In this article, we investigate the spatial decay estimates for the biharmonic conduction equations within a coupled wave-plate system incorporating thermal effects in a two-dimensional cylindrical domain. Using the method of a second-order differential inequality, we can obtain the spatial decay estimates result for these equations. When the distance tends to infinity, the energy can decay exponentially. This result shows us that the Saint-Venant principle is also valid for the hyperbolic-parabolic coupled system.
Citation: Jincheng Shi, Yan Liu. Spatial decay estimates for the coupled system of wave-plate type with thermal effect[J]. AIMS Mathematics, 2025, 10(1): 338-352. doi: 10.3934/math.2025016
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In this article, we investigate the spatial decay estimates for the biharmonic conduction equations within a coupled wave-plate system incorporating thermal effects in a two-dimensional cylindrical domain. Using the method of a second-order differential inequality, we can obtain the spatial decay estimates result for these equations. When the distance tends to infinity, the energy can decay exponentially. This result shows us that the Saint-Venant principle is also valid for the hyperbolic-parabolic coupled system.
Over the last fifty years, many authors studied the Saint-Venant principle in both applied mathematics and mechanics. The results of the classical Saint-Venant principle have been greatly expanded by a large number of investigations. In order to track the results about the Saint-Venant's principle, one could see articles by Horgan [1,2], Horgan and Knowles [3]. The Saint-Venant type theorem states that the energy expression can decay exponentially when the axial distance from the near end to infinity along a semi-infinite strip or cylinder for different type of equations. For example, in [4], the authors studied the spatial behavior for the high order equation. In [5], the the authors studied the spatial behavior for the transient heat conduction. In [6], the partial behavior for the primitive equations was studied. In [7, 8], the authors studied the spatial behavior for the fluid flow in porous medium. In order to obtain the decay results, they must impose a priori assumptions that the solutions algebraically decay to zero at infinity.
In recent years, the biharmonic equation is used to describe the behavior of a two-dimensional physical field within a plane. It can represent many different physical phenomenas, including sound waves, electric fields, and magnetic fields. Many important applications are studied in applied mathematics and mechanics. In order to obtain the Saint-Venant type result for the biharmonic equations, many studies and various methods have been proposed for researching the spatial behavior for the solutions of the biharmonic equations in a semi-infinite strip in R2. We mention the studies by Knowles [9,10], Flavin [11], Flavin and Knops [12], and Horgan [13]. We note that some time-dependent problems concerning the biharmonic operator were considered in the literature. We mention the papers by Liu and Lin [14], Knops and Lupoli [15], and Song [16,17] in connection with the spatial behavior of solutions for a fourth-order transformed problem associated with the slow flow of an incompressible viscous fluid along a semi-infinite strip. Other results for the Saint-Venant principle may be found in [18,19,20,21,22].
In [23], the authors studied the properties of solutions for the wave plate type. The equations were a coupled system with a thermal effect. They obtained the analytic property. The exponential stability was also obtained by using the method of a C0-semigroup. The equations have the following form:
{ρ1u,tt−△u−μ△u,t+a△v=0,ρ2v,tt+γ△2v+a△u+m△θ=0,τθ,t−k△θ−m△v,t=0. | (1.1) |
The above system was used to describe the system constituted by an elastic membrane and an elastic plate that are subject to a thermal effect (see [24]). Here, u represents the vertical deflection of the membrane, v represents the vertical deflection of the plate, and θ represents the difference of temperature. ρ1, ρ2, μ, a, γ, m, τ, and k are all nonnegative coefficients.
In the present paper, we take a=0. This is to say, the vertical deflection of the membrane does not have any effect on the system. Equation (1.1) turns to
{ρv,tt+γΔ2v+mΔθ=0,τθ,t−kΔθ−mΔv,t=0. | (1.2) |
Here △ is the harmonic operator, and △2 is the biharmonic operator. The comma is used to indicate partial differentiation and the differentiation, with respect to the direction xk is denoted as ,k, thus u,α denotes ∂u∂xα, and u,t denotes ∂u∂t.
Our problem is considered on the domain Ω0 which is an unbounded region defined by
Ω0:={(x1,x2)∣x1>0,0<x2<h}, | (1.3) |
with h being a fixed positive constant. We denote the notation
Lz={(x1,x2)∣x1=z≥0,0≤x2≤h}. | (1.4) |
The problem is considered in the time interval [0,T], where T is a fixed positive constant.
We must add some a priori asymptotic decay assumptions for solutions at infinity.
v(x1,x2,t),˙v(x1,x2,t),v,α(x1,x2,t),θ(x1,x2,t),θ,1(x1,x2,t)→0,v,αt(x1,x2,t),v,αβ(x1,x2,t),v,αββ(x1,x2,t)→0,(uniformlyinx2)asx1→∞. | (1.5) |
In this paper, we study the spatial decay estimates for the system (1.2). Since equations of (1.2) are hyperbolic-parabolic coupled equations, it is difficult to construct the energy function. How to control the energy function by its own differentiation will be the main difficulty of this article. We have never seen any results about the Saint-Venant principle for system (1.2). If we follow the previous method that the energy function is controlled by its own derivative, we cannot obtain the desired result for this system of equations. The weighted energy method will be used, and a second-order differential inequality will be derived. This method is firstly used in current research on the Saint-Venant principle. We think this method is applicable to the study of other biharmonic operators. From these points, the result obtained in this paper is new and interesting.
In this paper, we are concerned with the spatial decay estimates for the coupled system of wave-plate type with a thermal effect. We formulate some energy expressions in section 2. In section 3, we derive some important inequalities and formulate a second-order differential inequality. We derive our main spatial decay estimates for the solutions in section 4. The usual summation convention is employed with repeated Greek subscripts α summed from 1 to 2. Hence,
u,αα=2∑α=1∂2u∂x2α. |
A is an area element on the x1−x2 plane, dA=dx2dξ.
In the following, we will define some energy functions that will be used in deriving our result.
Multiplying both sides of (1.2)1 by exp(−ωη)v,η(ξ−z) and integrating, we have
0=∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,η(ρv,ηη+γv,ααββ+mθ,αα)dAdη=ω2ρ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v2,ηdAdη+ρ2∫∞z∫Lξexp(−ωt)(ξ−z)v2,tdA−γ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αββdAdη−γ∫t0∫∞z∫Lξexp(−ωη)v,ηv,1ββdAdη−m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηθ,αdAdη−∫t0∫∞z∫Lξexp(−ωη)v,ηθ,1dAdη=ωρ2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v2,ηdAdη+ρ2∫∞z∫Lξexp(−ωt)(ξ−z)v2,tdA+γω2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αβv,αβdAdη+γ∫t0∫∞z∫Lξexp(−ωη)v,αηv,α1dAdη−γ∫t0∫∞z∫Lξexp(−ωη)v,ηv,1ββdAdη−m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηθ,αdAdη−m∫t0∫∞z∫Lξexp(−ωη)v,ηθ,1dAdη+γ2∫∞z∫Lξexp(−ωt)(ξ−z)v,αβv,αβdA. | (2.1) |
We define a function
F1(z,t)=ωρ2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v2,ηdAdη+ρ2∫∞z∫Lξexp(−ωt)(ξ−z)v2,tdA+γω2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αβv,αβdAdη+γ2∫∞z∫Lξexp(−ωt)(ξ−z)v,αβv,αβdA−m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηθ,αdAdη. | (2.2) |
Inserting (2.2) into (2.1), we have
F1(z,t)=−γ∫t0∫∞z∫Lξexp(−ωη)v,αηv,α1dAdη+γ∫t0∫∞z∫Lξexp(−ωη)v,ηv,1ββdAdη+m∫t0∫∞z∫Lξexp(−ωη)v,ηθ,1dAdη. | (2.3) |
Multiplying both sides of (1.2)2 by exp(−ωη)(ξ−z)v,η and integrating, we have
0=∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,η(τθ,η−kθ,αα−mv,ααη)dAdη=τ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ηθ,ηdAdη+k∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηθ,αdAdη+k∫t0∫∞z∫Lξexp(−ωη)v,ηθ,1dAdη+m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αηdAdη−m∫t0∫∞z∫Lξexp(−ωη)v,ηv,1ηdAdη=τ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ηθ,ηdAdη+k∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηθ,αdAdη+k∫t0∫∞z∫Lξexp(−ωη)v,ηθ,1dAdη+m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αηdAdη+m2∫t0∫Lzexp(−ωη)v2,ηdx2dη. | (2.4) |
We define a function
F2(z,t)=τ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηθ,αdAdη+k∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ηθ,ηdAdη+m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αηdAdη. | (2.5) |
Inserting (2.5) into (2.4), we can also obtain
F2(z,t)=−k∫t0∫∞z∫Lξexp(−ωη)v,ηθ,1dAdη−m2∫t0∫Lzexp(−ωη)v2,ηdx2dη. | (2.6) |
Multiplying both sides of (1.2)1 by exp(−ωη)(ξ−z)θ and integrating,
0=∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ(ρv,ηη+γvααββ+mθαα)dAdη=−ρ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,ηv,ηdAdη+ρω∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θv,ηdAdη+ρ∫∞z∫Lξexp(−ωt)(ξ−z)θv,tdA−γ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αv,αββdAdη−γ∫t0∫∞z∫Lξexp(−ωη)θv,1ββdAdη−m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αθ,αdAdη−m∫t0∫∞z∫Lξexp(−ωη)θθ,1dAdη. | (2.7) |
We define a function
F3(z,t)=−ρ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,ηv,ηdAdη+ρ∫∞z∫Lξexp(−ωt)(ξ−z)θv,tdA+ωρ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θv,ηdAdη−m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αθ,αdAdη−γ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αv,αββdAdη. | (2.8) |
Inserting (2.8) into (2.7), we can also obtain another expression of F3(z,t).
F3(z,t)=γ∫t0∫∞z∫Lξexp(−ωη)θv,1ββdAdη+m2∫t0∫Lzexp(−ωη)θ2dx2dη. | (2.9) |
Multiplying both sides of (1.2)2 by exp(−ωη)(ξ−z)θ and integrating, we have
0=∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ(τθ,η−kθ,αα−mv,ααη)dAdη=τω2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ2dAdη+τ2∫∞z∫Lξexp(−ωt)(ξ−z)θ2dA+k∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αθ,αdAdη+k∫t0∫∞z∫Lξexp(−ωη)θθ,1dx2dη+m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αv,αηdAdη+m∫t0∫∞z∫Lξexp(−ωη)θv,1ηdAdη=τω2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ2dAdη+τ2∫∞z∫Lξexp(−ωt)(ξ−z)θ2dA+k∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αθ,αdAdη−k2∫t0∫Lzexp(−ωη)θ2dx2dη+m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αv,αηdAdη+m∫t0∫∞z∫Lξexp(−ωη)θv,1ηdAdη. | (2.10) |
We define a function
F4(z,t)=τω2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ2dAdη+τ2∫∞z∫Lξexp(−ωt)(ξ−z)θ2dA+k∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αθ,αdAdη+m∫t0∫∞z∫Lξexp(−ωη)θv,1ηdAdη+m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αv,αηdAdη. | (2.11) |
Inserting (2.11) into (2.10), we can obtain another expression of F4(z,t)
F4(z,t)=k2∫t0∫Lzexp(−ωη)θ2dx2dη. | (2.12) |
Multiplying both sides of (1.2)1 by exp(−ωη)(ξ−z)v,αα and integrating, we have
0=∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αα(ρv,ηη+γvααββ+mθββ)dAdη=−ρ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααηv,ηdAdη−m∫t0∫∞z∫Lξexp(−ωη)v,ααθ,βdAdη+ρ∫∞z∫Lξexp(−ωt)(ξ−z)v,ααv,ηdA−γ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβv,ααβdAdη−γ∫t0∫∞z∫Lξexp(−ωη)v,ααv,1ααdAdη−m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβθ,βdAdη+ρω∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααv,ηdAdη=ρ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αηdAdη+ρ∫t0∫∞z∫Lξexp(−ωη)v,1ηv,ηdAdη+ρω∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααv,ηdAdη+ρ∫∞z∫Lξexp(−ωt)(ξ−z)v,ααv,ηdA−γ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβv,ααβdAdη+γ2∫t0∫Lzexp(−ωη)v,ααv,ββdx2dη−m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβθ,βdAdη−m∫t0∫∞z∫Lξexp(−ωη)v,ααθ,βdAdη. | (2.13) |
We define a function
F5(z,t)=−ρ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αηdAdη−ρ∫∞z∫Lξexp(−ωt)(ξ−z)v,ααv,ηdA+γ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβv,ααβdAdη+m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβθ,βdAdη−ρω∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααv,ηdAdη. | (2.14) |
Inserting (2.14) into (2.13), we also have
F5(z,t)=−ρ2∫t0∫Lzexp(−ωη)v2,ηdx2dη+γ2∫t0∫Lzexp(−ωη)v,ααv,ββdx2dη−m∫t0∫∞z∫Lξexp(−ωη)v,ααθ,βdAdη. | (2.15) |
We now define a new function
F(z,t)=F1(z,t)+k1F2(z,t)+k1τρF3(z,t)+k2F4(z,t)+k3F5(z,t), | (2.16) |
with k1, k2, and k3 being positive constants that will be determined later.
A combination of (2.2), (2.5), (2.8), (2.11), (2.14), and (2.16) gives
F(z,t)=ωρ2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v2,ηdAdη+γω2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αβv,αβdAdη+ωρ2∫∞z∫Lξexp(−ωt)(ξ−z)v2,tdA+(k1k+m)∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηθ,αdAdη+~k1∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αηdAdη+~k2∫∞z∫Lξexp(−ωt)(ξ−z)θv,tdA+~k3ω∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θv,ηdAdη−~k4∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αv,αββdAdη+~k5∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αθ,αdAdη+k2τ2∫∞z∫Lξexp(−ωt)(ξ−z)θ2dA+~k6∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ2dAdη+~k7∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αv,αηdAdη+k2m∫t0∫∞z∫Lξexp(−ωη)θv,1ηdAdη−k3ρω∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααv,ηdAdη−k3ρ∫∞z∫Lξexp(−ωt)(ξ−z)v,ααv,ηdA+k3γ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβv,ααβdAdη+k3m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβθ,βdAdη+γ2∫∞z∫Lξexp(−ωt)(ξ−z)v,αβv,αβdA, | (2.17) |
with
~k1=(k1m−k3ρ),~k2=k1τρ,~k3=k1τρ,~k4=k1τργ, |
~k5=(k2k−k1τρm),~k6=k2τω2,~k7=k2m. |
From the definition of F(z,t) in (2.16), we can also get another expression of F(z,t) by combining Eqs (2.3), (2.6), (2.9), (2.12), and (2.15)
F(z,t)=−γ∫t0∫∞z∫Lξexp(−ωη)v,αηv,α1dAdη+γ∫t0∫∞z∫Lξexp(−ωη)v,ηv,1ββdAdη−(m+k)∫t0∫∞z∫Lξexp(−ωη)v,ηθ,1dAdη+k2∫t0∫Lzexp(−ωη)θ2dx2dη−(m+ρ)2∫t0∫Lzexp(−ωη)v2,ηdx2dη+γ∫t0∫∞z∫Lξexp(−ωη)θv,1ββdAdη+m2∫t0∫Lzexp(−ωη)θ2dx2dη−m∫t0∫∞z∫Lξexp(−ωη)v,ααθ,βdAdη+γ2∫t0∫Lzexp(−ωη)v,ααv,ββdx2dη. | (2.18) |
Equalities (2.17) and (2.18) will play important roles in deriving the main result of this paper in the next section.
We now begin to bound F(z,t) in (2.17).
Using the Schwarz inequality, the fourth term on the right side of (2.17) can be bounded by
|(k1k+m)∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηθ,αdAdη|≤k1m8∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αηdAdη+2(k1k+m)2k1m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αθ,αdAdη, | (3.1) |
the sixth term on the right side of (2.17) can be bounded by
|k1τρω∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θv,ηdAdη|≤k1τ2ρω∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ2dAdη+k1τ2ρω∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v2,ηdAdη, | (3.2) |
the seventh term on the right side of (2.17) can be bounded by
|k1τρ∫∞z∫Lξexp(−ωt)(ξ−z)θv,tdA|≤k1τ2ρ∫∞z∫Lξexp(−ωt)(ξ−z)θ2dA+k1τ2ρ∫∞z∫Lξexp(−ωt)(ξ−z)v2,tdA, | (3.3) |
the eighth term on the right side of (2.17) can be bounded by
|−k1τργ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αv,αββdAdη|≤k21τ2ρ2k3γ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αθ,αdAdη+k3γ4∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αββv,αββdAdη, | (3.4) |
the tenth term on the right side of (2.17) can be bounded by
|k2m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αv,αηdAdη|≤2k22mk1∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,αθ,αdAdη+k1m8∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αηdAdη, | (3.5) |
the eleventh term on the right side of (2.17) can be bounded by
|k2m∫t0∫∞z∫Lξexp(−ωη)θv,1ηdAdη|≤2(k2m)2k1m∫t0∫∞z∫Lξexp(−ωη)θ2dAdη+k1m8∫t0∫∞z∫Lξexp(−ωη)v2,1ηdAdη, | (3.6) |
the twelfth term on the right side of (2.17) can be bounded by
|k3ρω∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααv,ηdAdη|≤k3ρω2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααv,ββdAdη+k3ρω2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v2,ηdAdη, | (3.7) |
the thirteenth term on the right side of (2.17) can be bounded by
|k3ρ∫∞z∫Lξexp(−ωt)(ξ−z)v,ααv,tdA|≤k3ρ2∫∞z∫Lξexp(−ωt)(ξ−z)v,ααv,ββdA+k3ρ2∫∞z∫Lξexp(−ωt)(ξ−z)v2,tdA, | (3.8) |
and the fifteenth term on the right side of (2.17) can be bounded by
|k3m∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβθ,βdAdη|≤k3γ2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβv,ααβdAdη+k3m22γ∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,βθ,βdAdη. | (3.9) |
Combining (2.17) and (3.1)–(3.9), we obtain
F(z,t)≥(ωρ2−k3ωρ2−k1τω2ρ)∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v2,ηdAdη+(ρ2−τk12ρ−k3ρ2)∫∞z∫Lξexp(−ωt)(ξ−z)v2,tdA+(γω2−k3ρω2)∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αβv,αβdAdη+(γ2−k3ρ2)∫∞z∫Lξexp(−ωt)(ξ−z)v,αβv,αβdA+(3k1m4−k3ρ)∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αηdAdη+(k2τω2−k1τω2ρ−2(k2m)2k1m)∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ2dAdη+(k2τ2−k1τ2ρ)∫∞z∫Lξexp(−ωt)(ξ−z)θ2dA+˜k∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,βθ,βdAdη+k3γ4∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβv,ααβdAdη, | (3.10) |
with
˜k=(k2k−k1τmρ−2(k1k+m)2k1m−(k1τ)2ρ2k3γ−2k22mk1−k3m22γ). |
If we suggest
k>2(k1τmρ+2(k1k+m)2k1m+(k1τ)2ρ2k3γ+2k22mk1+k3m22γ)k2 |
and choose
k1=ρ24τ,k2=2k1ρ,k3=min{γ2ρ,14,mk14ρ},ω=16k2mk1τ, |
we have
F(z,t)≥ωρ4∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v2,ηdAdη+ρ4∫∞z∫Lξexp(−ωt)(ξ−z)v2,tdA+γω4∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αβv,αβdAdη+γ4∫∞z∫Lξexp(−ωt)(ξ−z)v,αβv,αβdA+k1m2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αηdAdη+k2τ4∫∞z∫Lξexp(−ωt)(ξ−z)θ2dA+k2τω8∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ2dAdη+k1k2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,βθ,βdAdη+k3γ4∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβv,ααβdAdη=G(z,t). | (3.11) |
In this part we will derive a second-order differential inequality to obtain our result.
Differentiating (2.17) with respect to z and using the same method as deriving (3.11), we have
−∂F(z,t)∂z≥ωρ4∫t0∫∞z∫Lξexp(−ωη)v2,ηdAdη+ρ4∫∞z∫Lξexp(−ωt)v2,tdA+γω4∫t0∫∞z∫Lξexp(−ωη)v,αβv,αβdAdη+γ4∫∞z∫Lξexp(−ωt)v,αβv,αβdA+k1m2∫t0∫∞z∫Lξexp(−ωη)v,αηv,αηdAdη+k2τω8∫t0∫∞z∫Lξexp(−ωη)θ2dAdη+k2τ4∫∞z∫Lξexp(−ωt)θ2dA+k1k2∫t0∫∞z∫Lξexp(−ωη)θ,βθ,βdAdη+k3γ4∫t0∫∞z∫Lξexp(−ωη)v,ααβv,ααβdAdη. | (3.12) |
Differentiating (2.17) again with respect to z, we also obtain
∂2F(z,t)∂z2≥ωρ4∫t0∫Lzexp(−ωη)v2,ηdx2dη+ρ4∫Lzexp(−ωt)v2,tdx2+γω4∫t0∫Lzexp(−ωη)v,αβv,αβdx2dη+γ4∫Lzexp(−ωt)v,αβv,αβdx2+k1m2∫t0∫Lzexp(−ωη)v,αηv,αηdx2dη+k2τω8∫t0∫Lzexp(−ωη)θ2dx2dη+k2τ4∫Lzexp(−ωt)θ2dx2+k1k2∫t0∫Lzexp(−ωη)θ,βθ,βdx2dη+k3γ4∫t0∫Lzexp(−ωη)v,ααβv,ααβdx2dη. | (3.13) |
Using the Schwarz inequality in (2.18) and combining (3.12) and (3.13), we can obtain
F(z,t)≤k4(−∂F(z,t)∂z)+k5∂2F(z,t)∂z2, | (3.14) |
with k4 and k5 being computable positive constants.
Inequality (3.14) is the key inequality that will be used in deriving our main result.
We will obtain the following theory in this paper.
Theorem 4.1. Let (u,v) be a classical solution (the solution is smooth and differentiable) of the initial boundary value problems (1.2)–(1.5). For the energy expression G(z,t) defined in (3.11), we can obtain the decay estimates
ωρ4∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v2,ηdAdη+ρ4∫∞z∫Lξexp(−ωt)(ξ−z)v2,tdA+γω4∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αβv,αβdAdη+γ4∫∞z∫Lξexp(−ωt)(ξ−z)v,αβv,αβdA+k1m2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,αηv,αηdAdη+k2τ4∫∞z∫Lξexp(−ωt)(ξ−z)θ2dA+k2τω8∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ2dAdη+k1k2∫t0∫∞z∫Lξexp(−ωη)(ξ−z)θ,βθ,βdAdη+k3γ4∫t0∫∞z∫Lξexp(−ωη)(ξ−z)v,ααβv,ααβdAdη≤F(0,t)e−k7z, | (4.1) |
where k7 is a positive constant that will be defined later.
Proof. We now rewrite (3.14) as the following inequality:
∂2F∂z2−k4k5∂F∂z−1k5≥0. | (4.2) |
Inequality (4.2) can be rewritten as
(∂∂z−k6)[∂φ(z,t)∂z+k7φ(z,t)]≥0, | (4.3) |
where k6 and k7 satisfy
k7−k6=−k4k5,k6k7=1k5. | (4.4) |
Solving (4.4), we have
k6=12(√(k4k5)2+4k5+k4k5),k7=12(√(k4k5)2+4k5−k4k5). |
Inequality (4.3) can be rewritten as
∂∂z[exp(−k6z)(∂φ(z,t)∂z+k7φ(z,t))]≥0. | (4.5) |
Integrating (4.5) from z to ∞, we have
∂φ(z,t)∂z+k7φ(z,t)≤0. | (4.6) |
Solving (4.6) and using (3.11), we can obtain the desired result (4.1).
Inequality (4.1) shows the spatial decay estimates result that the solutions can decay exponentially as the distance from the entry section tends to infinity. The result can be viewed as a version of Saint-Venant principle. Using the result (4.1), we can also obtain point-wise decay estimates for the solutions. This is the specific property for the biharmonic equation. Next, we will give a numerical simulation of solutions for these equations. What is more, the structural stability for these equations in an unbounded domain would be interesting. We will study it in another paper.
All authors of this article have contributed equally. All authors have read and approved the final version of the manuscript for publication.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The first author is supported by the Natural Science foundation of Guangzhou Huashang College(2024HSTS09). The second author is supported by Guangdong Natural Science foundation (2023A1515012044).
The authors declare there are no conflicts of interest.
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