Magneto-thermo-elasticity — large time behavior for linear systems^{∗}

Jaime E. Muñoz Rivera and Reinhard Racke

\underline{Abstract}: Initial and initial boundary value problems for linearized magneto-thermo-elastic models are considered. For the Cauchy problem in three space dimensions, a polynomial rate of decay as time tends to infinity is proved. In bounded domains a boundary condition of memory type is considered for the displacement. When the relaxation function satisfies dissipative properties and decays exponentially, we show that the solution of the magneto-thermo-elastic system decays exponentially. When the relaxation function decays polynomially, it is proved that the solution decays polynomially. Energy methods are used.

\underline{AMS subject classification}: 73 B 30, 35 Q 99

\underline{Keywords and phrases}: exponential stability, damping boundary condition

1 Introduction

We consider initial and initial boundary value problems for some linear magneto-thermo-elastic models describing elastic materials where reciprocal effects of the temperature, the magnetic field and the elastic displacement are taken into account. The linear differential equations — for the homogeneous, isotropic case — which govern these models in three space dimensions, are the following:

u_{tt} − Eu − α[∇ × h]× H^{⇀} +γ∇θ = 0,  (1.1)
h_{t} − Δh − β∇ × [u_{t}× H^{⇀}] = 0,  (1.2)
θ_{t} − κΔθ + γ div u_{t} = 0.  (1.3)

Here u = (u^{1}, u^{2}, u^{3})′ = u(t, x) is the displacement vector depending on the time variable t ≥ 0 and on x ∈ ℝ^{3}, h = (h^{1}, h^{2}, h^{3})′ = h(t, x) is the magnetic field, θ = θ(t, x) is the temperature difference with respect to a fixed reference temperature, E is the elasticity operator

Eu = [(C_{ijkl}u_{,l}^{k})_{,j}]_{i=1,2,3} ,  (1.4)

where C_{ijkl} (i, j, k, l = 1, 2, 3) are the elastic moduli being constant here and leading in the homogeneous isotropic case under consideration to

C_{ijkl} = λδ_{ij}δ_{kl} + μ(δ_{ik}δ_{jl} + δ_{jk}δ_{il})  (δ_{ij} : Kronecker delta)  (1.5)

∗Supported by a CNPq-DLR grant

and hence

Eu = μΔu + (λ + μ)∇div u,  (1.6)

with positive constants λ and μ. The coupling constants α, β satisfy αβ > 0 and γ ∈ ℝ. κ is a positive constant and H^{⇀}= (0, 0, H)′ is a constant vector with H ≠ 0, distinguishing the x_{3}-direction. The notation ”_{,j}” means differentiation with respect to x_{j}. Additionally, one has initial conditions

u(0, x) = u_{0}(x), u_{t}(0, x) = u_{1}(x), h(0, x) = h_{0}(x), θ(0, x) = θ_{0}(x).  (1.7)

The Cauchy problem (1.1)–(1.3), (1.7) will be considered in Section 2 giving a description of the asymptotic behavior of the solution as time tends to infinity in terms of polynomial decay rates depending on the smoothness of the initial data. The Cauchy problem in magneto-elasticity i.e. neglecting thermal effects, was recently studied by Andreou & Dassios [2] using spectral analysis and techniques provided from thermoelasticity; polynomial decay rates were obtained. Here we discuss the more complicated system of magneto-thermo-elasticity — indeed, it might be too complicated for the approach from [2] —, and we use energy methods constructing appropriate Lyapunov functionals. Note that our results are also valid for magneto-elasticity, i.e. for γ = 0, θ = 0.

For a derivation of the equations and earlier papers on magneto-thermo-elastic plane waves see Paria [12], Willson [17] or Chander [3] and the references in [2]. The bounded domain case will be studied in Section 3. Then x will vary in a bounded domain Ω ⊂ ℝ^{3} with smooth boundary Γ = ∂Ω and Γ = Γ_{0} ∪ Γ_{1} with Γ_{0} ∩ Γ_{1} ≠ ∅. We assume that there is x_{0} ∈ Ω such that

Γ_{0} = {x ∈ Γ| (x − x_{0}) · ν(x) ≤ 0},  (1.8)
Γ_{1} = {x ∈ Γ| (x − x_{0}) · ν(x) ≥ a > 0}  (1.9)

for some a > 0 and ν = ν(x) denoting the exterior normal vector in x ∈ Γ. This means that typically we have

[figure: Ω = Ω_{1} \ Ω_{0}; Ω_{0}; Ω_{1}; Γ_{0}; Γ_{1}; x_{0}]

The boundary conditions for h and θ are

ν × (∇ × h) = 0, ν · h = 0, θ = 0 on Γ  (1.10)

and the memory type boundary condition

u = 0 on Γ_{0}, u + r ∗ ∂_{ν}u = 0 on Γ_{1}  (1.11)

for u. We denote by ∂_{ν}u the abbreviation

∂_{ν}u := \underbrace{(C_{ijkl}u_{,l}^{k}ν^{j})_{i=1,2,3}}_{=: \frac{∂u}{∂ν_{A}}} −αHh^{3}ν + αHν^{3}h  (1.12)

which is the natural Neumann type boundary operator for the equations (1.1). By ∗ we denote the convolution in time,

(r ∗ f)(t) := ∫_{0}^{t} r(t − s)f(s)ds.

Differentiating the boundary condition on Γ_{1} we get

∂_{ν}u + \frac{r′}{r(0)} ∗ ∂_{ν}u = −\frac{u_{t}}{r(0)},  (1.13)

or, in terms of the associated resolvent kernel g,

∂_{ν}u = −τu_{t} − τg ∗ u_{t}  (1.14)

with

τ := \frac{1}{r(0)} > 0.  (1.15)

First we shall assume that g essentially decays exponentially, i.e. for t ≥ 0

\begin{matrix}0 < g(t) ≤ c_{0}e^{−g_{0}t}, \\ −c_{1}g(t) ≤ g′(t) ≤ −c_{2}g(t), \\ −c_{3}g′(t) ≤ g′′(t) ≤ −c_{4}g′(t), \end{matrix} \}  (1.16)

with positive constants g_{0}, c_{0}, c_{1}, c_{2}, c_{3}, c_{4}. The classical example g̃ is of course

g̃(t) = c_{0}e^{−g_{0}t}.

The exponential type kernel together with the ”damping” boundary condition on Γ_{1} will lead to an exponential decay result for (u, h, θ)(t). Second, we shall consider polynomially decaying kernels satisfying for t ≥ 0

\begin{matrix}0 < g(t) ≤ b_{0}(1 + t)^{−p}, \\ −b_{1}g(t)^{\frac{p+1}{p}} ≤ g′(t) ≤ −b_{2}g(t)^{\frac{p+1}{p}} , \\ −b_{3}|g′(t)|^{\frac{p+2}{p+1}} ≤ g′′(t) ≤ −b_{4}|g′(t)|^{\frac{p+2}{p+1}} , \end{matrix} \}  (1.17)

with positive constants b_{0}, b_{1}, b_{2}, b_{3}, b_{4} and p > 1. The typical example ḡ is

ḡ(t) = b_{0}(1 + t)^{−p}.

The result will be a polynomial decay for the solution. We present the first discussion of magneto-thermo-elastic initial boundary value problems that leads to a description of the asymptotic behavior in terms of decay rates as time tends to infinity. Perla Menzala & Zuaza studied the asymptotic behavior of magneto-elastic systems with u being zero everywhere on the boundary, see [13]. They proved decay results without giving decay rates but it should be observed that their system is more complicated to deal with because of the boundary conditions considered. In this connection, it should be mentioned that in the related thermo-elastic system exponential decay can only be expected in special situations like radial symmetry, for example, see Jiang & the authors [7], but not in general, see Koch [8] or Lebeau & Zuaza [9], In our case, the complexity of the system can be dealt with because of the memory type boundary condition. For a discussion of this kind of boundary conditions in connection with viscoelastic effects see e.g. the papers [14, 15] by Qin. In [6], Fabrizio & Morro also consider a boundary condition of memory type for an electromagnetic system (see also Ciarletta [4]). The boundary condition was also studied for a pure elastic system by Andrade & Muñoz Rivera [1]. Their approach and techniques from Muñoz Rivera & Barreto [11] are adapted here and modified for our purposes. In Section 2 we discuss the Cauchy problem and in Section 3 the bounded domain case is considered. In both sections, the existence and uniqueness of solutions with smoothness in relation to that of the initial data is simply assumed, its proof — e.g. via semigroup theory as in [13] for the magneto-elastic system — is rather standard and omitted here. We denote the Fourier transform by

(Fw)(η) ≡ ŵ(η) = \frac{1}{(2π)^{3/2}} ∫_{ℝ^{3}} e^{−ixη}w(x)dx

for a function w : ℝ^{3} → ℝ^{(3)}.

2 The Cauchy problem

We consider smooth solutions to the Cauchy problem (1.1)– (1.3), (1.7). The necessary regularity of the initial data will be specified below. Noting that

(∇ × h)× H^{⇀} = −H∇h^{3} + H \frac{∂h}{∂x_{3}} ,
∇ × (u_{t}× H^{⇀}) = H \frac{∂u_{t}}{∂x_{3}} − H(0, 0, div u_{t})′

and using (1.6), the system (1.1)–(1.3) is equivalent to

u_{tt} − μΔu − (μ + λ)∇div u + αH∇h^{3} − αH \frac{∂h}{∂x_{3}} + γ∇θ = 0,  (2.1)

h_{t} − Δh − βH \frac{∂u_{t}}{∂x_{3}} + βH(0, 0, div u_{t})′ = 0,  (2.2)
θ_{t} − κΔθ + γ div u_{t} = 0.  (2.3)

Taking the Fourier transform and denoting

v := û,  w := ĥ,  ψ := θ̂,

with Fourier variable η = (η_{1}, η_{2}, η_{3})′, we obtain from (2.1)–(2.3)

v_{tt} + μ|η|^{2}v + (μ + λ)(ηv)η − iαHηw^{3} + iαHη_{3}w − iγψη = 0,  (2.4)
w_{t} + |η|^{2}w + iβHη_{3}v_{t} − βH(0, 0, v_{t}η)′ = 0,  (2.5)
ψ_{t} + κ|η|^{2}ψ − iγv_{t}η = 0.  (2.6)

Multiplying equation (2.4) by \overline{u}_{t}, and (2.5) by \frac{α}{β} \overline{w} and (2.6) by \overline{ψ} we get

\frac{d}{dt}E_{1}(t) = −\frac{α}{β}|η|^{2}|w|^{2} − κ|η|^{2}|ψ|^{2}

where

E_{1}(t) := \frac{1}{2}\{|v_{t}|^{2} + μ|η|^{2}|v|^{2} + (μ + λ)(ηv)^{2} + \frac{α}{β}|w|^{2} + |ψ|^{2}\}(t).

To study the dissipative properties of the above system, we will consider the equations for the components separately. Denoting

z := η_{1}v^{1} + η_{2}v^{2}

we obtain

v_{tt}^{1} + μ|η|^{2}v^{1} + (μ + λ)η_{1}z + (μ + λ)η_{1}η_{3}v^{3} − iαHη_{1}w^{3} + iαHη_{3}w^{1} − iγψη_{1} = 0,  (2.7)
v_{tt}^{2} + μ|η|^{2}v^{2} + (μ + λ)η_{2}z + (μ + λ)η_{2}η_{3}v^{3} − iαHη_{2}w^{3} + iαHη_{3}w^{2} − iγψη_{2} = 0,  (2.8)
v_{tt}^{3} + μ|η|^{2}v^{3} + (μ + λ)η_{3}z + (μ + λ)η_{3}^{2}v^{3} − iγψη_{3} = 0,  (2.9)
w_{t}^{1} + |η|^{2}w^{1} + iβHη_{3}v_{t}^{1} = 0,  (2.10)
w_{t}^{2} + |η|^{2}w^{2} + iβHη_{3}v_{t}^{2} = 0,  (2.11)
w_{t}^{3} + |η|^{2}w^{3} − iβHz_{t} = 0.  (2.12)

Lemma 2.1 Let

Φ_{1}(t) := −Re\{\frac{i}{η_{3}} (\overline{v_{t}^{1}}w^{1} + \overline{v_{t}^{2}}w^{2})\}(t), (Re : real part)

Φ_{2}(t) := −Re \{\frac{i}{η_{3}} (\overline{v_{tt}^{1}}w_{t}^{1} + \overline{v_{tt}^{2}}w_{t}^{2})\}(t).

Then we have for η_{3} ≠ 0:

\frac{d}{dt}Φ_{1} ≤ −\frac{βH}{2} (|v_{t}^{1}|^{2} + |v_{t}^{2}|^{2}) + \frac{c|η|^{2}}{η_{3}^{2}} (1 + |η|^{2})|w|^{2}

+\frac{c|η|^{2}}{|η_{3}|} (|v^{1}| + |v^{2}|)|w| + c|η||v^{3}||w| + c\frac{|η|}{|η_{3}|}|ψ||w|,

\frac{d}{dt}Φ_{2} ≤ −\frac{βH}{2} (|v_{tt}^{1}|^{2} + |v_{tt}^{2}|^{2}) + \frac{c|η|^{2}}{η_{3}^{2}} (1 + |η|^{2})|w_{t}|^{2}

+c|η||v_{t}^{3}||w_{t}| + \frac{c|η|^{2}}{|η_{3}|} |\overline{v_{t}^{1}}w_{t}^{1} + \overline{v_{t}^{2}}w_{t}^{2}|.

Here and in the sequel c will denote positive constants not depending on η or on t. Proof: Multiplying equation (2.10) by \frac{−i}{η_{3}} \overline{v_{t}^{1}} and equation (2.11) by \frac{−i}{η_{3}} \overline{v_{t}^{2}} we get

\frac{d}{dt}Φ_{1} = Re \frac{i}{η_{3}} |η|^{2}(w^{1}\overline{v_{t}^{1}} + w^{2}v_{t}^{2}) − βH(|v_{t}^{1}|^{2} + |v_{t}^{2}|^{2}) − Re \frac{i}{η_{3}} (w^{1}v_{tt}^{1} + w^{2}v_{tt}^{2}).

Using the equations (2.7), (2.8) the conclusion for Φ_{1} follows. The estimate for Φ_{2} follows similarly after differentiating the equations with respect to t.

Q.e.d. Introducing the function

Φ_{3}(t) := Re (v_{tt}^{1}\overline{v_{t}^{3}})(t) and Φ_{4}(t) := Re (v_{tt}^{2}\overline{v_{t}^{3}})(t)

we have Lemma 2.2 Let for η_{3} ≠ 0 ≠ η_{1}

J(t) := \frac{η_{1}}{η_{3}} Φ_{3}(t) + \frac{η_{3}}{η_{1}} Φ_{3}(t) + \frac{η_{1}}{η_{3}} Φ_{4}(t).

Then

\frac{d}{dt}J ≤ c|η|^{4} (\frac{1}{η_{1}^{2}} + \frac{1}{η_{3}^{2}} ) (|v_{t}^{1}|^{2} + |v_{t}^{2}|^{2}) − \frac{μ + λ}{2}|η|^{2}|v_{t}^{3}|^{2}

+c|η|^{2} (\frac{1}{η_{1}^{2}} + \frac{1}{η_{3}^{2}} ) |w_{t}|^{2} (\frac{η_{3}}{η_{1}} + \frac{η_{2}}{η_{3}} + \frac{η_{1}}{η_{3}} ) (|η|^{2}|v| + |η| |ψ|)(|v_{tt}^{1}| + |v_{tt}^{2}|)

+c|η|^{2} (\frac{1}{η_{1}^{2}} + \frac{1}{η_{3}^{2}} ) |ψ_{t}|^{2}.

Proof: Differentiating Φ_{3} we get

\frac{d}{dt}Φ_{3} = (v_{ttt}^{1}\overline{v_{t}^{3}} + v_{tt}^{1}\overline{v_{tt}^{3}})
= −(μ|η|^{2}v_{t}^{1} + (μ + λ)η_{1}v_{t} + (μ + λ)η_{1}η_{3}v_{t}^{3} − iαHη_{1}w_{t}^{3}
+iαHη_{3}w_{t}^{1} + γη_{1}ψ_{t})\overline{v_{t}^{3}} + v_{tt}^{1}\overline{v_{tt}^{3}}

which implies

\frac{d}{dt} (\frac{η_{1}}{η_{3}} Φ) ≤ \frac{c|η|^{4}}{η_{3}^{2}} |v_{t}^{1}|^{2} + \frac{c|η|^{2}|z_{t}|^{2}}{η_{3}^{2}} − \frac{(μ + λ)}{2}η_{1}^{2}|v_{t}^{3}|^{2}

+\frac{c|η|^{2}}{η_{3}^{2}} (|w_{t}|^{2} + |ψ_{t}|^{2}) − \frac{η_{1}}{η_{3}} v_{tt}^{1}\overline{v_{tt}^{3}}.

Similarly we have

\frac{d}{dt} (\frac{η_{3}}{η_{1}} Φ_{3}) ≤ \frac{c|η|^{4}}{η_{1}^{2}} |v_{t}|^{2} + \frac{c|η|^{2}|z_{t}|^{2}}{η_{1}^{2}} − \frac{(μ + λ)}{2}η_{3}^{2}|v_{t}^{3}|^{2}

+\frac{c|η|^{2}}{η_{1}^{2}} (|w_{t}|^{2} + |ψ_{t}|^{2}) + \frac{η_{3}}{η_{1}} v_{tt}^{1}\overline{v_{tt}^{3}},

\frac{d}{dt} (\frac{η_{2}}{η_{3}} Φ_{4}) ≤ \frac{c|η|^{4}|v_{t}|^{2}}{η_{3}^{2}} + \frac{c|η|^{2}|z_{t}|^{2}}{η_{3}^{2}} − \frac{(μ + λ)}{2}η_{2}^{2}|v_{t}^{3}|^{2}

+\frac{c|η|^{2}}{η_{3}^{2}} (|w_{t}|^{2} + |ψ_{t}|^{2}) + \frac{η_{2}}{η_{3}} v_{tt}^{2}\overline{v_{tt}^{3}}.

Summing up these inequalities our conclusion follows.

Q.e.d. As a consequence of Lemma 2.1 and Lemma 2.2 we obtain Lemma 2.3 Let, for η_{1} ≠ 0 ≠ η_{3},

A ≡ A(η) := |η|^{2} (\frac{1}{η_{1}^{2}} + \frac{1}{η_{3}^{2}} ) ,

and let δ > 0 be sufficiently small. Then we have the estimates

\frac{d}{dt} \{(|η|^{2} + |η|^{2}A)Φ_{1} + δJ\} ≤ −\frac{βH}{4} (|η|^{2} + |η|^{2}A)(|v_{t}^{1}|^{2} + |v_{t}^{2}|^{2})

−δk_{0}|η|^{2}|v_{t}|^{2} + c(|η|^{2} + |η|^{2}A)\frac{|η|^{2}}{η_{3}^{2}} (1 + |η|^{2})(|w|^{2} + |ψ|^{2})

+cA|η|^{3}|v^{3}| |w| + \frac{c|η|^{4}}{|η_{3}|} A(|v^{1}| + |v^{2}|)|w| + cδA(|w_{t}|^{2} + |ψ_{t}|^{2})

+cδ |\frac{η_{1}}{η_{3}} + \frac{η_{3}}{η_{1}} + \frac{η_{2}}{η_{3}} | (|η|^{2}|v| + |η| |ψ|) (|v_{tt}^{1}| + |v_{tt}^{2}|),

where k_{0} > 0 is a constant only depending on the coefficients of the differential equation. Now let

Φ_{5}(t) := Re(v_{t}\overline{v})(t).

Using equation (2.1) it is not difficult to prove that

\frac{d}{dt}Φ_{5} ≤ |v_{t}|^{2} − \frac{μ}{2}|η|^{2}|v|^{2} − \frac{μ + λ}{2}(ηv)^{2} + c|η|(|w|^{2} + |ψ|^{2})

which, together with Lemma 2.3, yields

\frac{d}{dt} \{(|η|^{2} + |η|^{2}A)Φ_{1} + δJ + \frac{δk_{0}}{2} |η|^{2}Φ_{5}\} ≤

−\frac{βH}{4} |η|^{2}A(|v_{t}^{1}|^{2} + |v_{t}^{2}|^{2}) − \frac{δk_{0}}{2} |η|^{2}E_{1}

+c \{\frac{|η|^{4}}{η_{3}^{2}} (1 + |η|^{2})A + |η|^{2} + |η|^{2}A + \frac{|η|^{6}}{η_{3}^{2}} A\} (|w|^{2} + |ψ|^{2})

+c\frac{|η|^{4}}{|η_{3}|}A(|v^{1}| + |v^{2}|)|w| + cδA(|v_{tt}^{1}|^{2} + |v_{tt}^{2}|^{2})

cδA(|w_{t}|^{2} + |ψ_{t}|^{2}).  (2.13)

Notice that

\frac{d}{dt} \{(|η|^{2} + |η|^{2}A)Φ_{1} + δJ + \frac{δk_{0}}{2} |η|^{2}Φ_{5}\} ≤

−\frac{βH}{4} |η|^{2}A(|v_{t}^{1}|^{2} + |v_{t}^{2}|^{2}) − \frac{δk_{0}}{2} |η|^{2}E_{1} + c|η|^{6} \frac{1 + |η|^{2}}{η_{3}^{2}} (\frac{1}{η_{1}^{2}} + \frac{1}{η_{3}^{2}} ) (|w|^{2} + |ψ|^{2})

+cδA(|v_{tt}^{1}|^{2} + |v_{tt}^{2}|^{2}) + cδA(|w_{t}|^{2} + |ψ_{t}|^{2})

c\frac{|η|^{4}}{|η_{3}|}A(|v^{1}| + |v^{2}|)|w|  (2.14)

and that

c\frac{|η|^{4}}{|η_{3}|}A(|v^{1}| + |v^{2}|)|w| ≤ ε\frac{|η|^{4}}{|η_{3}|^{2}} (|v^{1}|^{2} + |v^{2}|^{2}) + c(ε)|η|^{4}A^{2}|w|^{2},  (2.15)

where ε > 0 will be chosen small enough in the sequel and c(ε) denotes a constant depending on ε. Let

Φ_{0}(t) := Re (v_{t}^{1}\overline{v^{1}} + v_{t}^{2}\overline{v^{2}}) (t).

Then, by

\frac{d}{dt}Φ_{6} = |v_{t}^{1}|^{2} + |v_{t}^{2}|^{2} + v_{tt}^{1}\overline{v^{1}} + v_{tt}^{2}\overline{v^{2}}
= |v_{t}^{1}|^{2} + |v_{t}^{2}|^{2} − μ|η|^{2}|v^{1}|^{2} − μ|η|^{2}|v^{2}|^{2} − (μ + λ)|z|^{2}
−(μ + λ)η_{3}v^{3}\overline{z} + iαH\overline{z}w^{3} − iαHη_{3}(w^{1}\overline{v^{1}} + w^{2}\overline{v^{2}}) − iγzψ,

we get

\frac{d}{dt}Φ_{6} ≤ |v_{t}^{1}|^{2} + |v_{t}^{2}|^{2} − \frac{μ}{2}|η|^{2}(|v^{1}|^{2} + |v^{2}|^{2}) − \frac{μ + λ}{2}|z|^{2}
+c(|w|^{2} + (ψ|^{2}) + cη_{3}^{2}|v^{3}|^{2}.  (2.16)

Taking ε small enough we conclude from (2.13)–(2.16)

\frac{d}{dt} \{(|η|^{2} + |η|^{2}A)Φ_{1} + δJ + \frac{δk_{0}}{2} |η|^{2}Φ_{5} + \frac{δk_{0}}{ψ} \frac{|η|^{2}}{η_{3}^{2}} Φ_{6}\}
≤ −\frac{βH}{8} |η|^{2}A(|v_{t}^{1}|^{2} + |v_{t}^{2}|^{2}) − \frac{δk_{0}}{8} |η|^{2}E_{1} + |η|^{6} \frac{1 + |η|^{2}}{η_{3}^{2}} (\frac{1}{η_{1}^{2}} + \frac{1}{η_{3}^{2}} ) (|w|^{2} + |ψ|^{2})

+cδA(|v_{tt}^{1}|^{2} + |v_{tt}^{2}|^{2}) + cδA(|w_{t}|^{2} + |ψ_{t}|^{2})

+c(ε)|η|^{4}A^{2}|w|^{2} − \frac{k_{0}}{4} \frac{|η|^{4}}{η_{3}^{2}} (|v^{1}|^{2} + |v^{2}|^{2}).  (2.17)

Note that

\frac{d}{dt}(AΦ_{2}) ≤ −\frac{βH}{2} A(|v_{tt}^{1}|^{2} + |v_{tt}^{2}|^{2}) + \frac{|η|^{2}}{η_{3}^{2}} A(1 + |η|^{2})|w_{t}|^{2}

+ \underbrace{c|η|A|v_{t}^{3}| |w_{t}|}_{=:I_{1}} + \underbrace{c\frac{|η|^{2}}{η_{3}^{2}} A|v_{t}^{1}\overline{w_{t}^{1}} + v_{t}^{2}w_{t}^{2}|}_{=:I_{2}}

+\frac{|η|}{|η_{3}|}A|ψ_{t}w_{t}|,  (2.18)

and

I_{1} ≤ \frac{δk_{0}}{4} |η|^{2}|v_{t}^{3}|^{2} + c(δ)A^{2}|w_{t}|^{2},  (2.19)

I_{2} ≤ δA|η|^{2}(|v_{t}^{1}|^{2} + |v_{t}^{2}|^{2}) + c(δ)\frac{|η|^{2}}{η_{3}^{2}} A|w_{t}|^{2}.  (2.20)

Denoting by Φ the following function

Φ(t) := (|η|^{2} + |η|^{2}A)Φ_{1}(t)(+δJ(t) + \frac{δk_{0}}{2} |η|^{2}Φ_{5}(t) + \frac{δk_{0}|η|^{2}}{4|η_{3}|^{2}} Φ_{6}(t) + AΦ_{2}(t)

we obtain from (2.13)–(2.20)

\frac{d}{dt}Φ ≤ −\frac{βH}{8} |η|^{2}A(|v_{t}^{1}|^{2} + |v_{t}^{2}|^{2}) − \frac{δk_{0}}{4} |η|^{2}E_{1}

−\frac{βH}{4} A(|v_{tt}^{1}|^{2} + |v_{tt}^{2}|^{2}) + c|η|^{6} \frac{1 + |η|^{2}}{η_{3}^{2}} (\frac{1}{η_{1}^{2}} + \frac{1}{η_{3}^{2}} ) (|w|^{2} + |ψ|^{2})

c(δ) \underbrace{(\frac{|η|^{2}}{η_{3}^{2}} A + A^{2})}_{=:I_{3}}(|w_{t}|^{2} + |ψ_{t}|^{2})

+c(ε)|η|^{4}A^{2}|w|^{2} − \frac{k_{0}|η|^{4}}{4η_{3}^{2}} (|v^{1}|^{2} + |v^{2}|^{2}).

Using

I_{3} ≤ 2A^{2}

we conclude

\frac{d}{dt}Φ ≤ −\frac{βH}{8} |η|^{2}A(|v_{t}^{1}|^{2} + |v_{t}^{2}|^{2}) − \frac{δk_{0}}{4} |η|^{2}E_{1}

−\frac{βH}{4} A(|v_{tt}^{1}|^{2} + |v_{tt}^{2}|^{2}) + c|η|^{2}(1 + |η|^{2})A^{2}(|w|^{2} + |ψ|^{2})

+c(δ)A^{2}(|w_{t}|^{2} + |ψ_{t}|^{2}) − \frac{k_{0}|η|^{4}}{ψη_{3}^{2}} (|v^{1}|^{2} + |v^{2}|^{2}).  (2.21)

Let

L(t) := Φ(t) + N_{1}(1 + |η|^{2})A^{2}E_{1}(t) + N_{2} \frac{A^{2}}{|η|^{2}} E_{2}(t),

where

E_{2}(t) ≡ E_{2}(t; v_{t}, w_{t}, ψ_{t})

if

E_{1}(t) = E_{1}(t; v, w, ψ),

and N_{1}, N_{2} are positive constants to be determined below. Observe that L is non-negative for large N_{1}, N_{2}. From the inequalities up to (2.21) we know that there is d_{0} > 0 such that

\frac{d}{dt}L ≤ −d_{0}N  (2.22)

where d_{0}, d_{1}, … will denote positive constants and

N (t) := |η|^{2}A(|v_{t}^{1}|^{2} + |v_{t}^{2}|^{2})(t) + |η|^{2}E_{1}(t) + A(|v_{tt}^{1}|^{2} + |v_{tt}^{2}|^{2})

+\frac{|η|^{4}}{η_{3}^{2}} (|v^{1}|^{2} + |v^{2}|^{2}) + (1 + |η|^{2})A^{2}(|w|^{2} + |ψ|^{2})

+\frac{A^{2}}{|η|^{2}} (|w_{t}|^{2} + |ψ_{t}|^{2}).

Since

|Φ| ≤ c̃((1 + |η|^{2})A^{2}E_{1} + \frac{A^{2}}{|η|^{2}} E_{2}) ,  (2.23)

for some c̃ > 0, and

E_{2}(t) ≤ c̃|η|^{2}(1 + |η|^{2})E_{1}(t)  (2.24)

we have

L(t) ≤ d_{1}(1 + |η|^{2})A^{2}E_{1}(t).  (2.25)

The relations (2.22)–(2.25) imply

\frac{d}{dt}L ≤ −d_{0}N
≤ −d_{0}|η|^{2}E_{1}
≤ − \underbrace{\frac{d_{0}}{d_{1}}}_{=:d_{2}} \underbrace{\frac{|η|^{2}}{(1 + |η|^{2})^{2}A^{2}}}_{=:k(η)} L

and hence

L(t) ≤ L(0)e^{−d_{2}k(η)t}.  (2.26)

Using (2.23), (2.24) and choosing

N_{2} := \frac{c̃}{2}

we obtain

L ≥ (N_{1} − 2N_{2})(1 + |η|^{2})AE_{1} − \frac{N_{2}}{2} \frac{A^{2}}{|η|^{2}} E_{2}
≥ (N_{1} − 2N_{2} − \frac{N_{2}}{2} − \frac{N_{2}}{2} ) (1 + |η|^{2})A^{2}E_{1},

that is, choosing

N_{1} := \frac{5}{2}N_{2},

L ≥ N_{2}(1 + |η|^{2})A^{2}E_{1}.  (2.27)

From (2.25) and (2.27) we have

N_{2}(1 + |η|^{2})A^{2}E_{1}(t) ≤ L(t) ≤ d_{1}(1 + |η|^{2})A^{2}E_{1}(t).

Therefore, (2.25) implies for the ”energy” term E_{1}

E_{1}(t) ≤ \frac{d_{1}}{N_{2}} E(0)e^{−d_{2}k(η)t}

which implies for m ∈ ℕ_{0}, t ≥ 0 :

∫_{0}^{t} s^{m}E_{1}ds ≤ c(m) (\frac{1}{k(η)})^{m} E(0).  (2.28)

This gives the desired polynomial decay in a standard manner:

\frac{d}{dt}tE_{1}(t) = E_{1}(t) + t\frac{d}{dt}E_{1}(t) ≤ E_{1}(t)

implies using (2.28) for m = 0:

E_{1}(t) = O(t^{−1}) as t → ∞.

The estimate

\frac{d}{dt}t^{2}E_{1}(t) ≤ 2tE_{1}(t)

implies

E_{1}(t) = O(t^{−2}),

and so on. Integrating (2.28) with respect to η — for appropriate initial data yielding a finite value of the integral on the right-hand side — gives the final result. Observing

E(0) = E(0, η), A = A(η)

we have Theorem 2.4 If the initial data satisfy

∫_{ℝ^{n}} [\frac{(1 + |η|^{2})A^{2}}{|η|^{2}} ]^{m} E_{1}(0, η)dη < ∞,

m ∈ ℕ_{0}, then the energy term

E(t) := \frac{1}{2} ∫_{ℝ^{n}} \{|u_{t}|^{2} + μ|∇u|^{2} + (μ + λ)|div u|^{2} + \frac{α}{β}|h|^{2} + |θ|^{2}\} (t, x) dx

associated to a solution (u, h, θ) to the Cauchy problem (1.1)–(1.3), (1.7) decays polynomially:

E(t) = O(t^{−m}) as t → ∞.

3 The bounded domain case

In a bounded domain Ω ⊂ ℝ^{3} described by (1.8), (1.9) we consider the differential equations (1.1)–(1.3) together with the initial conditions (1.7) and the boundary conditions (1.10), (1.11). The initial magnetic field h_{0} is assumed to satisfy

div h_{0} = 0  (3.1)

which implies by (1.2)

div h(t, ·) = 0 for all t ≥ 0.

Either the condition (1.16) or (1.17) on the resolvent kernel g is assumed to hold. Since we have formulated conditions in terms of the resolvent kernel, we point out the following relationship between the decay of a kernel and the decay of the associated resolvent kernel. Let

b(t) := −\frac{r′(t)}{r(0)},

Then b and g satisfy

b + g = −b ∗ g.

Lemma 3.1 (i) If g satisfies

∃ γ > 0 ∃ c_{g} > 0 ∀ t ≥ 0 : |g(t)| ≤ c_{g}e^{−γt},

and if for some 0 < ε < γ

c_{g} < γ − ε

holds, then we have

∀ t ≥ 0 : |b(t)| ≤ \frac{c_{g}(γ − ε)}{γ − ε − c_{g}} e^{−εt}.

(ii) If g satisfies

∃ p > 1 ∃c_{g} > 0 ∀ t ≥ 0 : |g(t)| ≤ c_{g}(1 + t)^{−p},

and if

\frac{1}{c_{g}} > c_{p} := sup_{0≤t<∞} ∫_{0}^{t}(1 + t)^{p}(1 + t − τ)^{−p}(1 + τ)^{−p}dτ

holds, then we have

∀ t ≥ 0 : |b(t)| ≤ \frac{c_{g}}{1 − c_{g}c_{p}} (1 + t)^{−p}.

Proof: (i): Let

g̃(t) := e^{εt}g(t),  b̃(t) := e^{εt}b(t).

Then

b̃ + g̃ = −b̃ ∗ g̃.

The operator G given by

G(h) := g̃ ∗ h,

acting on C^{0}([0, T]), T > 0 arbitrary, but fixed, has norm less than or equal to

\frac{c_{g}}{γ − ε},

hence

sup_{0≤t≤T} |b̃(t)| ≤ \frac{1}{1 − c_{g}/(γ − ε)} sup_{0≤t≤T} |g̃(t)| ≤ \frac{c_{g}(γ − ε)}{γ − ε − c_{g}} ,

which implies the assertion in (i). (ii): Let

g̃(t) := (1 + t)^{p}g(t),  b̃(t) := (1 + t)^{p}b(t).

Then

b̃ + g̃ = −k[g] ∗ b

with kernel

k[g](t, τ) := g̃(t − τ)(1 + t)^{p}(1 + t − τ)^{−p}(1 + τ)^{−p}.

The operator K[g] acting on C^{0}([0, T]) as

K[g](h) := k[g] ∗ h

has norm less than or equal to

c_{g} sup_{0≤t<∞} ∫_{0}^{t}(1 + t)^{p}(1 + t − τ)^{−p}(1 + τ)^{−p}dτ ≤ c_{g}c_{p},

hence

sup_{0≤t≤T} |b̃(t)| ≤ \frac{1}{1 − c_{g}c_{p}} sup_{0≤t≤T} |g̃(t)| ≤ \frac{c_{g}}{1 − c_{g}c_{p}} ,

which gives the assertion.

Q.e.d. Remarks: 1. For the finiteness of c_{p} compare Lemma 7.4 in [16] in a more general setting.

2. Since the resolvent kernel of the resolvent kernel is the orinigal kernel, it is clear that in (ii) of Lemma 3.1 no stronger uniform polynomial decay can be obtained. In this sense the characterization is sharp and shows that exponentially decaying kernels correspond to exponentially decaying resolvents, and polynomially decaying kernels correspond to polynomially decaying resolvents.

Taking the boundary condition (1.11) for u on Γ_{1} in the form (1.14), and performing an integration by parts, we get

∂_{ν}u = −τu_{t} − τg ∗ u_{t}
= −τu_{t} − τg(0)u − τg′ ∗ u + τgu_{0}.  (3.2)

As a non-negative energy function we define

F (t) := \frac{1}{2} ∫_{Ω} (|u_{t}|^{2} + C_{ijkl}u_{,l}^{k}u_{,j}^{i} + \frac{α}{β}|h|^{2} + |θ|^{2})(t, x)dx
−\frac{τ}{2} ∫_{Γ_{1}} (g′□u)(t, z)dz + \frac{τ}{2} g(t) ∫_{Γ_{1}} |u|^{2}(t, z)dz,  (3.3)

where

(f□φ)(t) := ∫_{0}^{t} f(t − s)|φ(t) − φ(s)|^{2}ds.

Lemma 3.2 For f, φ ∈ C^{1}([0, ∞), ℝ) we have

2(f ∗ φ)(t)φ_{t}(t) = (f′□φ)(t) + \frac{d}{dt} \{∫_{0}^{t} f(s)ds|φ(t)|^{2} − (f□φ)(t)\} − f(t)|φ(t)|^{2}.

Proof:

\frac{d}{dt}(f□φ) = f′□φ + 2 ∫_{0}^{t} f(t − s)(φ(t) − φ(s))ds φ_{t}

= f′□φ − 2f ∗ φφ_{t} + 2 ∫_{0}^{t} f(s)ds φφ_{t}

= f′□φ − 2f ∗ φφ_{t} + \frac{d}{dt} \{∫_{0}^{t} f(s)ds|φ|^{2}\} − f|φ|^{2}.

Q.e.d. Lemma 3.3

\frac{d}{dt}F ≤ −\frac{α}{β} ∫_{Ω} |∇ × h|^{2}dx − κ ∫_{Ω} (∇θ)^{2}dx − \frac{τ}{2} ∫_{Γ_{1}} |u_{t}|^{2}dz

−\frac{τ}{2} ∫_{Γ_{1}} g′′□u dz + \frac{τg′}{2} ∫_{Γ_{1}} |u|^{2}dz + τg^{2} ∫_{Γ_{1}} |u_{0}|^{2}dz.

Proof: Multiplying equation (1.1) by u_{t}, equation (1.2) by \frac{α}{β}h and (1.3) by θ, we obtain after integration and summation

\frac{1}{2} \frac{d}{dt} \underbrace{∫_{Ω} (|u_{t}|^{2} + C_{ijkl}u_{,l}^{k}u_{,j}^{i} + \frac{α}{β}|h|^{2} + |θ|^{2})dx}_{=:M=M(t)}

= ∫_{Γ_{1}} ∂_{ν}uu_{t}dz − \frac{α}{β} ∫_{Ω} |∇ × h|^{2}dx − κ ∫_{Ω} |∇θ|^{2}dz.  (3.4)

Using (3.2) and Lemma 3 we get

\frac{d}{dt}M = −\frac{α}{β} ∫_{Ω} (∇ × h|^{2}dx − κ ∫_{Ω} |∇θ|^{2}dx − τ ∫_{Γ_{1}} |u_{t}|^{2}dz

−τg(0) ∫_{Γ_{1}} uu_{t}dz − τ ∫_{Γ_{1}} (g′ ∗ u)u_{t} + τg ∫_{Γ_{1}} u_{0}u_{t}dz

= −\frac{α}{β} ∫_{Ω} |∇ × h|^{2}dx − κ ∫_{Ω} |∇θ|^{2}dx − τ ∫_{Γ_{1}} |u_{t}|^{2}dz

−\frac{1}{2} \frac{d}{dt}\{τg ∫_{Γ_{1}} |u|^{2}dz − τ ∫_{Γ_{1}} g′□u dz\}

−\frac{τ}{2} ∫_{Γ_{1}} g′′□u dz + \frac{τ}{2} g′ ∫_{Γ_{1}} |u|^{2}dz + τg ∫_{Γ_{1}} u_{0}u_{t}dz

which yields the assertion.

Q.e.d. Define

q(x) := x − x_{0}.

Lemma 3.4 Let

f := α(∇ × h)× H^{⇀} −γ∇θ.

Then

\frac{d}{dt} ∫_{Ω} u_{t}q^{k}u_{,k}dx = ∫_{Γ} \frac{∂u}{∂ν_{A}} q^{k}u_{,k}dz − ∫_{Ω} C_{ijml}u_{,l}^{m}q_{,j}^{k}u_{,k}^{i}dx

−\frac{1}{2} ∫_{Γ} q^{k}ν^{k}C_{ijml}u_{,l}^{m}u_{,j}^{i}dz + \frac{1}{2} ∫_{Γ} q^{k}ν^{k}|u_{t}|^{2}dz

+\frac{1}{2} ∫_{Ω} q_{,k}^{k}C_{ijml}u_{,l}^{m}u_{,j}^{i}dx − \frac{1}{2} ∫_{Ω} q_{,k}^{k}|u_{t}|^{2}dx + ∫_{Ω} fq^{k}u_{,k}dx.

Proof: Multiplying (1.1) by q^{k}u_{,k} and integrating yields

\frac{d}{dt} ∫_{Ω} u_{t}q^{k}u_{,k}dx = ∫_{Ω} u_{tt}q^{k}u_{,k}dx + ∫_{Ω} u_{t}q^{k}u_{t,k}dx
= ∫_{Ω} (C_{ijml}u_{,l}^{m})_{,j}q^{k}u_{,k}^{i}dx + \frac{1}{2} ∫_{Ω} q^{k}|u_{t}|_{,k}^{2}dx + ∫_{Ω} fq^{k}u_{,k}dx
= ∫_{Γ} C_{ijml}u_{,l}^{m}ν^{j}q^{k}u_{,k}^{i}dz − ∫_{Ω} C_{ijml}u_{,l}^{m}∂_{j}(q^{k}u_{,k}^{i})dx

+\frac{1}{2} ∫_{Γ} q^{k}ν^{k}|u_{t}|^{2}dz − \frac{1}{2} ∫_{Ω} q_{,k}^{k}|u_{t}|^{2}dx + ∫_{Ω} fq^{k}u_{,k}dx.

Using the symmetry of the moduli C_{ijml}, we conclude

\frac{d}{dt} ∫ u_{t}q^{k}u_{,k}dx = ∫_{Γ} \frac{∂u}{∂ν_{A}} q^{k}u_{,k}dz − ∫_{Ω} C_{ijml}u_{,l}^{m}q_{,j}^{k}u_{,k}^{i}dx

−\frac{1}{2} ∫_{Ω} q^{k}(C_{ijml}u_{,l}^{m}u_{,j}^{i})_{,k}dx + \frac{1}{2} ∫_{Γ} q^{k}ν^{k}|u_{t}|^{2}dz

−\frac{1}{2} ∫_{Ω} q_{,k}^{k}|u_{t}|^{2}dx + ∫_{Ω} fq^{k}u_{,k}dx.

Q.e.d. From Lemma 3.4 we get using q^{k} = x^{k} − x_{0}^{k}

\frac{d}{dt} ∫_{Ω} u_{t}q^{k}u_{,k}dx = ∫_{Γ} \frac{∂u}{∂ν_{A}} q^{k}u_{,k}dz − ∫_{Ω} C_{ijml}u_{,l}^{m}u_{,j}^{i}dx

−\frac{1}{2} ∫_{Γ} q^{k}ν^{k}C_{ijml}u_{,l}^{m}u_{,j}^{i}dz + \frac{1}{2} ∫_{Γ} q^{k}ν^{k}|u_{t}|^{2}dz

+\frac{3}{2} ∫_{Ω} C_{ijml}u_{,l}^{m}u_{,j}^{i}dx − \frac{3}{2} ∫_{Ω} |u_{t}|^{2}dx + ∫_{Ω} fq^{k}u_{,k}dx

which implies

\frac{d}{dt} ∫_{Ω} u_{t}q^{k}u_{,k}dx = ∫_{Γ} \frac{∂u}{∂ν_{A}} q^{k}u_{,k}dz − \frac{1}{2} ∫_{Ω} (u_{t}|^{2} + C_{ijml}u_{,l}^{m}u_{,k}^{i})_{k}

+ ∫_{Ω} (C_{ijml}u_{,l}^{m}u_{,k}^{i} − |u_{t}|^{2})dx − \frac{1}{2} ∫_{Γ} q^{k}ν^{k}C_{ijml}u_{,l}^{m}u_{,k}^{i}dz

+\frac{1}{2} ∫_{Γ} q^{k}ν^{k}|u_{t}|^{2}dz + ∫_{Ω} fq^{k}u_{,k}dx.  (3.5)

Using the differential equation (1.1) we obtain

\frac{d}{dt} ∫_{Ω} uu_{t}dx = ∫_{Ω} uu_{tt}dx + ∫_{Ω} |u_{t}|^{2}dx
= ∫_{Ω} u^{i} {(C_{ijml}u_{,l}^{m})_{,j} + f^{i}} dx + ∫_{Ω} |u_{t}|^{2}dx
= ∫_{Γ} u\frac{∂u}{∂ν_{A}} dz − ∫_{Ω} (C_{ijml}u_{,l}^{m}u_{,j}^{i} − |u_{t}|^{2})dx + ∫_{Ω} uf dx

which implies

∫_{Ω} (C_{ijml}u_{,l}^{m}u_{,j}^{i} − |u_{t}|^{2})dx = −\frac{d}{dt} ∫_{Ω} uu_{t}dx + ∫_{Γ} u\frac{∂u}{∂ν_{A}} dz + ∫_{Ω} ufdx.

Substituting this identity into (3.5) yields

\frac{d}{dt} \underbrace{\{∫_{Ω} u_{t}q^{k}u_{,k}dx + ∫_{Ω} uu_{t}dx\}}_{=:χ(t)}

= ∫_{Γ} \frac{∂u}{∂ν_{A}} q^{k}u_{,k}dx − \frac{1}{2} ∫_{Ω} (|u_{t}|^{2} + C_{ijml}u_{,l}^{m}u_{,j}^{i})dx

+ ∫_{Γ} u\frac{∂u}{∂ν_{A}} dz + ∫_{Ω} uf dx − \frac{1}{2} ∫_{Γ} q^{k}ν^{k}C_{ijml}u_{,l}^{m}u_{,j}^{i}dz

+\frac{1}{2} ∫_{Γ} q^{k}ν^{k}|u_{t}|^{2}dz + ∫_{Ω} fq^{k}u_{,k}dx.

Since u = 0 on Γ_{0} we have

∫_{Γ_{0}} \frac{∂u}{∂ν}q^{k}u_{,k}dz = ∫_{Γ_{0}} q^{k}ν^{k} |\frac{∂u}{∂ν}|^{2} dz,  (3.6)

∫_{Γ_{0}} div u νq^{k}u_{,k}dz = ∫_{Γ_{0}} q^{k}ν^{k}|div u|^{2}dz,  (3.7)

∫_{Γ_{0}} (∇u)νq^{k}u_{,k}dz = ∫_{Γ_{0}} q^{k}ν^{k}|div u|^{2}.  (3.8)

Observing that (1.5) yields

\frac{∂u}{∂ν_{A}} = λ div u ν + μ\frac{∂u}{∂ν} + μ(∇u)ν,

we conclude from (3.6)–(3.8)

∫_{Γ} \frac{∂u}{∂ν_{A}} q^{k}u_{,k}dz = ∫_{Γ_{0}} q^{k}ν^{k} \{μ |\frac{∂u}{∂ν}|^{2} + (μ + λ)|div u|^{2}\} dz

+ ∫_{Γ_{1}} \frac{∂u}{∂ν_{A}} q^{k}u_{,k}dz

which implies

\frac{d}{dt}χ(t) = ∫_{Γ_{0}} q^{k}ν^{k} \{μ |\frac{∂u}{∂ν}|^{2} + (μ + λ)|div u|^{2}\} dz

+ ∫_{Γ_{1}} \frac{∂u}{∂ν_{A}} q^{k}u_{,k}dz − \frac{1}{2} ∫_{Ω} \{|u_{t}|^{2} + C_{ijml}u_{,l}^{m}u_{,j}^{i}\}dx

−\frac{1}{2} ∫_{Γ} q^{k}ν^{k}C_{ijml}u_{,l}^{m}u_{,j}^{i}dz + ∫_{Γ} u\frac{∂u}{∂ν_{A}} dz

+ ∫_{Ω} uf dx + \frac{1}{2} ∫_{Γ_{1}} q^{k}ν^{k}|u_{t}|^{2}dz + ∫_{Ω} fq^{k}u_{,k}dx  (3.9)

Since on Γ_{1} we have

q^{k}ν^{k} ≥ a > 0

we get

∫_{Γ_{1}} \frac{∂u}{∂ν_{A}} q^{k}u_{,k}dz ≤ c ∫_{Γ_{1}} |\frac{∂u}{∂ν_{A}} |^{2} dz + \frac{1}{8} ∫_{Γ_{1}} q^{k}ν^{k}C_{ijml}u_{,l}^{m}u_{,j}^{i}dz,  (3.10)

∫_{Γ_{1}} u\frac{∂u}{∂ν_{A}} dz ≤ c ∫_{Γ_{1}} |\frac{∂u}{∂ν_{A}} |^{2} dz + \frac{1}{8} ∫_{Ω} C_{ijml}u_{,l}^{m}u_{,j}^{i}dx,  (3.11)

where c > 0 denotes various positive constants. Observing that on Γ_{0}

C_{ijml}u_{,l}^{m}u_{,j}^{i} = μ |\frac{∂u}{∂ν}|^{2} + (λ + μ)(div u)^{2}

and using (3.9)–(3.11) we obtain

\frac{d}{dt}χ(t) ≤ −\frac{1}{4} ∫_{Ω} (|u_{t}|^{2} + C_{ijml}u_{,l}^{m}u_{,j}^{i})dx + c ∫_{Γ_{1}} |\frac{∂u}{∂ν_{A}} |^{2} dz

+\frac{1}{2} ∫_{Γ_{1}} q^{k}ν^{k}|u_{t}|^{2}dz − \frac{1}{4} ∫_{Γ_{1}} q^{k}ν^{k}C_{ijml}u_{,l}^{m}u_{,j}^{i}dz

+ ∫_{Ω} uf dx + ∫_{Ω} fq^{k}u_{,k}dx.

On Γ_{1} we have using (3.2)

\frac{∂u}{∂ν_{A}} = ∂_{ν}u + αHh^{3}ν + αHhν^{3}

= −τu_{t} − τgu − τ ∫_{0}^{t} g′(t − s)(u(s, ·) − u(t, ·))ds
+τgu_{0} + αHh^{3}ν − αHhν^{3}.

Together with

∫_{0}^{t} g′(t − s)(u(s, ·) − u(0, ·))ds ≤ ( ∫_{0}^{t} |g′(s)|ds)^{1/2} (|g′|□u)^{1/2}

we arrive at

∫_{Ω} |f|^{2}dx ≤ c ∫_{Ω} |∇ × h|^{2}dx + c ∫_{Ω} |∇θ|^{2}dx,

∫_{Γ_{1}} |\frac{∂u}{∂ν_{A}} |^{2} dz ≤ c ∫_{Γ_{1}} (|u_{t}|^{2} + g^{2}|u|^{2} + |g′|□u)dx

+cg^{2}F (0) + c ∫_{Ω} |∇ × h|^{2}dx.

Using

∫_{Ω} |h|^{2}dx + ∫_{Ω} |∇h|^{2}dx ≤ c ∫_{Ω} |∇ × h|^{2}dx

in our situation because of the properties of h and the fact that Ω is simply connected (see page 358 in [5] or page 157 in [10]), we thus have proved

Lemma 3.5

\frac{d}{dt}χ(t) ≤ −\frac{1}{2} ∫_{Ω} (|u_{t}|^{2} + C_{ijml}u_{,l}^{m}u_{,j}^{i})dx − \frac{1}{4} ∫_{Γ_{1}} C_{ijml}u_{,l}^{m}u_{,j}^{i}dz

+c ∫_{Ω} (|∇ × h|^{2} + |∇θ|^{2})dx + c ∫_{Γ_{1}} |u_{t}|^{2}dz

+c ∫_{Γ_{1}} |g′|□u dz + cg^{2}F (0).

Now we can formulate and prove the main results of this section. First we assume the case where the kernel decays exponentially.

Theorem 3.6 Let g be an exponentially decaying resolvent kernel as in (1.16), and assume (3.1). Then the energy F defined in (3.3), which is associated to the solution of the intial boundary value problem (1.1)–(1.3), (1.7), (1.10), (1.11) decays exponentially, i.e.

∃d_{0}, d_{1} > 0 ∀t ≥ 0 : F (t) ≤ d_{0}e^{−d_{1}t}F (0).

Proof: Let

L(t) := NF (t) + χ(t),  (3.12)

N > 0 sufficiently large. Then there are positive constants k_{0}, k_{1} such that for all t ≥ 0

k_{0}F (t) ≤ L(t) ≤ k_{1}F (t).  (3.13)

Moreover, for N large enough, using the Lemmas 3.3 and 3.5,

\frac{d}{dt}L(t) ≤ −k_{2}F (t) + cg^{2}(t)F (0)

with a constant k_{2} > 0. Here we used the assumption (1.16) in order to conclude the following estimates

−\frac{τ}{2} ∫_{Γ_{1}} g′′□u dz ≤ c̃∫_{Γ_{1}} g′□u dz

and

\frac{τ}{2} ∫_{Γ_{1}} g′|u|^{2}dz ≤ −c̃∫_{Γ_{1}} g|u|^{2}dz

for the corresponding two terms appearing in Lemma 3.3, where c̃ is a positive constant. Thus we obtain

\frac{d}{dt}L(t) ≤ −\frac{k_{2}}{k_{1}} L(t) + cg^{2}(t)F (0).

Using the exponential decay of g we conclude

∃d̃_{0}, d̃_{1} > 0 ∀t ≥ 0 : L(t) ≤ d̃_{0}e^{−d̃_{1}t}L(0)

which implies the assertion by using (3.13) again.

Q.e.d. Finally, we consider the case where g decays polynomially as in (1.17).

Theorem 3.7 Let g be a polynomially decaying resolvent kernel as in (1.17), and assume (3.1). Then the energy F defined in (3.3), which is associated to the initial boundary value problem (1.1)–(1.3), (1.7), (1.10), (1.11), decays polynomially, i.e.

∃ d_{2} > 0 ∀ t ≥ 0 : F (t) ≤ \frac{d_{2}}{(1 + t)^{p}} F (0).

Proof: We define the functional L as in (3.12) and we have the equivalence to the energy term F as given in (3.13) again. A negative term

−cg(t) ∫_{Γ_{1}} |u|^{2}(t, z)dz

can be obtained from Lemma 3.5 and the estimate

g(t) ∫_{Γ_{1}} |u|^{2}(t, z)dz ≤ c ∫_{Ω} C_{ijml}u_{,l}^{m}u_{,j}^{i}(t, x)dx.

From Lemma 3.3 and Lemma 3.5, using the properties of g′′ from the assumption (1.17) for the term

−\frac{τ}{2} ∫_{Γ_{1}} g′′□u dz,

we obtain

\frac{d}{dt}L(t) ≤ −k_{3} (M(t) + g(t) ∫_{Γ_{1}} |u|^{2}dz + N ∫_{Γ_{1}} |g′|^{1+ \frac{1}{p+1}} □u dz)
+k_{4} ∫_{Γ_{1}} |g′|□u dz + cg^{2}(t)F (0),  (3.14)

where k_{3}, k_{4} denote positive constants and M = M(t) was defined in (3.4).

In the sequel we need several inequalities collected in the next three Lemmas which are based on those from [11], partially extending those.

Lemma 3.8 Let m and h be integrable functions, and let 0 ≤ r < 1 and q > 0. Then, for t ≥ 0:

∫_{0}^{t} |m(t − τ) h(τ)|dτ ≤

(∫_{0}^{t} |m(t − τ)|^{1+ \frac{1−r}{q}} |h(τ)|dτ)^{\frac{q}{q+1}} (∫_{0}^{t} |m(t − τ)|^{r}|h(τ)|dτ)^{\frac{1}{q+1}} .

Proof: Define

v(τ) := |m(t − τ)|^{1− \frac{r}{q+1}} |h(τ)|^{\frac{q}{q+1}} ,
w(τ) := |m(t − τ)|^{\frac{r}{q+1}} |h(τ)|^{\frac{1}{q+1}} .

Then, for fixed t ≥ 0,

|m(t − τ)h(τ)| = |v(τ) h(τ)|.

An application of Hölder’s inequality with exponents

δ = \frac{q}{q + 1} for v

δ^{∗} = q + 1 for w

gives the assertion of Lemma 3.8.

Q.e.d. Lemma 3.9 Let p > 1, 0 ≤ r < 1 and t ≥ 0. Then we have for r > 0:

∫_{Γ_{1}} |g′|□u dz ≤

2 (∫_{0}^{t} |g′(τ)|^{r}dτ∥u∥_{L^{∞}((0,t),L^{2}(Γ_{1}))}^{2})^{\frac{1}{1+(1−r)(p+1)}} (∫_{Γ_{1}} |g′|^{1+ \frac{1}{p+1}} □u dz)^{\frac{(1−r)(p+1)}{1+(1−r)(p+1)}} ,

and for r = 0:

∫_{Γ_{1}} |g′|□u dz ≤

2 (∫_{0}^{t} ∥u(τ, ·)∥_{L^{2}(Γ_{1})}^{2}dτ + t∥u(t, ·)∥_{L^{2}(Γ_{1})}^{2})^{\frac{1}{p+2}} (∫_{Γ_{1}} |g′|^{1+ \frac{1}{p+1}} □u dz)^{\frac{p+1}{p+2}} .

Proof: Apply Lemma 3.8 with m(τ) := |g′(τ)|, h(τ) := ∫_{Γ_{1}} |u(t)−u(τ)|^{2}dz and q := (1−r)(p+1) (t fixed). This proves Lemma 3.9.

Q.e.d. Lemma 3.10 Let f ≥ 0 be differentiable, let α > 0 and let f satisfy

f′(t) ≤ \frac{−c̄_{1}}{f(0)^{1/α}} f(t)^{1+ \frac{1}{α}} + \frac{c̄_{2}}{(1 + t)^{β}} f(0)

for t ≥ 0, positive constants c̄_{1}, c̄_{2} and

β ≥ α + 1.

Then there exists a constant c̄_{3} > 0 such that for t ≥ 0:

f(t) ≤ \frac{c̄_{3}}{(1 + t)^{α}} f(0).

Proof: Let t ≥ 0 and

F (t) := f(t) + \frac{2c̄_{2}}{α} (1 + t)^{−α}f(0).

Then

F′ = f′ − 2c̄_{2}(1 + t)^{−(α+1)}f(0)
≤ \frac{−c̄_{1}}{f(0)^{1/α}} f^{1+ \frac{1}{α}} − c̄_{2}(1 + t)^{−(α+1)}f(0),

where we used β ≥ α + 1. Hence

F′ ≤ \frac{−c}{f(0)^{1/α}} (f^{1+ \frac{1}{α}} + (1 + t)^{−(α+1)}f(0)^{1+ \frac{1}{α}} )
≤ \frac{−c}{F (0)^{1/α}} F^{1+ \frac{1}{α}} .

Integration yields

F (t) ≤ \frac{F (0)}{(1 + ct)^{α}} ≤ \frac{c}{(1 + t)^{α}} f(0)

whence

f(t) ≤ \frac{c̄_{3}}{(1 + t)^{α}} f(0)

follows for some c̄_{3} > 0, which proves Lemma 3.10.

Q.e.d. Applying Lemma 3.9 with r > 0 we get

∫_{Γ_{1}} |g′|^{1+ \frac{1}{p+1}} □u dz ≥

\frac{c}{(∫_{0}^{t} |g′|^{r}(τ)dτ)^{1+ \frac{1}{(1−r)(p+1)}} F (0)^{\frac{1}{(1−r)(p+1)}}} (∫_{Γ_{1}} |g′|□u dz)^{1+ \frac{1}{(1−r)(p+1)}}

(with c = c(r) as long as r is not yet fixed). On the other hand we have

(cg ∫_{Γ_{1}} |u|^{2}dz + M)^{1+ \frac{1}{(1−r)(p+1)}} ≤ cF (0)^{\frac{1}{(1−r)(p+1)}} (M + cg ∫_{Γ_{1}} |u|^{2}dz) .

We conclude from (3.14) using the last two inequalities:

\frac{d}{dt}L(t) ≤ \frac{−c}{F (0)^{\frac{1}{(1−r)(p+1)}}} [(M + cg ∫_{Γ_{1}} |u|^{2}dz)^{1+ \frac{1}{(1−r)(p+1)}}

+ (∫_{Γ_{1}} |g′|□u dz)^{1+ \frac{1}{(1−r)(p+1)}} ] + cg^{2}F (0)

if

r > \frac{1}{p + 1}

(such that ∫_{0}^{∞} |g′(τ)|^{r}dτ < ∞). This implies, using (3.13),

\frac{d}{dt}L(t) ≤ \frac{−c̃}{L(0)^{\frac{1}{(1−r)(p+1)}}} L(t)^{1+ \frac{1}{(1−r)(p+1)}} + cg^{2}(t)L(0)  (3.15)

with some c̃ > 0. Lemma 3.10 with f = L, α = (1 − r)(p + 1) and β = p^{2} gives

L(t) ≤ \frac{c}{(1 + t)^{(1−r)(p+1)}} L(0).  (3.16)

Choosing 1 ≥ r > \frac{1}{p+1} such that

(1 − r)(p + 1) > 1

or, equivalently,

\frac{1}{p + 1} < r < \frac{p}{p + 1},

we obtain from the inequalitity (3.16)

∫_{0}^{∞}F (τ)dτ ≤ c ∫_{0}^{∞}L(τ)dτ ≤ cL(0)  (3.17)

and

t∥u(t, ·)∥_{L^{2}(Γ_{1})}^{2} ≤ ctL(t) ≤ cL(0)  (3.18)

as well as

∫_{0}^{t} ∥u(τ, ·)∥_{L^{2}(Γ_{1})}^{2}dτ ≤ c ∫_{0}^{∞}L(τ)dτ ≤ cL(0).  (3.19)

With the estimates (3.17)–(3.19) we conclude, using Lemma 3.9 again, now with r = 0:

∫_{Γ_{1}} |g′|^{1+ \frac{1}{p+1}} □u dz ≥ \frac{c}{F (0)^{\frac{1}{p+1}}} (∫_{Γ_{1}} |g′|□u dz)^{1+ \frac{1}{p+1}} ,

and hence, with the same arguments as in the derivation of (3.15),

\frac{d}{dt}L(t) ≤ \frac{−c̄}{L(0)^{\frac{1}{p+1}}} L(t)^{1+ \frac{1}{p+1}} + cg^{2}(t)L(0).

This implies by applying Lemma 3.10 again

L(t) ≤ \frac{c}{(1 + t)^{p}} L(0)

and hence, by (3.13) and for some d_{2} > 0:

F (t) ≤ \frac{c}{(1 + t)^{p}} F (0)

which completes the proof of Theorem 3.7.

Q.e.d. Remark: For a thermoviscoelastic system discussed in [11] it was shown that a polynomial relaxation function cannot lead to an exponential decay. This gives a hint for the conjecture that the polynomial decay rate obtained here can not be replaced by an exponential decay result. Indeed, for the system of pure elasticity, with the memory type boundary as discussed here, the exponential decay for exponential kernels was shown in [1]. A merely polynomial kernel there can not lead to a general exponential decay result, which can be seen as follows.

In one space dimension the system of equations of elasticity in Ω := (0, 1) with memory type boundary condition reduces to

u_{tt} − αu_{xx} = 0,  (3.20)
u(t = 0) = u_{0},  u_{t}(t = 0) = u_{1},  (3.21)
u(x = 0) = 0,  (u + (u_{x} ∗ r))(x = 1) = 0,  (3.22)

where α is a positive constant. The energy is given by

E(t) := ∫_{0}^{1} (|u_{t}|^{2} + α|u_{x}|^{2}) (t, x) dx.

We will argue by contradiction. Assume

∃c > 0 ∃δ > 0 ∀t ≥ 0 : E(t) ≤ ce^{−δt}E(0),  (3.23)

and let us take

u_{0} = 0,  u_{1} ∈ C_{0}^{∞}(Ω) \ {0}  (3.24)

and for the kernel b with b(t) = −r′(t)/r(0):

b(t) = \frac{1}{(1 + t)^{p}}

for some p > 1. With the conditions above, v := u_{t} satisfies the same system (3.20)–(3.22) as u, due to the choice of the initial data in (3.24). Hence also the energy associated to v decays exponentially, which implies, using the differential equation and Sobolev’s imbedding theorem, that there is a constant c_{0} depending on the initial data such that for all t ≥ 0:

|u(t, 1)| + |u_{x}(t, 1)| ≤ c_{0}e^{−δt}.  (3.25)

The boundary condition (3.22) can also be stated as (cp. (1.13))

(−b ∗ u_{x} + u_{x} + τu)(x = 1) = 0,

which implies by (3.25)

| ∫_{0}^{t} \frac{1}{(1 + t − s)^{p}} u_{x}(s, 1) ds| ≤ c_{0}e^{−δt}.  (3.26)

On the other hand, by dividing the integral from 0 to t into two parts from 0 to t/2 and from t/2 to t, it can be easily seen that for any m > 1

| ∫_{0}^{t} \frac{1}{(1 + t − s)^{m}} u_{x}(s, 1) ds| ≤ \frac{c_{0}}{(1 + t)^{m}} .  (3.27)

For t ≥ 0 and β ≥ 0 let

G_{β}(t) := ∫_{t+β}^{∞} u_{x}(s, 1) ds.

Then

∫_{0}^{t} \frac{1}{(1 + t − s)^{p}} u_{x}(s, 1) ds = [\frac{1}{(1 + s)^{p}} G_{β}(t − s)]_{s=0}^{s=t} + p ∫_{0}^{t} \frac{1}{(1 + s)^{p+1}} G_{β}(t − s) ds
= \frac{G_{β}(0)}{(1 + t)^{p}} − G_{β}(t) + O (\frac{1}{(1 + t)^{p+1}} ) ,  (3.28)

where we used (3.27) for m = p + 1. Case 1: ∃β̃ ∈ [0, ∞] : Gβ̃(0) ≠ 0. Thus, from (3.28),

lim_{t→∞} |∫_{0}^{t} \frac{1}{(1 + t − s)^{p}} u_{x}(s, 1) ds| (1 + t)^{p} = Gβ̃(0) ≠ 0

which is a contradiction to (3.26). Case 2: ∀β ∈ [0, ∞] : G_{β}(0) = 0. This implies

∀t ≥ 0 : u_{x}(t, 1) = 0

and hence, using the boundary condition and the initial condition u_{0} = 0, we see that u satisfies

u_{tt} − αu_{xx} = 0,
u(t = 0) = 0, u_{t}(t = 0) = u_{1},
u(x = 0) = 0, u(x = 1) = 0,

which implies that the energy

E(t) = ∫_{0}^{1} |u_{1}|^{2}(x) dx

is constant being a contradiction to the assumption (3.23).

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Jaime E. Muñoz Rivera, Department of Research and Development, National Laboratory for Scientific Computation, Rua Getulio Vargas 333, Quitandinha, CEP 25651-070 Petrópolis, RJ, Brasil e-mail: rivera@lncc.br

Reinhard Racke, Department of Mathematics and Statistics, University of Konstanz, Box D 187, 78457 Konstanz, Germany e-mail: reinhard.racke@uni-konstanz.de