(y)ut(x)dydx
∣∣

≤
(∫

Ω

(∫
Ω

K(x, y)ut(y)dy
)2

dx
)1/2(∫

Ω

u2t (x)dx
)1/2

≤
(∫

Ω

(( ∫
Ω

K2(x, y)dy
)1/2( ∫

Ω

u2t (y)dy
)1/2)2

dx
)1/2

∥ut∥2

≤
(∫

Ω×Ω

K2(x, y)dxdy
)1/2

∥ut∥2

= ∥K∥L2(Ω×Ω)∥ut∥2.

(3.26)

When t ∈ [0, tmax), from (3.24), (3.26) and Young inequality, we conclude that

d

dt
X (u(t), ut(t)) ≤ −k∥ut∥p+2 + ∥K∥L2(Ω×Ω)∥ut∥2

≤ −k∥ut∥p+2 +
k

2
(∥ut∥2)

p+2
2 + C(∥K∥L2(Ω×Ω))

p+2
p

≤ −k∥ut∥p+2 +
k

2
∥ut∥p+2 + C ≤ C.

(3.27)

Integrating (3.27) on [0, t], we have

X (u(t), ut(t)) ≤ X (u0, u1) + Ct. (3.28)

If tmax < +∞, applying (3.20), (3.23) and (3.28), we obtain

∥(u(t), ut(t))∥2W ≤ X (u(t), ut(t))

≤ X (u0, u1) + Ct

≤ C(∥u1∥2 + ∥∆u0∥2 + ∥∆u0∥q+2 + 1 + tmax) ≤ +∞.

(3.29)

According to the definition of generalized solutions, (3.29) is still valid for generalized solutions.
Thus, the global well-posedness of strong solutions and generalized solutions are obtained.

In the third step, we prove that every generalized solution of (1.1) is also a weak solution. Let
u(t) be a generalized solution of the problem (1.1). By the definition of the generalized solution,



EJDE-2025/84 KIRCHHOFF TYPE PLATE EQUATIONS 11

there exists a sequence of strong solution {uj} to problem (1.1) with initial value (uj0, uj1) instead
of (u0, u1) such that

lim
j→∞

max
t∈[0,T ]

{|∂tu(t)− ∂tuj(t)|+ |A1/2(u(t)− uj(t))|} = 0 (3.30)

in C([0, T ];W). Now, we deduce that∫
Ω

ujt(t, x)ϕ(x)dx

=

∫
Ω

uj1ϕ(x)dx+

∫ t

0

[ ∫
Ω

h(x)ϕ(x)dx+

∫
Ω×Ω

K(x, y)ujt(τ, y)ϕ(x)dxdy

−
∫
Ω

∆uj(τ, x)∆ϕ(x)dx− k∥ujt(τ)∥p
∫
Ω

ujt(τ, x)ϕ(x)dx

+m(∥∇u∥2)
∫
Ω

∇uj(τ, x)∇ϕ(x)dx−
∫
Ω

g(uj(τ, x))ϕ(x)dx
]
dτ,

(3.31)

for all ϕ ∈ D(A1/2) and for almost all t ∈ [0, T ].
We define the mapping D : L2(Ω) → L2(Ω) as v 7→ ∥v∥pv, such that ∥D(ut)∥ = ∥ut∥p+1 ≤ C

for every ut ∈ L2(Ω), ∥ut∥ ≤ µ1 (µ1 > 0), then we conclude that D is quasi-weakly continuous
and bounded, and D is monotonic from (3.6). From the Corollary 2.3, we infer that D is strongly
and weakly continuous and

∥ujt(τ)∥p
∫
Ω

ujt(τ, x)ϕ(x)dx→ ∥ut(τ)∥p
∫
Ω

ut(τ, x)ϕ(x)dx (j → ∞). (3.32)

By (3.30), there is J ∈ N+ such that maxτ∈[0,T ] ∥ujt(τ)∥ ≤ maxτ∈[0,T ] ∥ut(τ)∥ + 1, for all j ≥ J ,
which implies ∣∣∣∥ujt(τ)∥p ∫

Ω

ujt(τ, x)ϕ(x)dx
∣∣∣ ≤ (

max
τ∈[0,T ]

∥ujt(τ)∥
)p+1

∥ϕ∥

≤
(

max
τ∈[0,T ]

∥ut(τ)∥+ 1
)p+1

∥ϕ∥ ≤ C.

(3.33)

Applying the Lebesgue Dominated Convergence Theorem, we infer from (3.32) and (3.33) that

lim
j→+∞

∫ t

0

[
∥ujt(τ)∥p

∫
Ω

ujt(τ, x)ϕ(x)dx
]
dτ =

∫ t

0

[
∥ut(τ)∥p

∫
Ω

ut(τ, x)ϕ(x)dx
]
dτ. (3.34)

Let j → ∞ with (3.31), combining (3.30) and (3.34), we obtain that u(t) holds in (3.1) and u(t)
is a weak solution. On the other hand, it is easy to conclude the energy equation (3.3). The proof
is complete. □

4. Existence of bounded absorbing sets

In this section, we study the dissipativity of the semigroup {S(t)}t≥0 corresponding to problem
(1.1), that is, we prove that it has a bounded absorbing set.

Theorem 4.1. Assuming that conditions (A1) and (A2) hold. Then the dynamical system
(W, S(t)) generated by (1.1) is dissipative in the space W = D(A1/2) × L2(Ω). That is, there
exists R > 0 such that for any bounded set B in W, there is t0 = t0(B) with ∥S(t)y∥W ≤ R for
all y ∈ B and t ≥ t0(B). In particular, the set

B0 = {(u, v) ∈ W; ∥(u, v)∥W ≤ R}
is the bounded absorbing set of system (W, S(t)).

Proof. Let

Qσ(t) =
1

2
∥ut∥2 +

1

2
∥∆u∥2 + 1

2
M(∥∇u∥2) +

∫
Ω

G(u)dx−
∫
Ω

hudx+ σ

∫
Ω

utudx,

H(t) =
1

2
∥ut∥2 +

1

2
∥∆u∥2 + 1

2
M(∥∇u∥2) +

(∫
Ω

G(u)dx+
λ+ λ1

4
∥u∥2 + C

)
.



12 L. XU, Y. WANG, B. YANG EJDE-2025/84

Obviously,

H(t) ≥ 1

2
(∥ut∥2 + ∥∆u∥2), ∀t ≥ 0. (4.1)

Applying Poincaré inequality, Young inequality and Hölder inequality, we can find a σ0 > 0 such
that ∣∣∣σ ∫

Ω

utudx
∣∣∣ ≤ σ∥ut∥∥u∥

≤ σ√
λ1

(1
2
∥ut∥2 +

1

2
∥∆u∥2

)
≤ 1

16

(
1− λ

λ1

)
(∥ut∥2 + ∥∆u∥2),

(4.2)

for all σ ≤ σ0. The hypothesis σ ∈ (0, σ0] is always true.
By (2.5), (3.19), (4.2) and Poincaré inequality, we infer that

Qσ(t) ≤
3

2
H(t) + C, (4.3)

and

Qσ(t) ≥
1

4
(1− λ

λ1
)H(t)− C. (4.4)

Multiplying (1.1) by ut + σu and integrating L2(Ω), we have

d

dt
Qσ(t) ≤ −k∥ut∥p+2 +

∫
Ω×Ω

K(x, y)ut(y)ut(x)dydx

+ σ
[
∥ut∥2 − k∥ut∥p

∫
Ω

utudx− ∥∆u∥2 −m(∥∇u∥2)∥∇u∥2

−
∫
Ω

g(u)udx+

∫
Ω

hudx+

∫
Ω×Ω

K(x, y)ut(y)u(x)dydx
]
.

(4.5)

Using (2.4), we know that there exists N > 0 such that

G(s) ≤ g(s)s+
λ

2
s2 + C, |s| > N. (4.6)

Combining Poincaré inequality and (4.6), we arrive at

−
∫
Ω

g(u)udx ≤ −
∫
Ω

G(u)dx+
λ

2

∫
Ω

u2dx+ C

≤ −
(∫

Ω

G(u)dx+
λ1 + λ

4

∫
Ω

u2dx+ C
)
+

1

4

(3λ
λ1

+ 1
)
∥∆u∥2 + 2C.

(4.7)

Using Young’s inequality and Hölder’s inequality, we obtain∣∣− k∥ut∥p
∫
Ω

utudx
∣∣ ≤ Ck∥ut∥p+1∥∆u∥

= Ck∥ut∥p+2∥∆u∥
p

p+1 +
1

12
(1− λ

λ1
)∥∆u∥2.

(4.8)

We conclude from (3.25) that∣∣ ∫
Ω×Ω

K(x, y)ut(y)u(x)dydx
∣∣ ≤ ∥K∥L2(Ω×Ω)∥ut∥∥u∥

≤ 1√
λ1

∥K∥L2(Ω×Ω)∥ut∥∥∆u∥

≤ 1

12
(1− λ

λ1
)∥∆u∥2 + C∥ut∥2.

(4.9)



EJDE-2025/84 KIRCHHOFF TYPE PLATE EQUATIONS 13

Because m ∈ C1(R+) and m(s) ≥ 0, combining (3.18), (3.19), (3.26), (4.1), (4.3), (4.4), (4.7)-(4.9)
and Young inequality, we deduce from (4.5) that

d

dt
Qσ(t) ≤ −k∥ut∥p+2

(
1− Cσ∥∆u∥

p
p+1

)
+
k

2
∥ut∥p+2 + C

− σ
[1
2

(
1− λ

λ1

)
(∥∆u∥2 + ∥ut∥2)−m(∥∇u∥2)∥∇u∥2

+
(∫

Ω

G(u)dx+
λ1 + λ

4

∫
Ω

u2dx+ C
)]

≤− k∥ut∥p+2
[1
2
− Cσ(2H(t))

p
2(p+1)

]
+ C −

(
1− λ

λ1

)
σH(t)

≤− k∥ut∥p+2
[1
2
− Cσ(Qσ(t) + C)

p
2(p+1)

]
− 2

3

(
1− λ

λ1

)
σQσ(t) + C.

(4.10)

To find an upper bound for Qσ(t) such that d
dtQσ(t) ≤ 0, we have

1

2
− Cσ(Qσ(t) + C)

p
2(p+1) ≥ 0, (4.11)

−2

3

(
1− λ

λ1

)
σQσ(t) + C ≤ 0. (4.12)

We deduce from (4.11) and (4.12) that, for each s ≥ 0 we have

Qσ(s) ≤ (2C)−
2(p+1)

p σ− 2(p+1)
p − 3

2
(1− λ

λ1
)−1Cσ−1 − C ≡ φ(σ). (4.13)

By (4.11) and (4.13), we have

Qσ(t) ≤ (2C)−
2(p+1)

p σ− 2(p+1)
p − C ≡ ψ(σ), ∀t ≥ s ≥ 0. (4.14)

Actually, because of Qσ(s) ≤ φ(σ) < ψ(σ) and the continuity of Qσ(t), there exists T > s such
that Qσ(t) ≤ ψ(σ) for all t ∈ [s, T ). Let T ′ = inf{t ≥ s| Qσ(t) ≥ ψ(σ)}. Apparently, there is
T ′ > s. If T ′ < +∞, then

Qσ(t) ≤ ψ(σ), ∀t ∈ [s, T ′], (4.15)

and
Qσ(T

′) = ψ(σ). (4.16)

Combining (4.10) and (4.11), we obtain

d

dt
Qσ(t) ≤ −2

3
(1− λ

λ1
)σQσ(t) + C, ∀t ∈ [s, T ′]. (4.17)

Using the Gronwall lemma in (4.17), we have

Qσ(t) ≤ e−
2
3 (1−

λ
λ1

)σ(t−s)Qσ(s) +
3

2
(1− λ

λ1
)−1Cσ−1, ∀t ∈ [s, T ′]. (4.18)

Applying t = T ′ to (4.13) and (4.18), we obtain

Qσ(T
′) < Qσ(s) +

3

2
(1− λ

λ1
)−1Cσ−1

≤ φ(σ) +
3

2
(1− λ

λ1
)−1Cσ−1

= ψ(σ).

(4.19)

The above formula contradicts (4.16). So T ′ = +∞, so we obtain (4.11).
The above results show that if ∥(u0, u1)∥W ≤ R, then

1

2
∥(u(t), ut(t))∥2W =

1

2
(∥∆u∥2 + ∥ut∥2) ≤ H(t) ≤ C(R), ∀t ≥ 0, (4.20)

for some R > 0. Actually, since ∥(u0, u1)∥W ≤ R, by Poincaré inequality and (3.22), we obtain

H(0) ≤ 1

2

[
∥u1∥2 + ∥∆u0∥2 +

M√
λ1

|∆u∥2
]



14 L. XU, Y. WANG, B. YANG EJDE-2025/84

+
[
C
(
∥∆u0∥2 + ∥∆u0∥q+2

)
+
λ1 + λ

4λ1
∥∆u0∥2 + C

]
≤ C(Rq+2 +R4 +R2 + 1),

recombining (4.3), we arrive at

Qσ(0) ≤
3

2
H(0) + C ≤ C(Rq+2 +R4 +R2 + 1). (4.21)

By (4.13), we obtain

φ′(σ) = −σ−2
[
(2C)−

2(p+1)
p

2(p+ 1)

p
σ− p+2

p − 3

2

(
1− λ

λ1

)−1

C
]
,

this means

φ′(σ)

{
≥ 0, σ ∈ [σ1,+∞),

< 0, σ ∈ (0, σ1),

where σ1 =
[
3
2 (1 − λ

λ1
)−1C(2C)

2(p+1)
p p

2(p+1)

]− p
p+2

. In addition, when σ → +∞, φ(σ) → −C
holds, and when σ → 0 , φ(σ) → +∞ holds. Therefore, there exists a constant σ2 > 0 such that
φ(σ2) = 0 and φ(σ) > 0 for σ ∈ (0, σ2). Then, the function φ limited to the interval (0, σ2) is
strictly decreasing, and there is an inverse function that is denoted by φ−1. Let

σ = min{σ0, φ−1(C(Rq+2 +R4 +R2 + 1))}. (4.22)

Using(4.21) and (4.22), we know φ(σ) ≥ C(Rq+2 + R4 + R2 + 1) ≥ Qσ(0), which is found when
s = 0 in (4.13). Hence, based on above conclusions, we have

Qσ(t) ≤ ψ(σ) = ψ(min{σ0, φ−1(C(Rq+2 +R4 +R2 + 1))}), ∀t ≥ 0. (4.23)

Combining (4.4) and (4.23), for all t ≥ 0, we have

H(t) ≤ 4
(
1− λ

λ1

)−1

(Qσ(t) + C)

≤ 4
(
1− λ

λ1

)−1

[ψ(min{σ0, φ−1(C(Rq+2 +R4 +R2 + 1))}) + C]

= C(R),

(4.24)

that is (4.20) holds.
The inequality ∥(u0, u1)∥W ≤ R holds. Let Ψ(ε) = min{σ0, φ−1( 32ε + C)}. Obviously, Ψ is

continuously decreasing. For any s ≥ 0, assuming σ = Ψ(H(s)) and recombining (4.3), we obtain
φ(σ) ≥ 3

2H(s) + C ≥ Qσ(s), this means that (4.13) holds. Besides, there is σ ≤ σ0. Therefore,
according to the above conclusions, (4.11) is true. Substituting σ = Ψ(H(s)) and (4.11) into
(4.10), we arrive at

d

dt
Qσ(t) ≤ −k∥ut∥p+2

[1
2
− Cσ(Qσ(t) + C)

p
2(p+1)

]
− 2

3

(
1− λ

λ1

)
σQσ(t) + C

≤ −2

3

(
1− λ

λ1

)
Ψ(H(s))Qσ(t) + C, ∀t ∈ [s,+∞).

(4.25)

Employing the Gronwall lemma to (4.25), for any t ∈ [s,+∞), we infer that

Qσ(t) ≤ e−
2
3 (1−

λ
λ1

)Ψ(H(s))(t−s)Qσ(s) +
2

3

(
1− λ

λ1

)−1

C[Ψ(H(s))]−1. (4.26)

Applying (4.3), (4.4) and (4.26), for all t ≥ s ≥ 0, we obtain

1

4

(
1− λ

λ1

)
H(t)− C ≤ e−

2
3 (1−

λ
λ1

)Ψ(H(s))(t−s)
[3
2
H(s) + C

]
+

3

2

(
1− λ

λ1

)−1

C[Ψ(H(s))]−1.

(4.27)

Because Ψ is decreasing, by (4.20), we obtain

Ψ(H(s)) ≥ Ψ(C(R)), ∀s ≥ 0. (4.28)



EJDE-2025/84 KIRCHHOFF TYPE PLATE EQUATIONS 15

Substituting (4.28) into (4.27), for all t ≥ s ≥ 0, we obtain

1

4

(
1− λ

λ1

)
H(t)−C ≤ e−

2
3 (1−

λ
λ1

)Ψ(C(R))(t−s)
[3
2
H(s) +C

]
+

3

2

(
1− λ

λ1

)−1

C[Ψ(H(s))]−1, (4.29)

we conclude from (4.29) that

1

4

(
1− λ

λ1

)
sup

∥(u0,u1)∥W≤R

H(t)− C ≤ e−
2
3 (1−

λ
λ1

)Ψ(C(R))(t−s)
[3
2

sup
∥(u0,u1)∥W≤R

H(s) + C
]

+
3

2

(
1− λ

λ1

)−1

C[Ψ( sup
∥(u0,u1)∥W≤R

H(s))]−1,

(4.30)

for all t ≥ s ≥ 0, this indicates that

1

4

(
1− λ

λ1

)
lim sup
t→+∞

sup
∥(u0,u1)∥W≤R

H(t)− C ≤ 3

2

(
1− λ

λ1

)−1

C[Ψ( sup
∥(u0,u1)∥W≤R

H(s))]−1, (4.31)

for all s ≥ 0.
Using the continuity of Ψ and(4.31), we derive that

1

4

(
1− λ

λ1

)
lim sup
t→+∞

sup
∥(u0,u1)∥W≤R

H(t) ≤ 3

2

(
1− λ

λ1

)−1

C[Ψ(lim sup
s→+∞

sup
∥(u0,u1)∥W≤R

H(s))]−1 + C.

(4.32)
Suppose that

f(W ) =
3
2 (1−

λ
λ1
)−1[Ψ(W )]−1 + 1

W
,

so that (4.32) can be rewritten as

f
(
lim sup
t→+∞

sup
∥(u0,u1)∥W≤R

H(t)
)
≥ 1

4

(
1− λ

λ1

)
C−1. (4.33)

Although, by definition we have

lim
W→+∞

f(W ) = lim
W→+∞

3
2 (1−

λ
λ1
)−1[φ−1( 32W + C)]−1 + 1

W

= lim
ε→+∞

3
2 (1−

λ
λ1
)−1ε+ 1

2
3 (φ(ε

−1)− C)
= 0.

(4.34)

By (4.33) and (4.34), there exists R0 > 0 (independent of R) such that

lim sup
t→+∞

sup
∥(u0,u1)∥W≤R

H(t) ≤ R0. (4.35)

Also, since 1
2∥(u0, u1)∥

2
W ≤ H(t), the dynamical system generated by the problem (1.1) is dissi-

pative, which completes the proof. □

5. Asymptotic smoothness

In this paper, the asymptotic smoothness of the dynamical system is proved by using the
energy reconstruction method of Chueshov and Lasiecka (see[7]). Heretofore, a priori estimate is
established.

Lemma 5.1. Under assumptions (A1) and (A2), w(t) and v(t) are strong solutions of problem
(1.1) corresponding to (w(0), wt(0)) = (w0, w1), (v(0), vt(0)) = (v0, v1) with different initial values,



16 L. XU, Y. WANG, B. YANG EJDE-2025/84

then there exist T0 > 0 and a constant C > 0 (independent of T ) such that

TIm(T ) +

∫ T

0

Im(t)dt

≤ C(R)
{∫ T

0

∥ιt(t)∥2dt+ k

∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt

+ k

∫ T

0

|(∥wt∥pwt − ∥vt∥pvt, ι)|dt+
∫ T

0

∥∇ι∥2dt+
∫ T

0

dt

∫ T

t

∥∇ι(τ)∥2dτ

+

∫ T

0

dt

∫ T

t

∥∇ι(τ)∥∥ιt(τ)∥dτ +
∣∣∣ ∫ T

0

(N (ιt), ιt)dt
∣∣∣+ ∣∣∣ ∫ T

0

(N (ιt), ι)dt
∣∣∣

+
∣∣∣ ∫ T

0

dt

∫ T

t

(N (ιt), ιt)dτ
∣∣∣+ ∣∣∣ ∫ T

0

(g(w)− g(v), ιt)dt
∣∣∣+ ∣∣∣ ∫ T

0

(g(w)− g(v), ι)dt
∣∣∣

+
∣∣∣ ∫ T

0

dt

∫ T

t

(g(w)− g(v), ιt(τ))dτ
∣∣∣}, ∀T ≥ T0,

(5.1)

where ι(t) = w(t)− v(t), N (ιt) =
∫
Ω
K(x, y)ιt(y)dy, ((w0, w1), (v0, v1)) ∈ D(A1/2)×D(A1/2) and

Im(t) =
1

2
(∥ιt(t)∥2 + ∥∆ι(t)∥2 +m(∥∇w∥2)∥∇ι(t)∥2).

Proof. According to Theorem 4.1 and m ∈ C1(R+), there exists a constant C(R, ∥∇w0∥) such
that

m(∥∇w∥2)∥∇ι∥2 ≤ C(R, ∥∇w0∥)∥∇ι∥2,

where ι(t) = w(t)− v(t). Applying interpolation inequality, we deduce

∥∇ι∥2 ≤ ∥∆ι∥2 + c∥ι∥2,

where c is a positive constant, then

1

2
(∥ιt(t)∥2 + ∥∇ι(t)∥2) = Iι(t) ≤ Im(t) ≤ C(R, ∥∇w0∥2)Iι(t),

and

Im(t) ∼ Iι(t) =
1

2
∥(ι(t), ιt(t))∥2W . (5.2)

Since ι(t) = w(t)− v(t) satisfies the equation

ιtt +∆2ι−m(∥∇w∥2)∆ι− (m(∥∇w∥2)−m(∥∇v∥2))∆v
+ k(∥wt∥pwt − ∥vt∥pvt) + g(w)− g(v)

= N (ιt).

(5.3)

Multiplying (5.3) by ιt(t) on L
2(Ω) yields

(ιtt, ιt) + (∆2ι, ιt)− (m(∥∇w∥2)∆ι, ιt) + (k(∥wt∥pwt − ∥vt∥pvt), ιt)
= ((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)− (g(w)− g(v), ιt) + (N (ιt), ιt).

(5.4)

Also

(m(∥∇w∥2)∆ι, ιt) = −1

2

d

dt
m(∥∇w∥2)∥∇ι∥2 −m′(∥∇w∥2)∥∇ι∥2(∆w,wt).

Substituting the above formula into (5.4), we have

1

2

d

dt
(∥ιt(t)∥2 + ∥∆ι(t)∥2 +m(∥∇w∥2)∥∇ι∥2) + (k(∥wt∥pwt − ∥vt∥pvt), ιt)

= −m′(∥∇w∥2)∥∇ι∥2(∆w,wt) + ((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)
− (g(w)− g(v), ιt) + (N (ιt), ιt).

(5.5)



EJDE-2025/84 KIRCHHOFF TYPE PLATE EQUATIONS 17

Then integrating on [t, T ] by (5.5), we obtain

Im(T ) + k

∫ T

t

(∥wt∥pwt − ∥vt∥pvt, ιt)dτ = Im(t)−
∫ T

t

m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dτ

+

∫ T

t

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dτ

−
∫ T

t

(g(w)− g(v), ιt)dτ +

∫ T

t

(N (ιt), ιt)dτ.

(5.6)

Multiplying (5.3) by ι(t) on L2(Ω), we arrive at

1

2
(∥ιt∥2 + ∥∆ι∥2 +m(∥∇w∥2)∥∇ι∥2) + 1

2

d

dt
(ιt, ι)

= ∥ιt∥2 −
1

2
k(∥wt∥pwt − ∥vt∥pvt, ι) +

1

2
((m(∥∇w∥2)−m(∥∇v∥2))∆v, ι)

− 1

2
(g(w)− g(v), ι) +

1

2
(N (ιt), ι).

(5.7)

Integrating on [0, T ] by (5.7), we have

2

∫ T

0

Im(t)dt+ (ιt, ι)|T0 = 2

∫ T

0

∥ιt∥2dt− k

∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ι)dt

+

∫ T

0

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ι)dt

−
∫ T

0

(g(w)− g(v), ι)dt+

∫ T

0

(N (ιt), ι)dt.

(5.8)

Applying Sobolev embedding theorem (D(A1/2) ↪→ L2(Ω)), Poincaré inequality, Young inequality
and Hölder inequality, we conclude that

|(ιt, ι)| ≤ ∥ιt∥∥ι∥ ≤ 1

2
(∥ιt∥2 + ∥ι∥2) ≤ CIm(t). (5.9)

Substituting (5.9) into (5.8), we obtain

2

∫ T

0

Im(t)dt ≤ C(Im(0)− Im(T )) + 2

∫ T

0

∥ιt∥2dt− k

∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt

+

∫ T

0

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ι)dt

−
∫ T

0

(g(w)− g(v), ι)dt+

∫ T

0

(N (ιt), ι)dt.

(5.10)

In (5.6), letting t = 0, we obtain

Im(0) = Im(T ) + k

∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt

+

∫ T

0

m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dt

−
∫ T

0

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dt

+

∫ T

0

(g(w)− g(v), ιt)dt−
∫ T

0

(N (ιt), ιt)dt.

(5.11)



18 L. XU, Y. WANG, B. YANG EJDE-2025/84

Integrating on [0, T ] by (5.6), we deduce from the monotonicity of the damping operator that

TIm(T ) ≤
∫ T

0

Im(t)dt−
∫ T

0

dt

∫ T

t

m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dτ

+

∫ T

0

dt

∫ T

t

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dτ

−
∫ T

0

dt

∫ T

t

(g(w)− g(v), ιt)dτ +

∫ T

0

dt

∫ T

t

(N (ιt), ιt)dτ.

(5.12)

Combining (5.10)-(5.12), we derive

TIm(T ) +

∫ T

0

Im(t)dt

≤ C
[
k

∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt+
∫ T

0

m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dt

−
∫ T

0

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dt+
∫ T

0

(g(w)− g(v), ιt)dt

−
∫ T

0

(N (ιt), ιt)dt
]
+ 2

∫ T

0

∥ιt∥2dt− k

∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ι)dt

+

∫ T

0

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dt−
∫ T

0

(g(w)− g(v), ι)dt

+

∫ T

0

(N (ιt), ι)dt−
∫ T

0

dt

∫ T

t

m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dτ

+

∫ T

0

dt

∫ T

t

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dτ

−
∫ T

0

dt

∫ T

t

(g(w)− g(v), ιt)dτ +

∫ T

0

dt

∫ T

t

(N (ιt), ιt)dτ

≤ C
{∫ T

0

∥ιt∥2dt+ k

∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt

+ k

∫ T

0

|(∥wt∥pwt − ∥vt∥pvt, ι)|dt+
∣∣∣ ∫ T

0

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dt
∣∣∣

+
∣∣∣ ∫ T

0

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ι)dt
∣∣∣+ ∣∣∣ ∫ T

0

m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dt
∣∣∣

+
∣∣∣ ∫ T

0

dt

∫ T

t

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dτ
∣∣∣

+
∣∣∣ ∫ T

0

dt

∫ T

t

m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dτ
∣∣∣

+
∣∣∣ ∫ T

0

(N (ιt), ιt)dt
∣∣∣+ ∣∣∣ ∫ T

0

(N (ιt), ι)dt
∣∣∣

+
∣∣∣ ∫ T

0

dt

∫ T

t

(N (ιt), ιt)dτ
∣∣∣+ ∣∣∣ ∫ T

0

(g(w)− g(v), ιt)dt
∣∣∣

+
∣∣∣ ∫ T

0

(g(w)− g(v), ι)dt
∣∣∣+ ∣∣∣ ∫ T

0

dt

∫ T

t

(g(w)− g(v), ιt)dτ
∣∣∣}.

(5.13)

According to Theorem 4.1 and the boundedness of 1
2∥(u, ut)∥

2
W , then when ∥(u(0), ut(0))∥W ≤ R,

we have

∥ut(t)∥2 + ∥∆u(t)∥2 ≤ C(R),∀t ≥ 0. (5.14)



EJDE-2025/84 KIRCHHOFF TYPE PLATE EQUATIONS 19

Combiningm ∈ C1(R+), (5.14), mean value theorem and Sobolev embedding theorem (D(A1/2) ↪→
H1

0 (Ω)), we know

m(∥∇w∥2) ≤ C(R), (5.15)

|m′(∥∇w∥2)∥∇ι∥2(∆w,wt)| ≤ C(R)∥∇ι∥2, (5.16)

|m(∥∇w∥2)−m(∥∇v∥2)| ≤ C(R)∥∇ι∥, (5.17)

|(m(∥∇w∥2)−m(∥∇v∥2))(∆v, ι)| ≤ C(R)∥∇ι∥2, (5.18)

|(m(∥∇w∥2)−m(∥∇v∥2))(∆v, ιt)| ≤ C(R)∥∇ι∥∥ιt∥. (5.19)

Therefore, by (5.13) and (5.15)-(5.18) we obtain (5.1). The proof is complete. □

Lemma 5.2. Let u, v ∈ H, (·, ·) and ∥ · ∥H denote the inner product and norm of Hilbert space
H, respectively. Then there exists a p dependent constant Cp such that

(
∥u∥p−2

H u− ∥v∥p−2
H v, u− v

)
≥

{
Cp∥u− v∥pH , p ≥ 2,

Cp
∥u−v∥2

H

(∥u∥H+∥v∥H)2−p , 1 ≤ p ≤ 2.

Proposition 5.3. Suppose that (A1), (A2) hold. Then the dynamical system (W, S(t)) generated
by (1.1) is asymptotically smooth in the space W.

Proof. It is known from Theorem 4.1 that B0 is a bounded absorbing set in the dynamical system
(W, S(t)). By definition, there exists t0 ≥ 0 such that S(t)B0 ⊆ B0 for any t ≥ t0. Let B =⋃

t≥t0
S(t)B0, then B is a closed bounded positive invariant set of the system. Since for any

bounded set B makes S(t)B ⊂ B for any t ≥ t(B), B is also absorbing set of the system. Let the
two weak solutions of the problem (1.1) be w(t) and v(t), which correspond to two different initial
values in B, namely

(w(t), wt(t)) = S(t)y0, (v(t), vt(t)) = S(t)y1, y0, y1 ∈ B. (5.20)

Since B is a bounded positive invariant set of the system, it follows that

∥(w(t), wt(t))∥W ≤ C, ∥(v(t), vt(t))∥W ≤ C, (5.21)

for all t > 0, y0, y1 ∈ B.
Note that ι(t) = w(t)− v(t), N (ut(t, x)) =

∫
Ω
K(x, y)ut(y)dy and ι(t) satisfies

ιtt +∆2ι−m(∥∇w∥2)∆ι− (m(∥∆w∥2)−m(∥∇v∥2))∆v
+ k(∥wt∥pwt − ∥vt∥pvt) + g(w)− g(v)

= N (ιt).

(5.22)

Similar to (5.6) of method, for any t ∈ [0, T ], we have

Im(T ) + k

∫ T

t

(∥wt∥pwt − ∥vt∥pvt, ιt)dτ

= Im(t)−
∫ T

t

m′(∥∇w∥2)∥∇ι∥2(∆w,wt)dτ +

∫ T

t

((m(∥∇w∥2)−m(∥∇v∥2))∆v, ιt)dτ

−
∫ T

t

(g(w)− g(v), ιt)dτ +

∫ T

t

(N (ιt), ιt)dτ.

(5.23)



20 L. XU, Y. WANG, B. YANG EJDE-2025/84

The first step is energy reconstruction. Let

OT (w, v) =

∫ T

0

∥∇ι∥2dt+
∫ T

0

dt

∫ T

t

∥∇ι(τ)∥2dτ

+

∫ T

0

dt

∫ T

t

∥∇ι(τ)∥∥ιt(τ)∥dτ +
∣∣∣ ∫ T

0

(N (ιt), ιt)dt
∣∣∣

+
∣∣∣ ∫ T

0

(N (ιt), ι)dt
∣∣∣+ ∣∣∣ ∫ T

0

dt

∫ T

t

(N (ιt), ιt)dτ
∣∣∣

+
∣∣∣ ∫ T

0

(g(w)− g(v), ιt)dt
∣∣∣+ ∣∣∣ ∫ T

0

(g(w)− g(v), ι)dt
∣∣∣

+
∣∣∣ ∫ T

0

dt

∫ T

t

(g(w)− g(v), ιt)dτ
∣∣∣.

(5.24)

Substituting (5.24) in (5.1) yields

TIm(T ) +

∫ T

0

Im(t)dt ≤ C(R)
{∫ T

0

∥ιt∥2dt+ k

∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt

+ k

∫ T

0

|(∥wt∥pwt − ∥vt∥pvt, ι)|dt+OT (w, v)
}
.

(5.25)

By definition of OT (w, v), we know

OT (w, v) ≤ C
{∫ T

0

∥∇ι∥2dt+
∫ T

0

∥∇ι∥∥ιt∥dt

+

∫ T

0

∥N (ιt)∥∥ιt∥dt+
∫ T

0

∥N (ιt)∥∥ι∥dt

+

∫ T

0

∥g(w)− g(v)∥∥ι∥dt+
∫ T

0

∥g(w)− g(v)∥∥ιt∥dt
}
.

(5.26)

By Young inequality, Cauchy inequality and (D(A1/2) ↪→↪→ D(A 1
2−γ) ↪→↪→ D(A1/4)), we obtain

that there is a minimal constant 0 < γ < 1
4 such that∫ T

0

∥∇ι∥2dt+
∫ T

0

∥∇ι∥∥ιt∥dt ≤
∫ T

0

∥∇ι∥2dt+
∫ T

0

(
1

2ε
∥∇ι∥2 + ε

2
∥ιt∥2)dt

≤ C

∫ T

0

∥A 1
2−γι∥2dt+ ε

∫ T

0

Im(t)dt.

(5.27)

According to Theorem 4.1 and the growth condition (2.3) of assumption (A2), and if n > 4 , we
take r = n

(n−4)q and r̄ = n
n−(n−4)q , then when q < 4

n−4 , obviously there is 1
r + 1

r̄ = 1, if n ≤ 4, we

take r largely enough and use Sobolev embedding theorem to deduce that

∥g(w)− g(v)∥2 =

∫
Ω

|g(w)− g(v)|2dx

=

∫
Ω

[ ∫ 1

0

g′(v + ϑ(w − v))ι dϑ
]2
dx

≤ C

∫
Ω

(1 + |v + ϑ(w + v)|q)2|ι|2dx

≤ C

∫
Ω

(1 + |w|2q + |v|2q)|ι|2dx

≤ C
[ ∫

Ω

(1 + |w|2q + |v|2q)rdx
]1/r(∫

Ω

|ι|2r̄dx
)1/r̄

≤ C(R)∥ι∥2L2r̄(Ω)

≤ C(R)∥A 1
2−ηι∥2,

(5.28)



EJDE-2025/84 KIRCHHOFF TYPE PLATE EQUATIONS 21

where 0 < ϑ < 1 and η is a properly small constant. By (5.28), we arrive at∫ T

0

∥g(w)− g(v)∥∥ι∥dt+
∫ T

0

∥g(w)− g(v)∥∥ιt∥dt

≤ C

∫ T

0

∥g(w)− g(v)∥2dt+ ε

∫ T

0

Im(t)dt

≤ C

∫ T

0

∥A 1
2−ηι∥2dt+ ε

∫ T

0

Im(t)dt.

(5.29)

Using Hölder inequality, we infer that

∥N (ιt)∥2 =
∥∥∥∫

Ω

K(x, y)ιt(y)dy
∥∥∥2

=

∫
Ω

(∫
Ω

K(x, y)ιt(y)dy
)2

dx

≤
∫
Ω

[( ∫
Ω

K2(x, y)dy
)1/2

∥ιt(y)∥
]2
dx

≤
∫
Ω×Ω

K2(x, y)dxdy · ∥ιt∥2

≤ ε2

8
∥ιt∥2.

(5.30)

Furthermore, ∫ T

0

∥N (ιt)∥∥ιt∥dt+
∫ T

0

∥N (ιt)∥∥ι∥dt

≤ 2

ε

∫ T

0

∥N (ιt)∥2dt+
ε

4

∫ T

0

∥ιt∥2dt+
ε

4

∫ T

0

∥ι∥2dt

≤ C

∫ T

0

∥A 1
2−β∥2dt+ ε

∫ T

0

Im(t)dt,

(5.31)

where β is a properly small positive constant. Combining (5.27), (5.29) and (5.31), we take
η̃ = min{γ, η, β} such that

OT (w, v) ≤ C(T )

∫ T

0

∥A 1
2−η̃ι∥2dt+ 3ε

∫ T

0

Im(t)dt, ε > 0. (5.32)

According to Lemma 5.2, we write

S0(s) = C
−2
p+2
p s

2
p+2 , p ≥ 0.

It is also known that S0(s) is a strictly increasing concave function and S0 ∈ C(R+), S0(0) = 0,
then

S0[(∥w + v∥p(w + v)− ∥w∥pw, v)] ≥ S0(Cp∥v∥p+2) = ∥v∥2, w, v ∈ D(A1/2). (5.33)

Combining this with the Jensen inequality, we have∫ T

0

∥ιt∥2dt ≤
∫ T

0

S0[(∥wt∥pwt − ∥vt∥pvt, ιt)]dt

≤ TS0

( 1

T

∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt
)

= S0

(∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt
)
,

(5.34)



22 L. XU, Y. WANG, B. YANG EJDE-2025/84

where S0(s) = TS0

(
s
T

)
. Using Lemma 5.1, (5.25), (5.32) and (5.34), when ε > 0 and small

enough, we infer that

TIm(T ) +
1

2

∫ T

0

Im(t)dt

≤ C
{
(S0 + kI)

(∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt
)

+ k

∫ T

0

|(∥wt∥pwt − ∥vt∥pvt, ι)|dt+ C

∫ T

0

∥A 1
2−η̃ι∥2dt

}
, ∀T ≥ T0.

(5.35)

In addition, using the Cauchy inequality and Sobolev embedding theorem, we know that there
exists a suitable small constant 0 < α < 1

2 such that

|(∥wt∥pwt − ∥vt∥pvt, ι)| =
∣∣∣ ∫

Ω

(∥wt∥pwt − ∥vt∥pvt)ι dx
∣∣∣

≤
(∫

Ω

(∥wt∥pwt − ∥vt∥pvt)2dx
)1/2

∥ι∥

≤ C(∥wt∥2p∥wt∥2 + ∥vt∥2p∥vt∥2)1/2∥ι∥
≤ C∥ι∥

≤ C∥A 1
2−αι∥.

(5.36)

Inserting (5.36) into (5.35), we arrive at

TIm(T ) +
1

2

∫ T

0

Im(t)dt ≤ C
{
(S0 + kI)

(∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt
)

+ k

∫ T

0

∥A 1
2−αι∥dt+ C

∫ T

0

∥A 1
2−η̃ι∥dt

}
, ∀T ≥ T0.

(5.37)

The second step is the treatment of damping. Let α̃ = min{α, η̃}, and rewrite (5.37) as

Im(T ) ≤ C(S0 + kI)
(∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt
)
+ C

∫ T

0

∥A 1
2−α̃ι∥dt

≤ C(S0 + kI)
(∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt
)
+ C sup

t∈[0,T ]

∥A 1
2−α̃ι∥.

(5.38)

Let Y0(s) = (S0 + kI)−1
(

s
2C

)
, and Y0(s) is a strictly increasing concave function. For ∀s ≥ 0, we

gain (S0 + kI)−1(s) ≤ s. By (5.38),

Y0(Im(T ))

= (S0 + kI)−1
(Im(T )

2C

)
≤ (S0 + kI)−1

{1

2
(S0 + kI)

(∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt
)
+

1

2
sup

t∈[0,T ]

∥A 1
2−α̃ι∥

}
≤ 1

2

∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt+
1

2
(S0 + kI)−1{ sup

t∈[0,T ]

∥A 1
2−α̃ι∥}

≤ 1

2

∫ T

0

(∥wt∥pwt − ∥vt∥pvt, ιt)dt+
1

2
sup

t∈[0,T ]

∥A 1
2−α̃ι∥.

(5.39)

Thus, substituting(5.39) into (5.38), we conclude that

Im(T ) + 2kY0(Im(T )) ≤ Im(0) + C sup
t∈[0,T ]

∥A 1
2−α̃ι∥+ C(T ). (5.40)



EJDE-2025/84 KIRCHHOFF TYPE PLATE EQUATIONS 23

Combining the interpolation inequality and (5.15), C > 0 and θ′ = 1
2 , we have

m(∥∇w∥2)∥∇ι(t)∥2 ≤ C∥∆ι(t)∥θ
′
∥ι(t)∥1−θ′

≤ ε∥∆ι(t)∥2 + C∥ι(t)∥2

≤ ε∥∆ι(t)∥2 + C sup
t∈[0,T ]

∥A 1
2−α̃ι(t)∥.

(5.41)

Then according to the definition of Im(t), we have

Im(T ) + 2kY0(Im(T )) ≤ (1 + ε)Iι(0) + C sup
t∈[0,T ]

∥A 1
2−α̃ι(t)∥. (5.42)

Since ι(t) is uniformly bounded in D(A1/2), and there exists a tight embedding relationship

(D(A1/2) ↪→↪→ D(A 1
2−α̃) ↪→↪→ L2(Ω)), once more we use interpolation inequality to obtain

∥A 1
2−α̃ι(t)∥ ≤ ∥∆ι(t)∥θ1∥ι(t)∥1−θ1 ≤ C(R)∥ι(t)∥1−θ1 , θ1 ∈ (0, 1). (5.43)

Substituting (5.43) into (5.42), for some θ2 ∈ (0, 1], we have

Im(T ) + 2kY0(Im(T )) ≤ (1 + ε)Iι(0) + C sup
t∈[0,T ]

∥ι(t)∥θ2 . (5.44)

And because

Iι(t) =
1

2
(∥ιt(t)∥2 + ∥∆ι(t)∥2) = ∥S(T )y1 − S(T )y2∥2W ≤ Im(t).

It follows that

∥S(T )y1 − S(T )y2∥2W ≤ 2[I + 2kY0]
−1

[1
2
(1 + ε)∥y1 − y2∥2 + C sup

t∈[0,T ]

∥ι(t)∥θ2
]

≤ 2[I + 2kY0]
−1

[1
2

(
(1 + ε)1/2∥y1 − y2∥+ C sup

t∈[0,T ]

∥ι(t)∥θ3
)2]

,

(5.45)

where θ3 ∈ (0, 12 ]. By (5.45),

∥S(T )y1−S(T )y2∥W ≤
√
2
{
[I+2kY0]

−1
[1
2

(
(1+ε)1/2∥y1−y2∥+C sup

t∈[0,T ]

∥ι(t)∥θ3
)2]}1/2

, (5.46)

namely,

∥S(T )y1 − S(T )y2∥W ≤ q
(
(1 + ε)1/2∥y1 − y2∥+ ρTB({Sτy1}, {Sτy2})

)
, (5.47)

where q(s) =
√
2
(
[I + 2kY0]

−1( s
2

2 )
)1/2

and ρTB(Sτy1, Sτy2) = C supt∈[0,T ] ∥ι(t)∥θ3 . Thus, the

function q(s) satisfies all the conditions in Theorem 2.14. Denote by FB,T the set of all solutions in
the equation (1.1) on [0, T ] with the initial value on B. Next, we only need to prove that the pseudo-
metric ρTB is quasi-compact in the set FB,T . In the space C([0, T ];D(A1/2)) ∩ C1([0, T ];L2(Ω)),
for any bounded set G has the constant C such that

∥∆u(t)∥+ ∥ut(t)∥ ≤ C, ∀u(t) ∈ G(t) = {u(t) : u ∈ G}. (5.48)

Applying the compact embedding theorem (D(A1/2) ↪→↪→ L2(Ω)), we obtain from that G(t) is
relatively compact in L2(Ω), for any 0 < t < T . Additionally, for any ε > 0, u ∈ G, we have

∥u(t)− u(t1)∥ ≤
∫ t

t1

∥ut(τ)∥dτ

≤ (t− t1)
1/2(

∫ t

t1

∥ut(τ)∥2dτ)1/2

≤ C(t− t1)
1/2 ≤ Cε,

(5.49)

for any 0 ≤ t < t1 ≤ T satisfies |t − t1| ≤ ε2, according to the Ascoli theorem, it is deduced that
G is uniformly equicontinuous. Furthermore, there is a tight embedding relationship

C([0, T ];D(A1/2)) ∩ C1([0, T ];L2(Ω)) ⊂ C([0, T ];L2(Ω)).



24 L. XU, Y. WANG, B. YANG EJDE-2025/84

Therefore, the pseudo-metric ρTB is quasi-compact on the set FB,T . According to Theorem 2.13,
we obtain that the asymptotic smoothness of (W, S(t)) in space W. The proof is complete. □

6. Existence of global attractors

Theorem 6.1. Under assumptions (A1), (A2) (F1)− (F3), the dynamical system (W, S(t)) gen-
erated by the problem (1.1) has a global attractor.

The above theorem follows from Theorem 4.1 and Proposition 5.3.

Acknowledgements. This research was supported by the National Natural Science Foundation
of China (1210502,12371198), by the University Innovation Project of Gansu Province (2023B-
062), and by the Gansu Province Basic Research Innovation Group Project (23JRRA684).

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Ling Xu (corresponding author)

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China.

Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics, Lanzhou, Gansu
730070, China

Email address: xuling@nwnu.edu.cn

Yanni Wang

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China.

Gansu Provincial Research Center for Basic Disciplines of Mathematics and Statistics, Lanzhou, Gansu
730070, China

Email address: 2833986227@qq.com

Bianxia Yang

College of Science, Northwest A and F University, Yangling, Shaanxi 712100, China

Email address: bxyang@nwafu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Well-posedness
	4. Existence of bounded absorbing sets
	5. Asymptotic smoothness
	6. Existence of global attractors
	Acknowledgements

	References