Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form

Yan Dong

doi:10.1515/math-2022-0565

Artikel Open Access

Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form

Yan Dong

Veröffentlicht/Copyright: 31. März 2023

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Abstract

The author of this article concerns with the existence, uniqueness, and stability of the weak solution to the variation-inequality problem. The Kirchhoff operator is a non-divergence form with space variable parameter. The existence of generalized solution is proved by the Leray-Schauder principle and parabolic regularization. The uniqueness and stability of the solution are also discussed by contradiction.

Keywords: parabolic variation-inequality problems; weak solution; variable parameter Kirchhoff operators not in divergence form; existence; uniqueness; stability

MSC 2010: 35K99; 97M30

1 Introduction

Let Ω ⊂ R N ( N ≥ 2 ) be a smooth bounded domain and T > 0 . Consider the variation-inequality problems

(1) min { L u , u − u 0 } = 0 , ( x , t ) ∈ Ω T ,

with the p ( x ) -Laplacian Kirchhoff operators,

L u = ∂ t u − u div ( ( 1 + ‖ ∇ u ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ∣ p ( x ) − 2 ∇ u ) − γ ∣ ∇ u ∣ p ( x )

and the initial-boundary value conditions

u ( 0 , x ) = u 0 ( x ) , x ∈ Ω ; u ( t , x ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) .

Throughout the article, the exponent p ( x ) is continuous on Ω and satisfies

2 ≤ inf x ∈ Ω p ( x ) = p − ≤ p ( x ) ≤ p + = sup x ∈ Ω p ( x ) < ∞ .

In finance, variational inequality (1) is often used to study the value of convertible bonds (for details, see [1–5]). Because the holders of convertible bonds can converting bonds into shares at any time to obtain benefits, the pricing of convertible bonds is essentially a variational inequality problem.

Theoretical research on variational inequalities have received considerable attention in the recent reference. The previous work [6] studies weak solutions of variational inequalities (1) with fourth-order p -Laplacian Kirchhoff operators,

L ϕ = ∂ t ϕ − Δ ( ( 1 + λ ‖ Δ ϕ ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ Δ ϕ ∣ p ( x ) − 2 Δ ϕ ) + γ ϕ ,

in which the existence and uniqueness of solution are studied by the penalty method (for details, see [7]). The author in [8] study 2D variational inequality systems

min { L i ϕ , ϕ i − ϕ i 0 } = 0 , ( x , t ) ∈ Q T , ϕ i ( 0 , x ) = ϕ i , 0 ( x ) , x ∈ Ω , ϕ i ( t , x ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) ,

that L i ϕ is a quasilinear degenerate parabolic operators,

L i ϕ i = ∂ t ϕ − Δ ( ∣ Δ ϕ ∣ p ( x ) − 2 Δ ϕ ) − f i ( x , t , ϕ 1 , ϕ 2 ) .

With the help of monotone iteration technique and the method of regularization to deal with the auxiliary problem, the existence and uniqueness of solution are also considered. Recently, several interesting existence results can be found in [9,10]. Their characteristic is that scholars used the mountain pass theorem to deal with the auxiliary problem.

In this article, we inspired by [1,2] study a kind of variation-inequality initial-boundary value problem with variable parameter Kirchhoff operators not in divergence form such that the parameter is spatial dependent. To overcome the inequality limitation of problem (1), we introduce a penalty function to turn problem (1) into an obstacle problem. Since [11–22] study the problem under the parabolic equation model, this article studies the problem of variation-inequality, which is the first innovation. Because of the coupled problem in Kirchhoff operator, it is unrealistic to obtain the norm boundedness of penalty problem. In this article, we construct an operator by Larey-Schauder lemma to prove the existence of solution, which is the second innovation. Further, some estimates obtained from the auxiliary problem constructed by Larey-Schauder lemma also prove the uniqueness and stability of the solution.

2 Statement of the problem and main results

This article consider a more general case, and we will study the existence, uniqueness, and stability of the solution of model (1). In doing so, we give the maximal monotone operator [1,2,23], which will be used throughout this article:

(2) G ( x ) = 0 , x > 0 , 1 , x = 0 .

Moreover, the following result will be used repeatedly [7].

Lemma 1

Define M ( s ) = ∣ s ∣ p ( x ) − 2 s , then for any ξ , η ∈ R N ,

( M ( ξ ) − M ( η ) ) ( ξ − η ) ≥ C ∣ ξ − η ∣ p ( x ) ,

where C is a positive constant depending on p + and p − .

Here, we do not expect problem (1) to have a classical solution, so the generalized solution of problem (1) is defined as follows.

Definition 1

A pair ( u , ξ ) is called a generalized solution of (1), if

u ∈ L ∞ ( 0 , T , W 1 , p ( x ) ( Ω ) ) , ∂ t u ∈ L 2 ( Ω T ) ,
u ( x , t ) ≥ u 0 ( x ) , u ( x , 0 ) = u 0 ( x ) for any ( x , t ) ∈ Ω T ,
ξ ∈ G ( u − u 0 ) for any ( x , t ) ∈ Ω T ,
for any φ ∈ C 1 ( Ω ¯ T ) ,
∫ ∫ Ω T ∂ t u φ + u ( 1 + λ ‖ ∇ u ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ∣ p ( x ) − 2 ∇ u ∇ φ d x d t + ∫ ∫ Ω T ( 1 − γ + λ ‖ ∇ u ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ∣ p ( x ) φ d x d t = ∫ 0 t ∫ Ω ξ φ d x d t .

Since ‖ ∇ u ‖ L p ( x ) ( Ω ) p ( x ) and ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε are coupled in L u . We cannot use Minkowski inequality as usual, in proving the existence. In doing so, we will use the Leray-Schauder principle, first define a map

(3) M : L ∞ ( 0 , T ; W 0 1 , p ( x ) ( Ω ) ) × [ 0 , 1 ] → L ∞ ( 0 , T ; W 0 1 , p ( x ) ( Ω ) ) ,

for any ω ∈ L ∞ ( 0 , T ; W 0 1 , p ( x ) ( Ω ) ) and fixed θ ∈ [ 0 , 1 ] , u = M ( ω , θ ) is a solution of the equation

(4) L ε θ , ω u ε + g ( u ε ) = − θ β ε ( u ε − u 0 ) , ( x , t ) ∈ Ω T , u ε ( x , 0 ) = u 0 ε ( x ) = u 0 + ε , x ∈ Ω , u ε ( x , t ) = ε , ( x , t ) ∈ ∂ Ω T ,

with an operator

L ε θ , ω u ε = ∂ t u ε − u ε div ( ( 1 + θ λ ‖ ∇ ω ε ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε ) − γ ∣ ∇ u ∣ p ( x ) .

Here, the penalty function β ε ( ⋅ ) satisfies

(5) ε ∈ ( 0 , 1 ) , β ε ( ⋅ ) ∈ C 2 ( R ) , β ε ( x ) ≤ 0 , β ε ′ ( x ) ≥ 0 , β ε ″ ( x ) ≤ 0 , β ε ( x ) = 0 , x ≥ ε , − 1 2 , x = 0 , lim ε → 0 + β ( x ) = 0 , x > 0 , − 1 2 , x = 0 .

So the existence of problem (1) is equivalent to the existence of the fixed point of operator M ( ⋅ , 1 ) in L ∞ ( 0 , T ; W 0 1 , p ( x ) ( Ω ) ) .

3 Some preliminaries

This section gives some useful preliminaries to prove the main results. From [5], problem (4) admits a unique solution u ε which satisfies

(6) ∫ Ω ∣ ∇ u ε ∣ p ( x ) d x + ∫ Ω u ε 2 d x ≤ C , ∀ t ∈ ( 0 , T ) ,

(7) ∥ ∂ t u ε ∥ L 2 ( Q T ) ≤ C ( p , T , ∣ Ω ∣ ) .

In fact, multiplying the first line of (4) by u ε and integrating the value over Ω t ,

(8) ∫ 0 t ∫ Ω u ε ( 1 + λ ‖ ∇ ω ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ε ∣ p ( x ) + ( 1 − γ + λ ‖ ∇ ω ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ε ∣ p ( x ) d x d τ = − θ ∫ 0 t ∫ Ω β ε ( u ε − u 0 ) u ε d x d τ − ∫ 0 t ∫ Ω ∂ τ u ε u ε d x d τ .

Using Cauchy and Young inequalities, (10) follows immediately. Further, by replacing u ε with ∂ t u ε , (8) can be changed into

∫ 0 t ∫ Ω ∣ ∂ t u ε ∣ 2 d x d t = − A 1 − A 2 + A 3 ,

where

A 1 = ∫ 0 t ∫ Ω u ε ( 1 + λ ‖ ∇ ω ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε ∇ ∂ t u ε d x d t , A 2 = ∫ 0 t ∫ Ω T ( 1 − γ + λ ‖ ∇ ω ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ε ∣ p ( x ) ∂ t u ε d x d t , A 3 = − θ ∫ 0 t ∫ Ω T β ε ( u ε − u 0 ) ∂ t u ε d x d t .

One, using Cauchy, Holder, and Young inequalities, can prove (7) by amplifying the values of A 1 , A 2 , and A 3 . Following the similar way of [1,2], we also give the following inequalities:

(9) u 0 ε ≤ u ε ≤ ∣ u 0 ∣ ∞ + ε ,

(10) u ε 1 ≤ u ε 2 for ε 1 ≤ ε 2 .

From (6) and (9), one, using Leray-Schauder theorem, can infer that

(11) deg ( I − G ( 1 , θ ) , B R , 0 ) = 1 ,

where B R denotes the ball with radius R centred at the origin. And, there exists a small positive constant r satisfying r < R

deg ( I − G ( 1 , θ ) , B r , 0 ) = 0 ,

which were used extensively in [5,7]. One can deduce from (11) and aforementioned equation that

(12) deg ( I − G ( 1 , θ ) , B R \ B r , 0 ) = 0 .

Lemma 2

The map M : L ∞ ( 0 , T ; W 0 1 , p ( x ) ( Ω ) ) × [ 0 , 1 ] → L ∞ ( 0 , T ; W 0 1 , p ( x ) ( Ω ) ) is compact.

Proof

Note that u ε ∈ L ∞ ( 0 , T ; W 1 , p ( x ) ( Ω ) ) , ∂ t u ε ∈ L ∞ ( 0 , T ; L 2 ( Q ) ) from (6) and (7). Recall that L 2 ( Ω ) ⊂ W 1 , p ( x ) ( Ω ) is compact, and W 1 , p ( x ) ( Ω ) is continuous. Thus, Lemma 3.4 from Aubin-Lions [5, 7] follows.

Following a similar way in [6,12], we infer that problem

(13) ∂ t u ε − ∇ ( ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε ) − γ u ε + y = 0 ,

with Dirichlet boundary condition, admits a unique solution in L ∞ ( 0 , T ; W 0 1 , p ( x ) ( Ω ) ) for any y ∈ L ∞ ( 0 , T ; W 0 1 , p ( x ) ( Ω ) ) . By the properties of the Leray-Schauder degree, together with (11)–(13) and Lemmas 3.1, the following initial boundary problem

(14) L u ε = − β ε ( u ε − u 0 ) , ( x , t ) ∈ Q T , u ε ( x , 0 ) = u 0 ε ( x ) , x ∈ Ω , u ε ( x , t ) = ε , ( x , t ) ∈ ∂ Q T ,

also has a solution u ε ∈ L ∞ ( 0 , T ; W 1 , p ( x ) ( Ω ) ) , ∂ t u ε ∈ L 2 ( Ω T ) which satisfies the following equation:

∫ ∫ Ω T ( ∂ t u ε φ + ( 1 + λ ‖ ∇ u ε ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε ∇ φ + γ u ε φ ) d x d t = − ∫ ∫ Ω β ε ( u ε − u 0 ) φ d x d t ,

with φ ∈ C 1 ( Ω ¯ T ) . Furthermore, the solution of (14) satisfies (9) and (10), that is,

(15) u 0 ε ≤ u ε ≤ ∣ u 0 ∣ ∞ + ε , u ε 1 ≤ u ε 2 for ε 1 ≤ ε 2 .

4 Proof of the existence

This section is devoted to the existence of generalized solution. Hence, by Lemmas 3.2 and 3.3, we see that there exists a function u ∈ L ∞ ( 0 , T ; W 1 , p ( x ) ( Ω ) ) and a subsequence (without loss of generality, one denote it by { u ε , ε ≥ 0 } ), such that as ε → 0

(16) u ε ⟶ weak u in L ∞ ( 0 , T ; W 0 1 , p ( x ) ( Ω ) ) , ∂ t u ε ⟶ weak ∂ t u in L 2 ( Ω T ) ,

where ⟶ weak stands for weak convergence.

Lemma 3

For any fixed t ∈ ( 0 , T ) , the following estimate holds

∫ Ω ∣ u ε − u ∣ p ( x ) d x → 0 as ε → 0 .

Proof

In formula (8), replace u ε with u ε − 1 ( u ε − ε − u ε ) to obtain

(17) ∫ Ω ∂ t u ε u ε − 1 ( u ε − ε − u ε ) d x = ∫ Ω u ε ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε ∇ [ u ε − 1 ( u ε − ε − u ) ] d x + ( γ − 1 ) ∫ Ω u ε − 1 ∣ ∇ u ε ∣ p ( x ) ( u ε − ε − u ) d x .

By suing progressive integral, (17) becomes

∫ Ω ∂ t u ε u ε − 1 ( u ε − ε − u ε ) d x = ∫ Ω ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε ∇ ( u ε − u ) d x + γ ∫ Ω u ε − 1 ∣ ∇ u ε ∣ p ( x ) ( u ε − ε − u ) d x .

From (15), ∫ Ω u ε − 1 ∣ ∇ u ε ∣ p ( x ) ( u ε − ε − u ) d x is not positive, so one can obtain

(18) ∫ Ω ∂ t u ε u ε − 1 ( u ε − ε − u ε ) d x ≤ − ∫ Ω ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε ∇ ( u ε − u ) d x .

Thus, by adding ∫ Ω ( ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε − ∣ ∇ u ∣ p ( x ) − 2 ∇ u ) ∇ ( u ε − u ) d x from both sides of (18), it can be easily verified that

(19) ∫ Ω ∂ t u ε u ε − 1 ( u ε − ε − u ε ) d x + ∫ Ω ( ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε − ∣ ∇ u ∣ p ( x ) − 2 ∇ u ) ∇ ( u ε − u ) d x ≤ − ∫ Ω ∣ ∇ u ∣ p ( x ) − 2 ∇ u ∇ ( u ε − u ) d x .

Using Cauchy inequality, it follows from (11), (10), and (16) that as ε → 0 ,

(20) ∫ Ω ∂ t u ε u ε − 1 ( u ε − ε − u ε ) d x ≤ C ∫ Ω ∣ ∂ t u ε ∣ 2 d x ∫ Ω ( u ε − ε − u ε ) 2 d x → 0 .

Using Holder inequality with parameters p ( x ) − 2 p ( x ) and 2 p ( x ) , one, from (6), can obtain

(21) ∫ Ω ∣ ∇ u ∣ p ( x ) − 2 ∇ u ∇ ( u ε − u ) d x ≤ ∫ Ω ∣ ∇ u ∣ p ( x ) d x p + p − − 1 ∫ Ω ∣ ∇ u ε − ∇ u ∣ p ( x ) d x 1 p − − 1 → 0 ( ε → 0 ) .

By combining (19), (20), and (21), it is easy to see that

(22) lim ε → 0 ∫ Ω ( ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε − ∣ ∇ u ∣ p ( x ) − 2 ∇ u ) ∇ ( u ε − u ) d x ≤ 0 .

From Lemma 2.1, we know that for any fixed ε ∈ ( 0 , 1 )

(23) ( ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε − ∣ ∇ u ∣ p ( x ) − 2 ∇ u ) ∇ ( u ε − u ) ≥ C ( p + , p − ) ∣ ∇ u ε − ∇ u ∣ p ( x ) .

This combining with (22) implies that Lemma 4.1 follows.□

Lemma 4

For any fixed t ∈ ( 0 , T ) , u ε ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε converges to u ∣ ∇ u ∣ p ( x ) − 2 ∇ u in L 1 ( Ω ) as ε → 0 , that is, to say,

(24) ∫ Ω ∣ u ε ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε − u ∣ ∇ u ∣ p ( x ) − 2 ∇ u ∣ d x → 0 .

Proof

We first analyze

(25) ∫ Ω u ε ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε − u ∣ ∇ u ∣ p ( x ) − 2 ∇ u d x ≤ I 1 + I 2 ,

where

A 4 = ∫ Ω ∣ u ε − u ∣ ∣ ∇ u ε ∣ p ( x ) − 1 d x , A 5 = ∫ Ω u ∣ ∣ ∇ u ε ∣ p ( x ) − 1 − ∣ ∇ u ∣ p ( x ) − 1 ∣ d x .

By using Holder inequality, it follows from (6) that as ε → 0 ,

(26) A 4 ≤ ∫ ∫ Q T ∣ ∇ u ε ∣ p ( x ) d x d t p − − 1 p + ∫ ∫ Q T ∣ u ε − u ∣ p ( x ) d x d t 1 p + → 0 .

By using trigonometric inequality, one from Lemma 4.1 obtains

(27) A 5 ≤ C ( u 0 ) ∫ Ω ∣ ∇ u ε − ∇ u ∣ p ( x ) − 1 d x ≤ ∥ ∇ u ε − ∇ u ∥ L p ( x ) ( Ω ) p − − 1 → 0 ( ε → 0 ) .

By combining (25)–(27), we derive (24) immediately.□

From Lemmas 4.1 and 4.2, we also know that as ε → 0 ,

( 1 + λ ‖ ∇ u ε ‖ L p ( x ) ( Ω ) p ( x ) ) u ε ∣ ∇ u ε ∣ p ( x ) − 2 ∇ u ε → ( 1 + λ ‖ ∇ u ‖ L p ( x ) ( Ω ) p ( x ) ) u ∣ ∇ u ∣ p ( x ) − 2 ∇ u ,

with norm ∥ ∥ L 1 ( Ω ) for any fixed t ∈ ( 0 , T ] . In fact, we can construct trigonometric inequality to complete the proof as in (25)–(27), which is omitted here. Further, following the similar way of Lemma 4.1 in [12], one obtains the following result

(28) − β ε ( u ε − u 0 ) → ξ ∈ G ( u − u 0 ) as ε → 0

for all ( x , t ) ∈ Ω T . Moreover, passing the limit ε → 0 in (4) and (9), it is easy to verify that

(29) u ( x , 0 ) = u 0 ( x ) in Ω , u ( x , t ) ≥ u 0 ( x ) in Ω T .

Combining Lemma 4.1 and (29), we infer that ( u , ξ ) satisfies the initial and boundary condition and the integrating expression. Thus ( u , ξ ) is a generalized solution of (1), so we summarize the following theorem.

Theorem 1

Assume that u 0 ∈ W 1 , p ( x ) ( Ω ) , γ ∈ [ 0 , 1 ) . Then (1) admits a solution u within the class of Definition 2.1.

5 Proof of the stability and uniqueness

In this section, we study the stability and uniqueness of generalized solution and consider a generalized solution ( u i , ξ i ) of (1) with two different initial conditions

u ( 0 , x ) = u 0 , i ( x ) , x ∈ Ω , i = 1 , 2 .

Note that λ = 0 in this section and define φ = u 1 − u 2 . We, from Definition 2.1, infer that

(30) ∫ Ω ∂ t φ φ + ( u 1 ∣ ∇ u 1 ∣ p ( x ) − 2 ∇ u 1 − u 2 ∣ ∇ u 2 ∣ p ( x ) − 2 ∇ u 2 ) ∇ φ d x + ( 1 − γ ) ∫ Ω ( ∣ ∇ u 1 ∣ p ( x ) − ∣ ∇ u 2 ∣ p ( x ) ) φ d x = ∫ 0 t ∫ Ω ( ξ 1 − ξ 2 ) φ d x d t .

Recall that u ∣ ∇ u ∣ p ( x ) − 2 ∇ u is a strictly increased function with u , such that for any fixed x ∈ Ω

u 1 ∣ ∇ u 1 ∣ p ( x ) − 2 ∇ u 1 − u 2 ∣ ∇ u 2 ∣ p ( x ) − 2 ∇ u 2 ≥ min { A 6 , A 7 } ,

where

A 6 = u 1 ∣ ∇ u 1 ∣ p + − 2 ∇ u 1 − u 2 ∣ ∇ u 2 ∣ p + − 2 ∇ u 2 , A 7 = u 1 ∣ ∇ u 1 ∣ p − − 2 ∇ u 1 − u 2 ∣ ∇ u 2 ∣ p − − 2 ∇ u 2 .

So we can choose an appropriate one between p + and p − (denote by ρ ) to ensure

u 1 ∣ ∇ u 1 ∣ p ( x ) − 2 ∇ u 1 − u 2 ∣ ∇ u 2 ∣ p ( x ) − 2 ∇ u 2 ≥ u 1 ∣ ∇ u 1 ∣ ρ − 2 ∇ u 1 − u 2 ∣ ∇ u 2 ∣ ρ − 2 ∇ u 2 .

It is difficult to reduce the value of the left hand side of (30) through

∫ Ω ( ∣ ∇ u 1 ∣ p ( x ) − ∣ ∇ u 2 ∣ p ( x ) ) φ d x .

In doing so, define

sgn δ ( z ) = sgn ( z ) , x ∉ [ − δ , δ ] , sin ( z / z δ ) , x ∈ [ − δ , δ ] , δ > 0 , f ( x ) = ln x , γ = 0 , γ − 1 x γ , γ > 0 ,

g ( z ) = z 1 + γ / ( ρ − 1 ) [ 1 + γ / ( ρ − 1 ) ] − 1 .

Since u 1 and u 2 are in L ∞ ( 0 , T ; W 0 1 , p ( x ) ( Ω ) ) , one obtains

( u 1 1 − γ − u 2 1 − γ ) + sgn δ ( g ( u 1 ) − g ( u 2 ) ) + ∈ L ∞ ( 0 , T ; W 0 1 , p ( x ) ( Ω ) ) .

Replacing ( u 1 1 − γ − u 2 1 − γ ) + sgn δ ( g ( u 1 ) − g ( u 2 ) ) + with φ in (30), we infer that

A 8 + ∫ Ω ∂ t ( f ( u 1 ) − f ( u 2 ) ) sgn δ ( g ( u 1 ) − g ( u 2 ) ) + d x ≤ ∫ Ω ( ξ 1 − ξ 2 ) ( u 1 1 − γ − u 2 1 − γ ) + sgn δ ( g ( u 1 ) − g ( u 2 ) ) + d x ,

where

A 8 = ∫ Ω ( ∣ ∇ g ( u 1 ) ∣ p ( x ) − 2 ∇ g ( u 1 ) − ∣ ∇ g ( u 2 ) ∣ p ( x ) − 2 ∇ g ( u 2 ) ) ∇ ( g ( u 1 ) − g ( u 2 ) ) + sgn δ ′ ( g ( u 1 ) − g ( u 2 ) ) + d x .

From (23), it is easy to verify that

1 ≥ sgn δ ′ ( g ( u 1 ) − g ( u 2 ) ) + ≥ 0 , ∫ Ω ( ∣ ∇ g ( u 1 ) ∣ p ( x ) − 2 ∇ g ( u 1 ) − ∣ ∇ g ( u 2 ) ∣ p ( x ) − 2 ∇ g ( u 2 ) ) ∇ ( g ( u 1 ) − g ( u 2 ) ) + ≥ 0 .

Combining the two inequalities above implies that A 8 ≥ 0 . Now, we consider

∫ Ω ( ξ 1 − ξ 2 ) ( u 1 1 − γ − u 2 1 − γ ) + sgn δ ( u 1 − u 2 ) + d x

and give the following result

(31) ∫ Ω ( ξ 1 − ξ 2 ) ( u 1 1 − γ − u 2 1 − γ ) + sgn δ ( u 1 − u 2 ) + d x ≤ 0 , ∀ t ∈ [ 0 , T ] .

Indeed, if u 1 ( x , t ) > u 2 ( x , t ) , then u 1 ( x , t ) > u 1 , 0 ( x ) which, combining with (28) and (29), infer that

ξ 1 = 0 ≤ ξ 2 , ( ξ 1 − ξ 2 ) φ = ( ξ 1 − ξ 2 ) ( u 1 − u 2 ) ≤ 0 .

At this time, sgn δ ( u 1 − u 2 ) + = 1 , then (31) follows. On the contrary, if u 1 ( x , t ) < u 2 ( x , t ) , sgn δ ( u 1 − u 2 ) + = 0 , (31) still holds. Further, we remove the non-negative term A 8 and non-positive term (31) to arrive at

∫ Ω ∂ t ( f ( u 1 ) − f ( u 2 ) ) sgn δ ( g ( u 1 ) − g ( u 2 ) ) + d x ≤ 0 .

By passing the limitation, one obtains

∫ Ω ∂ t ( f ( u 1 ) − f ( u 2 ) ) sgn ( g ( u 1 ) − g ( u 2 ) ) + d x ≤ 0 .

Note that both f and g are increasing functions, and then using

sgn ( ( g ( u 1 ) − g ( u 2 ) ) + ) = sgn ( ( f ( u 1 ) − f ( u 2 ) ) + ) ,

we infer that

∂ t ∫ Ω ∣ f ( u 1 ) − f ( u 2 ) ∣ d x ≤ 0 .

This implies that

(32) ∫ Ω ∣ f ( u 1 ) − f ( u 2 ) ∣ d x ≤ ∫ Ω ∣ f ( u 1 , 0 ) − f ( u 2 , 0 ) ∣ d x .

Final, we analyze the uniqueness of solution. Suppose that there are at least two solutions ( u 1 , ξ 1 ) and ( u 2 , ξ 2 ) with initial condition

u ( 0 , x ) = u 0 ( x ) , x ∈ Ω .

By choosing u 0 , 1 ( x ) = u 0 , 2 ( x ) in (32), uniqueness of solution can be found easily. Thus, we summarize the following theorem.

Theorem 2

Let ( u i , ξ i ) be a generalized solution of (1) with u ( 0 , x ) = u 0 , i ( x ) , x ∈ Ω , i = 1 , 2 . If λ = 0 ,

‖ u 1 γ − u 2 γ ∣ ∣ L 1 ( Ω ) ≤ ‖ u 0 , 1 γ ( ⋅ ) − u 0 , 2 γ ( ⋅ ) ∣ ∣ L 1 ( Ω ) , γ ∈ ( 0 , 1 ) , ‖ ln u 1 − ln u 2 ∣ ∣ L 2 ( 0 , T ; W 1 , p ( x ) Ω T ) ≤ C ‖ ln u 0 , 1 ( ⋅ ) − ln u 0 , 2 ( ⋅ ) ∣ ∣ L 2 ( Ω ) , γ = 0 .

Furthermore, problem (1) admits a unique solution in the sense of Definition 2.1.

6 Numerical illustration

Here, we discretize problems (20)–(22) on the time-space domain and assume Ω = [ 0 , 1 ] . Define uniform time and space mesh:

x i = i × h , h = 1 M , i = 0 , 1 , 2 , … , M , t k = k × Δ t , Δ t = 1 N , k = 0 , 1 , 2 , … , N ,

where M and N are integers. The derivative ∂ x u and ∂ t u can be approximated at point ( x i , t k ) with notation u i k = u ( x i , t k ) ,

∂ x u ∣ ( x i , t k ) = δ x u i k + O ( h ) = u i + 1 k − u i − 1 k 2 h + O ( h 2 ) , ∂ t u ∣ ( x i , t k ) = δ t u i k + O ( Δ t ) = u i k − u i k − 1 Δ t + O ( Δ t ) .

By neglecting the truncation errors, we obtain the following estimate of ‖ ∂ x u ‖ L p ( x ) ( Ω ) p ( x ) at point ( x i , t k ) ,

(33) ‖ δ x u ‖ L p ( x ) ( Ω ) p ( x ) = h ∑ i = 1 M ∣ δ x u i k ∣ p ( x i ) .

Define F = u div ( ( 1 + ‖ ∇ u ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ∣ p ( x ) − 2 ∇ u ) − γ ∣ ∇ u ∣ p ( x ) and F can be formulated at point ( x i , t k ) as follows:

(34) F i k = u i k ( 1 + ‖ δ x u i k ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ h u i k ∣ p ( x i ) − 2 ∇ h u i k − ∣ ∇ h u i − 1 k ∣ p ( x i ) − 2 ∇ h u i − 1 k h + γ ∣ ∇ h u i k ∣ p ( x i ) .

By inserting (33) and (34) into L u = 0 , one obtains the following difference scheme

δ t u i k = F k − 1 .

By rearranging the aforementioned difference scheme, one can infer that

(35) u i k = Δ t F k − 1 + u i k − 1 , k = 1 , 2 , … , M .

According to document [1,2,12], the difference scheme of variation-inequality (1) satisfies

(36) u i k = max { Δ t F k − 1 + u i k − 1 , u i 0 } , k = 1 , 2 , … , N .

Now we present some numerical experiments for variation-inequality (1) based on scheme (36). We first consider the stability of variation-inequality (1) at p ( x ) = 3 , u ( x , 0 ) = sin ( π x ) , T = 1 , N = 10 , M = 100 . Figure 1 shows the simulation results at x = 0.5 with initial error 0 , 0.1 , 0.2 , 0.3 , and 0.4. It can be seen from Figure 1 that the initial error gradually decreases with time. This shows that variational inequality (1) is stable with respect to initial value, which is consistent with Theorem 5.1. Figure 2 shows the effect of different values of p on the simulation results. It can be seen that with the increase of p , the peak value of simulation results decreases gradually.

$Figure 1 Simulation results of u u under different initial errors.$

Figure 1

Simulation results of u under different initial errors.

$Figure 2 Simulation results of u u under different values of p p .$

Figure 2

Simulation results of u under different values of p .

7 Conclusion

In this article, we study the variation-inequality problems

min { L u , u − u 0 } = 0 , ( x , t ) ∈ Ω T ,

with the p ( x ) -Laplacian Kirchhoff operators

L u = ∂ t u − u div ( ( 1 + ‖ ∇ u ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ∣ p ( x ) − 2 ∇ u ) − γ ∣ ∇ u ∣ p ( x ) .

Note that there is a minimum operator in problem (1), which makes it difficult to construct norm estimates. Therefore, we construct a penalty function to make problem (1) turn into an obstacle problem.

Since 1 + ‖ ∇ u ‖ L p ( x ) ( Ω ) p ( x ) and ∣ ∇ u ∣ p ( x ) − 2 ∇ u are coupled in Kirchhoff operator, it is hard to obtain the norm estimate of penalty problem. In this article, we construct an operator suitable for Larey-Schauder lemma to prove the existence of solution and prove the existence, uniqueness and stability of solution to problem (1).

So far, this article has used some strong assumptions. In proving Lemma 4.2, we used Holder inequality, which requires that p ( x ) > 2 . When 1 < p ( x ) < 2 , the conclusion cannot be guaranteed. On the another hand, u 1 − λ is required to be an increasing function with respect to u in section 5. At this time, it can only be assumed that γ ≤ 1 . According to [5], u ε ( 1 + λ ‖ ∇ ω ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ε ∣ p ( x ) and ( 1 − γ + λ ‖ ∇ ω ‖ L p ( x ) ( Ω ) p ( x ) ) ∣ ∇ u ε ∣ p ( x ) of formula (9) must be positive, so we need the restriction that γ ≥ 0 . In the future, we will continue to study this problem and consider gradually removing these restrictions.

Acknowledgment

Author has sincerely grateful to the reference and the associate editor handling the article for their valuable comments.

Funding information: This work was supported by the Key R&D Projects of Weinan Science and Technology Bureau (No. 2020ZDYF-JCYJ-162).
Author contributions: This is a single-author article. The author read and approved the final manuscript.
Conflict of interest: Author states no conflict of interest.
Data availability statement: No applicable.

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Received: 2022-12-08

Revised: 2023-02-04

Accepted: 2023-02-06

Published Online: 2023-03-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/math-2022-0565

Schlagwörter für diesen Artikel

parabolic variation-inequality problems; weak solution; variable parameter Kirchhoff operators not in divergence form; existence; uniqueness; stability

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