Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 4, 939–951

Nonlinear Hyperbolic Equations with

Dissipative Temporal and Spatial Non-Local Memory

F. Mošna and J. Nečas

Abstract. The equation governing the evolution of a displacement vector in an elastic body with dissipative temporal and spatial non-local memory is considered. The memory term is generated by a singular but integrable kernel. The existence of a global weak solution to the associated initial-boundary problem is established by constructing Galerkin approximations and deriving a suitable energy estimate.

Keywords: Singular viscoelasticity, energy estimates, weak solutions AMS subject classification: 45 K 05, 73 F 15

1. Introduction

In this paper, the equation governing the evolution of a displacement vector in an elastic body is investigated. The body is assumed to occupy a reference domain Ω ⊂ ℝ^{N} at an initial time and to have unit density. The vector u = (u_{1}, …, u_{N}) represents the displacement and from Newton’s laws of motion we obtain for it the wave equation

ü = div σ + f  (1)

where σ_{ij} is the Cauchy stress tensor and f = (f_{1}, …, f_{N}) is the external body force per unit mass.

This equation holds for both the elastic and the plastic cases. The properties of a material are expressed by the constitutive law, which describes the relation between the stress σ and the infinitesimal strain tensor e_{ij}u = \frac{1}{2}(\frac{∂u_{i}}{∂x_{j}} + \frac{∂u_{j}}{∂x_{i}} ). The stress is a function of eu for an elastic material, while it depends on the velocity of eu in the case of the plastic one. In the linear cases, the relation in an elastic body is expressed by Hook’s law (then (1) is a hyperbolic equation), while for a plastic material Newton’s law holds (then (1) is a parabolic equation).

The stress tensor usually depends on the instantaneous strain

σ_{ij}(x, t) = \frac{∂W}{∂e_{ij}} (eu(x, t))  (2)

F. Mošna: Charles Univ., Fac. Educ., M. D. Rettigové 4, 116 39 Praha 1, Czech Republic J. Nečas: Charles Univ., Fac. Math. & Phys., Sokolovská 83, 186 00 Praha 8, Czech Republic, and Nothern Illinois Univ., Dept. Math. Sci., DeKalb, Illinois 60115, USA ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

where W = W (e_{ij}) is a function of free energy. In the case of one space dimension the problem has been solved. The global existence of a weak solution of the mixed problem follows from a recent work of Di Perna [5] and is based on compensated compactness arguments. In spite of intensive efforts of many mathematicians the question of global existence of a solution to the general nonlinear elastic problem remains open. There are several results in some special cases for dimension N ≥ 2.

Experience indicates that certain materials have memory. It means that the stress depends not only on the strain at the present time t, but also on the entire history of the strain from zero to time t. In this case, the instantaneous stress (2) in equation (1) is extended by the memory part, which usually has the form

∫_{0}^{t} h(t − τ)(eu(x, t) − eu(x, τ))dτ

(h denotes a suitable kernel). At first sight it is surprising that the existence of a solution to such an equation can be proved (see [2, 12]). However, there are other materials where the stress depends not only on the history of the strain at given x, but also on the history at all points located in a neigbourhood of x, more generally on the history at all points of Ω. Our work generalizes the results from [2] for such a type of nonlinear elasticity memory choices, both time and spatial ones. (The singularity in the memory part of stress approaches the Dirac function.) It is the purpose of the present paper to prove global existence of a weak solution.

We will consider the equation

ü_{i}(x, t) − \frac{∂}{∂x_{j}} σ_{ij}(x, t) = f_{i}(x, t) on Ω × (0, ∞) (i = 1, 2, …, N)  (3)

where the stress consists of both the instantaneous and the memory part σ = σ^{I} + σ^{M},

σ_{ij}^{I} = \frac{∂W}{∂e_{ij}} (eu)  (4)

and

σ_{ij}^{M} = −λ ∫_{0}^{t} ∫_{Ω} (e_{ij}u(ξ, τ) − e_{ij}u(ξ, t))\frac{h(t − τ)}{|x − ξ|^{α}} dξdτ,  (5)

with boundary conditions

u(x, ·) = 0  (x ∈ ∂Ω)  (6)

and initial conditions

\begin{matrix}u(·, 0) & = u^{0} \\ u̇(·, 0) & = u^{1}. \end{matrix} \}  (7)

Let the domain Ω ⊂ ℝ^{N} (N ≥ 2) be bounded and let it possess a Lipschitz continuous boundary ∂Ω. We assume that the function W : ℝ^{2N} → ℝ is continuous, has bounded second derivatives, W (0) = \frac{∂W}{∂e_{ij}} (0) = 0 and the condition of ellipticity is satisfied, i.e. there exists a real number κ > 0 such that

\frac{∂^{2}W}{∂e_{ij} ∂e_{kl}} (q) a_{ij}a_{kl} ≥ κ ∥a∥^{2}  (8)

holds for every a, q ∈ ℝ^{2N}. We also suppose h(t) = e^{−t}t^{−ν} where 0 < ν < \frac{1}{2}, N − 1 < α < N, λ > 0, and

f ∈ W^{\frac{ν}{2} ,2}((0, ∞); W^{1− \frac{N−α}{2} ,2}(Ω; ℝ^{N}))
∩ L^{2}((0, ∞); W^{−1,2}(Ω; ℝ^{N})) ∩ L^{∞}((0, ∞); L^{2}(Ω; ℝ^{N}))
u^{0} ∈ W_{0}^{1,2}(Ω; ℝ^{N})
u^{1} ∈ L^{2}(Ω; ℝ^{N}).

We use the Galerkin approximation. The operator \frac{∂}{∂x_{j}} σ_{ij} is compact both in time and space. The memory part of the stress tensor allows us to establish the basic estimates.

We will deal with spaces of functions with non-integer derivatives (see [1, 3, 10, 18]).

Definition 1. Let 0 ≤ s < 2 and let u : Ω → B be a function, where B is a Banach space and Ω ⊂ ℝ^{N} is a domain with Lipschitz continuous boundary. We define

∥u∥_{W^{s,2}(Ω;B)}^{2}

= \{ \begin{matrix}∥u∥_{L^{2}(Ω;B)}^{2} + ∫∫_{Ω×Ω} \frac{∥u(x)−u(y)∥_{B}^{2}}{|x−y|^{N+2s}} dxdy & if 0 < s < 1 \\ ∥u∥_{W^{1,2}(Ω;B)}^{2} + ∑_{i=1}^{N} ∫∫_{Ω×Ω} \frac{∥ \frac{∂u}{∂x_{i}} (x)− \frac{∂u}{∂x_{i}} (y)∥_{B}^{2}}{|x−y|^{N+2(s−1)}} dxdy & if 1 < s < 2. \end{matrix}

The space W^{s,2}(Ω; B) contains the functions u satisfying ∥u∥_{W^{s,2}(Ω;B)} < ∞, W^{0,2}(Ω; B) = L^{2}(Ω; B), and W^{1,2}(Ω; B) is introduced as usual. (If B = ℝ, then we denote W^{s,2}(Ω) = W^{s,2}(Ω; ℝ).) The space W_{0}^{s,2}(Ω) can be introduced as the closure of D(Ω) (test functions) in W^{s,2}(Ω) and we denote the dual space W^{−1,2}(Ω) = (W_{0}^{1,2}(Ω))^{∗}. For −\frac{1}{2} < s < +\frac{1}{2} we have W^{s,2}(Ω) = W_{0}^{s,2}(Ω).

Let {w^{n}}_{n≥1} be a basis in W_{0}^{1,2}(Ω) which is orthonormal in L^{2}(Ω) consisting of the eigenfunctions of the equation

Δw + θw = 0 on Ω

and let {λ_{n}}_{n≥1} denote the corresponding eigenvalues. Then in the space W_{0}^{s,2}(Ω) an equivalent norm can be given by

∥u∥_{W_{0}^{s,2}(Ω)}^{2} ≈ ∑_{i=1}^{∞} λ_{i}^{s} c_{i}^{2} where c_{i} = ∫_{Ω} uw^{i}dx

and

∥u∥_{W_{0}^{s,2}(ℝ^{N})}^{2} ≈ ∫_{ℝ^{N}} (|ξ|^{s}|û(ξ)|)^{2}dξ

where û means the Fourier transform of the function u:

û(ξ) = ∫_{ℝ^{N}} u(x) e^{−i (ξ_{1}x_{1}+...+ξ_{N} x_{N})}dx (ξ ∈ ℝ^{n}).

942 F. Mošna and J. Nečas

We shall use the Parseval equality

∫_{ℝ^{N}} u v̄ dx = \frac{1}{(2π)^{N}} ∫_{ℝ^{N}} û ¯ v̂dξ (u, v ∈ L^{2}(ℝ^{N})), (9)

and rules for the Fourier transform of the convolution and derivatives:

{ \atop u ∗ v} = û v̂,  (10)
ˆ\frac{∂u}{∂x_{j}} (ξ) = −i ξ_{j} û(ξ) (u ∈ S^{∗}(ℝ^{N}), v ∈ L^{2}(ℝ^{N})). (11)

Here the space of temperate distributions S^{∗}(ℝ^{N}) means the dual space to

S(ℝ^{N}) = {φ ∈ C^{∞}(ℝ^{n}) : sup_{x∈ℝ^{N}} |x^{β}D^{α}φ(x)| < ∞ for all multi-indexes α, β ∈ ℕ^{N}}

and the convolution of u and v is introduced by the formula

(u ∗ v)(x) = ∫_{ℝ^{N}} u(ξ) v(x − ξ) dξ.  (12)

We shall need also the Fourier transformation of the power \frac{1}{|·|^{α}} for \frac{N−1}{2} < α < N:

\begin{matrix} \\ ( & \frac{1}{| · |^{α}} & ) \end{matrix}(ξ) = (2π)^{\frac{N}{2}} 2^{\frac{N}{2} −α} \frac{Γ (\frac{N−α}{2})}{Γ (\frac{α}{2})}(\sqrt{ξ_{1}^{2} + … + ξ_{N}^{2}})^{α−N}  (13)

(see [4, 6, 9]).

2. Galerkin approximation

After defining a weak solution of our problem we will construct its approximants by the Galerkin method.

Definition 2. A weak solution to the mixed problem (3) - (7) is a function u ∈ L^{∞}((0, ∞); W_{0}^{1,2}(Ω; ℝ^{N})), for which

\begin{matrix}u̇ & ∈ L^{∞}((0, ∞); L^{2}(Ω; ℝ^{N})) \\ ü & ∈ L^{2}((0, T); W^{−1,2}(Ω; ℝ^{N})) for all T > 0 \end{matrix} \}

and for all v ∈ W_{0}^{1,2}(Ω; ℝ^{N}) and for almost all T > 0 the equality

\begin{matrix}∫_{0}^{T}∫_{Ω} ü_{i}(x, t)v_{i}(x) dxdt + ∫_{0}^{T}∫_{Ω} \frac{∂W}{∂e_{ij}} (eu(x, t)) e_{ij}v(x) dxdt \\ − λ ∫_{0}^{T}∫_{Ω} (∫_{0}^{t}∫_{Ω} (e_{ij}u(ξ, τ) − e_{ij}u(ξ, t))\frac{h(t − τ)}{|x − ξ|^{α}} dξdτ) e_{ij}v(x) dxdt \\ = ∫_{0}^{T}∫_{Ω} f_{i}(x, t)v_{i}(x) dxdt \end{matrix}  (14)

holds (it is necessary to understand the integrals in the sense of distributions).

There exists a basis {w^{n}}_{n≥1} in the space W_{0}^{1,2}(Ω) which is orthonormal in L^{2}(Ω). We construct Galerkin approximants u^{n} of the form

u^{n}(x, t) = ∑_{k=1}^{n} c_{k}^{(n)}(t)w^{k}(x) (n ∈ ℕ).

Using successively w^{1}, …, w^{n} as test functions in (14), we get the following conditions for the functions of time c_{1}^{(n)}, c_{2}^{(n)}, …, c_{n}^{(n)}:

c̈_{m}^{(n)}(t) + ∫_{Ω} \frac{∂W}{∂e_{ij}} (∑_{k=1}^{n} c_{k}^{(n)}(t)ew^{k}(x)) e_{ij}w^{m}(x) dx

− λ ∑_{k=1}^{n} ∫∫_{Ω×Ω} \frac{e_{ij}w^{k}(ξ)e_{ij}w^{m}(x)}{|x − ξ|^{α}} dξdx · ∫_{0}^{t} (c_{k}^{(n)}(τ) − c_{k}^{(n)}(t))h(t − τ) dτ

= ∫_{Ω} f_{i}(x, t)w_{i}^{m}(x) dx

with initial conditions

\begin{matrix}c_{m}^{(n)}(0) & = ∫_{Ω} u_{i}^{0}w_{i}^{m}dx \\ ċ_{m}^{(n)}(0) & = ∫_{Ω} u_{i}^{1}w_{i}^{m}dx \end{matrix} \} (m = 1, …, n).

This problem possesses a unique solution on an interval [0, δ) (δ is the maximal time of existence of the solution, see [8] or [11]). Thus there exist approximate solutions u^{n} satisfying the equation

\begin{matrix}∫_{Ω} ü_{i}^{n}v_{i}dx + ∫_{Ω} \frac{∂W}{∂e_{ij}} (eu^{n}) e_{ij}vdx \\ − λ ∫_{Ω} (∫_{0}^{t}∫_{Ω} (e_{ij}u^{n}(ξ, τ) − e_{ij}u^{n}(ξ, t))\frac{h(t − τ)}{|x − ξ|^{α}} dξdτ) e_{ij}v(x) dx \\ = ∫_{Ω} f_{i}v_{i} dx \end{matrix}  (15)

for all v ∈ sp {w^{1}, …, w^{n}} (the subspace spanned by w^{1}, …, w^{n}).

3. Basic estimates

For the approximate solution introduced in the previous section we may establish the following estimates.

Lemma 1. For any T ∈ [0, δ) the solution u^{n} of (15) satisfies for some C_{1} > 0 the inequality

\frac{1}{2} ∫_{Ω} ∥u̇^{n}(T)∥^{2}dx + ∫_{Ω} W (eu^{n}(T)) dx

+ λC_{1} ∫_{0}^{T}∫_{0}^{T}∫_{ℝ^{N}} (|ξ|^{1− \frac{N−α}{2}} |u^{n}(ξ, t) − u^{n}(ξ, τ)|)^{2}dξ \frac{dτdt}{(t − τ)^{1+ν}}

≤ \frac{1}{2} ∫_{Ω} ∥u^{1}∥^{2}dx + ∫_{Ω} W (eu^{0}) dx + ∫_{0}^{T}∫_{Ω} f_{i}u̇_{i}^{n}dxdt.

Proof. Let us extend u^{n} by zero outside Ω. We put the time derivatives u̇^{n}(·, t) as test functions into expression (15) and integrate over (0, T):

\begin{matrix}\frac{1}{2} ∫_{Ω} ∥u̇^{n}(T)∥^{2}dx − \frac{1}{2} ∫_{Ω} ∥u̇^{n}(0)∥^{2}dx + ∫_{Ω} W (eu^{n}(T)) dx − ∫_{Ω} W (eu^{n}(0)) dx \\ + λ ∫_{0}^{T}∫_{Ω} (∫_{0}^{t}∫_{Ω} (e_{ij}u^{n}(ξ, t) − e_{ij}u^{n}(ξ, τ))\frac{h(t − τ)}{|x − ξ|^{α}} dξdτ) e_{ij}u̇^{n}(x, t) dxdt \\ = ∫_{0}^{T}∫_{Ω} f_{i}u̇_{i}^{n} dxdt. \end{matrix} (16)

We can write the last integral on the left-hand side of (16) as a convolution (12), then use the Parseval equality (9) and the properties of the Fourier transform of a convolution, derivatives and powers (10), (11) and (13) to get

∫∫_{Ω×Ω} \frac{e_{ij}u^{n}(ξ, t) − e_{ij}u^{n}(ξ, τ)}{|x − ξ|^{α}} e_{ij}u̇^{n}(x, t) dξdx

= ∫_{ℝ^{N}} [(e_{ij}u^{n}(t) − e_{ij}u^{n}(τ)) ∗ \frac{1}{| · |^{α}} ] \overline{e_{ij}u̇^{n}(t)} dx

= (2π)^{\frac{N}{2}} 2^{\frac{N}{2} −α} \frac{Γ (\frac{N−α}{2})}{Γ (\frac{α}{2})} \frac{1}{2}[∫_{ℝ^{N}} |ξ|^{2+α−N}(û_{i}^{n}(ξ, t) − û_{i}^{n}(ξ, τ))\overline{ˆu̇_{i}^{n}(ξ, t)} dξ

+ ∫_{ℝ^{N}} |ξ|^{α−N}ξ_{i}ξ_{j}(û_{i}^{n}(ξ, t) − û_{i}^{n}(ξ, τ)) \overline{ˆu̇_{j}^{n}(ξ, t)} dξ].

Denoting by n = (n_{t}, n_{τ}) the outer normal to ∂M_{T}, where

M_{T} = {(t, τ) : 0 < t < τ and 0 < τ < T },

we compute that for some d_{1} > 0

2 ∫_{0}^{T}∫_{0}^{t}∫_{ℝ^{N}} |ξ|^{2+α−N}(û_{i}^{n}(ξ, t) − û_{i}^{n}(ξ, τ)) \overline{ˆu̇_{i}^{n}(ξ, t)} dξ h(t − τ) dτdt

= ∫_{ℝ^{N}} |ξ|^{2+α−N} ∫_{M_{T}} \frac{d}{dt}|û^{n}(ξ, t) − û^{n}(ξ, τ)|^{2}h(t − τ) dτdtdξ

= ∫_{ℝ^{N}} |ξ|^{2+α−N}( ∫_{∂M_{T}} |û^{n}(ξ, t) − û^{n}(ξ, τ)|^{2}h(t − τ)n_{t} dS

− ∫_{M_{T}} \frac{d}{dt}|û^{n}(ξ, t) − û^{n}(ξ, τ)|^{2}h′(t − τ) dτdt)dξ

≥ d_{1} ∫_{0}^{T}∫_{0}^{T}∫_{ℝ^{N}} (|ξ|^{1− \frac{N−α}{2}} |û^{n}(ξ, t) − û^{n}(ξ, τ)|)^{2} dξ \frac{dτdt}{(t − τ)^{1+ν}}

holds (we use integration by parts and take into account that h′ is negative on (0, ∞)). Similarly

∫_{0}^{T}∫_{0}^{t}∫_{ℝ^{N}} |ξ|^{α−N}ξ_{i}ξ_{j}(û_{i}^{n}(ξ, t) − û_{i}^{n}(ξ, τ)) \overline{ˆu̇_{j}^{n}(ξ, t)}h(t − τ) dτdt

= ∫_{ℝ^{N}} |ξ|^{α−N} ∫_{M_{T}} \frac{d}{dt} [∑_{i=1}^{N} ξ_{i}(û_{i}^{n}(ξ, t) − û_{i}^{n}(ξ, τ))]^{2} h(t − τ) dτdtdξ

≥ 0

and the lemma is proved

It follows from Lemma 1 that the approximants u^{n} are defined on the whole interval [0, ∞).

Using the Gronwall lemma we obtain the following corollary.

Corollary. There exists a constant C_{2} > 0 such that

\begin{matrix}∥u^{n}∥_{L^{∞}((0;∞);W_{0}^{1,2}(Ω;ℝ^{N}))} & ≤ C_{2} \\ ∥u̇^{n}∥_{L^{∞}((0;∞);L^{2}(Ω;ℝ^{N}))} & ≤ C_{2} \end{matrix} \} .

For any T > 0 there exists a constant C_{3}(T) > 0 such that

∥u^{n}∥_{W ^{\frac{ν}{2} ,2}((0;T);W^{1− \frac{N−α}{2} ,2}(Ω;ℝ^{N}))} ≤ C_{3}(T).

Lemma 2. For any T > 0 there exists a constant C_{4}(T) > 0 such that the solution u^{n} of (15) satisfies

∥ü^{n}∥_{W ^{\frac{ν}{2} ,2}((0;T);W^{−1− \frac{N−α}{2} ,2}(Ω;ℝ^{N}))} ≤ C_{4}(T).

Proof. Let ε = \frac{N−α}{2} and denote by R^{n} the projection operator mapping the space W_{0}^{1+ε,2}(Ω; ℝ^{N}) to sp (w^{1}, …, w^{n}). The starting point of our consideration will be the definition

∥ü^{n}∥_{W ^{\frac{ν}{2} ,2}((0;T);W^{−1−ε,2}(Ω;ℝ^{N}))}^{2} ≈ ∫_{0}^{T}∫_{0}^{T} \frac{∥ü^{n}(t_{1}) − ü^{n}(t_{2})∥_{W^{−1−ε,2}}^{2}}{|t_{1} − t_{2}|^{1+ν}} dt_{1}dt_{2}

and

∥ü^{n}(·, t_{1}) − ü^{n}(·, t_{2})∥_{W^{−1−ε,2}(Ω;ℝ^{N})}

= sup_{∥ψ∥_{W{1+ε,2 \atop 0}} ≤1} ∫_{Ω} (ü_{i}^{n}(t_{1}) − ü_{i}^{n}(t_{2}))ψ_{i} dx

= sup_{∥ψ∥_{W{1+ε,2 \atop 0}} ≤1} ∫_{Ω} (ü_{i}^{n}(t_{1}) − ü_{i}^{n}(t_{2}))(R^{n}ψ)_{i} dx.

If we extend any function φ ∈ W_{0}^{1+ε,2}(Ω; ℝ^{N}) by zero outside Ω, then

∥φ∥_{W_{0}^{s,2}(ℝ^{N})}^{2} ≈ ∫_{ℝ^{N}} (|ξ|^{s}|φ̂(ξ)|)^{2}dξ ( − \frac{3}{2} < s < +\frac{3}{2}).

We choose any function ψ ∈ W_{0}^{1+ε,2}(Ω; ℝ^{N}) with ∥ψ∥_{W_{0}^{1+ε,2}(Ω;ℝ^{N})} ≤ 1, denote φ = R^{n}ψ ∈ sp (w^{1}, …, w^{n}) and use equality (15) for u^{n}(t_{1}) and u^{n}(t_{2}). First we estimate

\begin{matrix}|∫_{Ω} (\frac{∂W}{∂e_{ij}} (eu^{n}(x, t_{1})) − \frac{∂W}{∂e_{ij}} (eu^{n}(x, t_{2}))) \frac{∂φ_{i}}{∂x_{j}} (x) dx| \\ ≤ d_{2} ∑_{i,j,k,l} |∫_{ℝ^{N}} \frac{∂}{∂x_{l}} (u_{k}^{n}(x, t_{1}) − u_{k}^{n}(x, t_{2}))\frac{∂φ_{i}}{∂x_{j}} (x) dx| \\ ≤ d_{3} ∑_{i,k} ∫_{ℝ^{N}} (|ξ|^{1−ε} |û_{k}^{n}(ξ, t_{1}) − û_{k}^{n}(ξ, t_{2})|)^{2} (|ξ|^{1+ε} |φ̂_{i}(ξ)|)^{2}dξ \\ ≤ d_{4} ∥u^{n}(t_{1}) − u^{n}(t_{2})∥_{W^{1−ε,2}}^{2}∥φ∥_{W^{1+ε,2}}^{2}. \end{matrix}  (17)

Let us remark that ∥u^{n}∥_{W ^{\frac{ν}{2}} ((0,T);W^{1−ε,2}(Ω;ℝ^{N}))} ≤ C_{3}(T) by Corollary. We can proceed similarly in the case of the difference of the right-hand sides of equation (15). It is sufficient to look at the term in (15) generated by the memory portion of the stress σ_{ij}^{M}. For 0 ≤ t_{2} < t_{1} < T we have

\begin{matrix}∫_{Ω} [ ∫_{0}^{t_{1}}∫_{Ω} \frac{e_{ij}u^{n}(ξ, t_{1}) − e_{ij}u^{n}(ξ, τ)}{|x − ξ|^{α}} h(t_{1} − τ) dξdτ \\ − ∫_{0}^{t_{2}}∫_{Ω} \frac{e_{ij}u^{n}(ξ, t_{2}) − e_{ij}u^{n}(ξ, σ)}{|x − ξ|^{α}} h(t_{2} − σ) dξdσ]e_{ij}φ(x) dx \\ = ∫∫_{Ω×Ω} [ ∫_{0}^{t_{1}} (e_{ij}u^{n}(ξ, t_{1}) − e_{ij}u^{n}(ξ, t_{1} − s)) h(s) ds \\ − ∫_{0}^{t_{2}} (e_{ij}u^{n}(ξ, t_{2}) − e_{ij}u^{n}(ξ, t_{2} − s)) h(s) ds]\frac{e_{ij}φ(x)}{|x − ξ|^{α}} dξdx \\ = ∫_{ℝ^{N}} \{ ∫_{0}^{t_{2}} (e_{ij}u^{n}(ξ, t_{1}) − e_{ij}u^{n}(ξ, t_{2})) h(s) ds \\ − ∫_{0}^{t_{2}} (e_{ij}u^{n}(ξ, t_{1} − s) − e_{ij}u^{n}(ξ, t_{2} − s)) h(s) ds \\ + ∫_{t_{2}}^{t_{1}} (e_{ij}u^{n}(ξ, t_{1}) − e_{ij}u^{n}(ξ, t_{1} − s)) h(s) ds\}(e_{ij}φ ∗ \frac{1}{| · |^{α}} )(ξ) dξ. \end{matrix} (18)

The last line of (18) contains three parts. We can write the symmetric parts of the gradient and estimate the corresponding integrals (similarly as in (17)) as

\begin{matrix}|∫_{0}^{t_{2}}∫_{ℝ^{N}} (\frac{∂φ_{i}}{∂x_{j}} ∗ \frac{1}{| · |^{α}} ) (\frac{∂u_{k}^{n}}{∂x_{l}} (t_{1}) − \frac{∂u_{k}^{n}}{∂x_{l}} (t_{2})) dξ h(s) ds| \\ ≤ d_{5}(T) ∥φ∥_{W^{1+ε,2}}∥u^{n}(t_{1}) − u^{n}(t_{2})∥_{W^{1−ε,2}} \\ ≤ d_{6}(T) ∥u^{n}(t_{1}) − u^{n}(t_{2})∥_{W^{1−ε,2}}. \end{matrix}  (19)

The second integral can be estimated as

\begin{matrix}|∫_{0}^{t_{2}}∫_{ℝ^{N}} (\frac{∂φ_{i}}{∂x_{j}} ∗ \frac{1}{| · |^{α}} ) (\frac{∂u_{k}^{n}}{∂x_{l}} (t_{1} − s) − \frac{∂u_{k}^{n}}{∂x_{l}} (t_{2} − s)) dξ h(s) ds| \\ ≤ d_{7}(T) ∥φ∥_{W^{1+ε,2}} ∫_{0}^{t_{2}} ∥u^{n}(t_{1} − s) − u^{n}(t_{2} − s)∥_{W^{1−ε,2}}h(s) ds \\ ≤ d_{8}(T) \{∫_{0}^{t_{2}} ∥u^{n}(t_{1}) − u^{n}(t_{2})∥_{W^{1−ε,2}}^{2}ds\}^{\frac{1}{2}} \end{matrix}  (20)

and

∫_{0}^{T}∫_{0}^{t_{1}}∫_{0}^{t_{2}} \frac{∥u^{n}(t_{1} − s) − u^{n}(t_{2} − s)∥_{W^{1−ε,2}}^{2}}{|(t_{1} − s) − (t_{2} − s)|^{ν+1}} ≤ d_{9}(T) ∥u^{n}∥_{W ^{\frac{ν}{2}} ((0,T);W^{1−ε,2}(Ω;ℝ^{N}))}.

Analogously we get

∫_{t_{2}}^{t_{1}}∫_{ℝ^{N}} (\frac{∂φ_{i}}{∂x_{j}} ∗ \frac{1}{| · |^{α}} ) (\frac{∂u_{k}^{n}}{∂x_{l}} (t_{1}) − \frac{∂u_{k}^{n}}{∂x_{l}} (t_{1} − s)) dξ h(s) ds

≤ d_{10}(T) ∫_{t_{2}}^{t_{1}} ∥u^{n}(t_{1}) − u^{n}(t_{1} − s)∥_{W^{1−ε,2}}h(s) ds

≤ d_{10}(T) ∫_{t_{2}}^{t_{1}} ∥u^{n}(t_{1}) − u^{n}(t_{1} − s)∥_{W^{1,2}}h(s) ds.

Hence

\begin{matrix}∫ & {T \atop 0} & ∫ & {t_{1} \atop 0} & [∫_{t_{2}}^{t_{1}} ∥u^{n}(t_{1}) − u^{n}(t_{1} − s)∥_{W^{1,2}}h(s) ds] & 2 & \frac{dt_{2}dt_{1}}{(t_{1} − t_{2})^{ν+1}} \\ ≤ sup_{τ∈[0,T]} ∥u^{n}(τ)∥_{W^{1,2}}^{2} ∫_{0}^{T}∫_{0}^{t_{1}} (∫_{t_{2}}^{t_{1}} h(s) ds)^{2} \frac{dt_{2}dt_{1}}{(t_{1} − t_{2})^{ν+1}} \\ ≤ d_{11}(T) ∥u^{n}∥_{L^{∞}((0,T);W^{1,2}(Ω;ℝ^{N}))}^{2}. \end{matrix}  (21)

Lemma 2 follows now from definitions and estimates (19) - (21) and Corollary

4. Interpolation

Let 1 < μ < \frac{3}{2} and −\frac{3}{2} < β < +\frac{3}{2}. We can introduce spaces W^{μ,2}((0, T); W_{0}^{β,2}(Ω)) by Definition 1. Then v ∈ W^{μ,2}((0, T); W_{0}^{β,2}(Ω)) may be expanded into the double Fourier series

v = ∑_{i=0}^{∞} ∑_{j=1}^{∞} c_{i,j} h_{i}(t)w^{j}(x)

where h_{0}(t) = \frac{1}{\sqrt{T}} and h_{i}(t) = \sqrt{\frac{2}{T}} cos \frac{iπ}{T} t (i ∈ ℕ). We use the equivalent norm

∥v∥_{W^{μ,2}((0,T);W_{o}^{β,2}(Ω))}^{2} ≈ ∑_{i=0}^{∞} ∑_{j=1}^{∞} c_{i,j}^{2}(1 + i^{2})^{μ}λ_{j}^{β}.

Lemma 3. Let 0 < δ < \frac{1}{2}, 0 < ε < \frac{1}{2} and 0 ≤ γ ≤ 1. Then there exists a constant C_{5} > 0 such that

∥v∥_{W^{(1+δ)γ,2}((0,T);W^{−(1+ε)γ,2}(Ω))} ≤ C_{5}∥v∥_{L^{2}((0,T);L^{2}(Ω))}^{1−γ}∥v∥_{W^{1+δ,2}((0,T);W^{−1−ε,2}(Ω))}^{γ}.

Proof. We compute directly that

∥v∥_{W^{(1+δ)γ,2}((0,T);W^{−(1+ε)γ,2}(Ω))}

= ∑_{i=0}^{∞} ∑_{j=1}^{∞} c_{ij}^{2}(1 + i^{2})^{(1+δ)γ}λ_{j}^{−(1+ε)γ}

= ∑_{i=0}^{∞} ∑_{j=1}^{∞} c_{ij}^{2γ}(1 + i^{2})^{γ(1+δ)} · λ_{j}^{−γ(1+ε)}c_{ij}^{2(1−γ)}

≤ C_{5} ∑_{i=0}^{∞} ∑_{j=1}^{∞} (c_{ij}^{2}(1 + i^{2})^{(1+δ)}λ_{j}^{−(1+ε)})^{γ} ∑_{i=0}^{∞} ∑_{j=1}^{∞} (c_{ij}^{2}(1 + i^{2})^{0}λ_{j}^{0})^{1−γ}

= C_{5}∥v∥_{L^{2}((0,T);L^{2}(Ω))}^{1−γ}∥v∥_{W^{1+δ,2}((0,T);W^{−1−ε,2}(Ω))}^{γ}

and the lemma is proved

5. Existence of a weak solution

The following theorem establishes the Lipschitz continuity of the operator σ^{M}.

Theorem 1. There exists p > 0, independent of T, such that

∥σ^{M}u − σ^{M}v∥_{L^{2}((0,T);L^{2}(Ω;ℝ^{N^{2}} ))} ≤ λp ∥eu − ev∥_{L^{2}((0,T);L^{2}(Ω;ℝ^{N^{2}} ))}.

Proof. Using twice the Schwarz inequality we get

\frac{1}{λ^{2}} (σ_{ij}^{M}u − σ_{ij}^{M}v)^{2}

= (∫_{0}^{t}∫_{Ω} [(e_{ij}u(ξ, t) − e_{ij}v(ξ, t)) − (e_{ij}u(ξ, τ) − e_{ij}v(ξ, τ))]\frac{h(t − τ)}{|x − ξ|^{α}} dξdτ)^{2}

≤ ∫_{Ω} (e_{ij}u(ξ, t) − e_{ij}v(ξ, t))^{2} \frac{dξ}{|x − ξ|^{α}} ∫_{Ω} \frac{dξ}{|x − ξ|^{α}} (∫_{0}^{t} h(t − τ) dτ)^{2}

+ ∫_{Ω} ∫_{0}^{t} (e_{ij}u(ξ, τ) − e_{ij}v(ξ, τ))^{2} \frac{h(t − τ)}{|x − ξ|^{α}} dτdξ ∫_{Ω} \frac{dξ}{|x − ξ|^{α}} ∫_{0}^{t} h(t − τ) dτ.

We denote p_{1} = ∫_{0}^{∞} h(τ) dτ and p_{2} = ∫_{B(0,diam Ω)} \frac{1}{|x|^{α}} dx. Changing the order of integration we go on to compute

\frac{1}{λ^{2}} ∥σ^{M}u − σ^{M}v∥_{L^{2}(L^{2})}^{2}

≤ p_{1}^{2}p_{2} ∫_{Ω} ( ∫_{0}^{T} (e_{ij}u(ξ, t) − e_{ij}v(ξ, t))^{2}dt) (∫_{Ω} \frac{dx}{|x − ξ|^{α}} ) dξ

+ p_{1}p_{2} ∫_{Ω} ∫_{0}^{T} (e_{ij}u(ξ, τ) − e_{ij}v(ξ, τ))^{2}( ∫_{τ}^{T} h(t − τ) dt) (∫_{Ω} \frac{d(ξ)}{|x − ξ|^{α}} ) dτdξ

≤ 2p_{1}^{2}p_{2}^{2} ∥eu − ev∥_{L^{2}(L^{2})}^{2}

and the theorem is proved

Theorem 2 (Existence of weak solutions). Let us consider equation (3) − (5) with boundary and initial conditions (6) and (7). Let the just introduced assumptions be satisfied. Moreover, let ν > N − α and c_{0}κ > pλ where c_{0} is the constant in Korn’s inequality. Then problem (3)−(7) possesses weak solutions u on (0, ∞). These solutions satisfy

\begin{matrix}u & ∈ L^{∞}((0, ∞); W_{0}^{1,2}(Ω; ℝ^{N})) \\ u̇ & ∈ L^{∞}((0, ∞); L^{2}(Ω; ℝ^{N})) \\ ü & ∈ L^{2}((0, T); W^{−1,2}(Ω; ℝ^{N})) \\ u & ∈ W^{\frac{ν}{2} ,2}((0, T); W^{1− \frac{N−α}{2} ,2}(Ω; ℝ^{N})) \\ ü & ∈ W^{\frac{ν}{2} ,2}((0, T); W^{−1− \frac{N−α}{2} ,2}(Ω; ℝ^{N})) \end{matrix} \}

for all T > 0.

Proof. Let us choose any T > 0. As u̇_{i}^{n} is a bounded sequence in L^{2}((0, T); L^{2}(Ω)) and W^{1+ \frac{ν}{2} ,2}((0, T); W^{−1− \frac{N−α}{2} ,2}(Ω)), we get from Lemma 3 (putting v = u̇_{i}^{n}) that u̇_{i}^{n} is bounded also in the space W^{γ(1+ \frac{ν}{2} ),2}((0, T); W^{−γ(1+ \frac{N−α}{2} ),2}(Ω)) where \frac{1}{1+ \frac{ν}{2}} < γ <

\frac{1}{1+ \frac{N−α}{2}} . It is possible to choose such γ, because ν > N −α. The space W^{γ(1+ \frac{ν}{2} ),2}((0, T);

W^{−γ(1+ \frac{N−α}{2} ),2}(Ω)) is compactly embedded into W^{1,2}((0, T); W^{−1,2}(Ω)). Thus we can choose a subsequence u^{n_{k}} which converges to a certain function u in the following sense:

\begin{matrix}u^{n_{k}} & ⇀ u  in L^{2}((0, T); W_{0}^{1,2}(Ω; ℝ^{N})) \\ u̇^{n_{k}} & ⇀ u̇  in L^{2}((0, T); L^{2}(Ω; ℝ^{N})) \\ u^{n_{k}} & ⇀ u  in W^{\frac{ν}{2} ,2}((0, T); W^{1− \frac{N−α}{2} ,2}(Ω; ℝ^{N})) \\ ü^{n_{k}} & ⇀ ü  in W^{\frac{ν}{2} ,2}((0, T); W^{−1− \frac{N−α}{2} ,2}(Ω; ℝ^{N})) \\ ü^{n_{k}} & → ü  in L^{2}((0, T); W^{−1,2}(Ω; ℝ^{N})) \end{matrix} \} .

Now, let P_{n} be the projection operator from L^{2}((0, T); W_{0}^{1,2}(Ω; ℝ^{N})) to the space spanned by the vectors c_{j}(t) w^{j}(x) where c_{j} ∈ L^{2}(0, T) (j = 1, …, n). We have P_{n}u → u in L^{2}((0, T); W_{0}^{1,2}(Ω; ℝ^{N})). We put v = u^{n} − P_{n}u as a test function into equality (15) to obtain

∫_{0}^{T}∫_{Ω} ü_{i}^{n_{k}}(u_{i}^{n_{k}} − (P_{n_{k}}u)_{i}) dxdt + ∫_{0}^{T}∫_{Ω} \frac{∂W}{∂e_{ij}} (eu^{n_{k}}) e_{ij}(u^{n_{k}} − P_{n_{k}}u) dxdt

− λ ∫_{0}^{T}∫_{Ω} [ ∫_{0}^{t}∫_{Ω} (e_{ij}u^{n_{k}}(ξ, τ) − e_{ij}u^{n_{k}}(ξ, t)) \frac{dξ}{|x − ξ|^{α}} h(t − τ) dτ]

× e_{ij}(u^{n_{k}} − P_{n_{k}}u) dxdt

= ∫_{0}^{T}∫_{Ω} f_{i} (u_{i}^{n_{k}} − (P_{n_{k}}u)_{i}) dxdt.

The first and the last integrals tend to 0. We obtain a lower estimate from the condition of ellipticity (8) and Korn’s inequality:

\begin{matrix}∫_{0}^{T}∫_{Ω} \frac{∂W}{∂e_{ij}} (e(u^{n_{k}} − P_{n_{k}}u))e_{ij}(u^{n_{k}} − P_{n_{k}}u) dxdt \\ ≥ κc_{0} ∥u^{n_{k}} − P_{n_{k}}u∥_{L^{2}((0,T);W_{o}^{1,2}(Ω;ℝ^{n}))}^{2}. \end{matrix}

As σ^{M} is Lipschitz continuous we have

\begin{matrix}∫_{0}^{T}∫_{Ω} σ_{ij}^{M}(u^{n_{k}} − P_{n_{k}}u) e_{ij}(u^{n_{k}} − P_{n_{k}}u) dxdt \\ ≤ λp ∥u^{n_{k}} − P_{n_{k}}u∥_{L^{2}((0,T);W_{0}^{1,2}(Ω;ℝ^{N}))}^{2}. \end{matrix}

As

\begin{matrix}∫_{0}^{T}∫_{Ω} \frac{∂W}{∂e_{ij}} (eP_{n_{k}}u) e_{ij}(u^{n_{k}} − P_{n_{k}}u) dxdt & → 0 \\ ∫_{0}^{T}∫_{Ω} σ_{ij}^{M}(P_{n_{k}}u) e_{ij}(u^{n_{k}} − P_{n_{k}}u) dxdt & → 0 \end{matrix} \}

and κc_{0} − λp > 0, then u^{n_{k}} − P_{n_{k}}u → 0 and also u^{n_{k}} → u in L^{2}((0, T); W_{0}^{1,2}(Ω; ℝ^{N})). The existence of the required weak solutions follows as a direct consequence of (15)

Nonlinear Hyperbolic Equations 951

References

[1] Adams, R. A.: Sobolev Spaces. New York - San Francisco - London: Acad. Press 1975.

[2] Bellout, H., Bloom, F. and J. Nečas: Existence of global weak solutions to the dynamical problem for a three-dimensional elastic body with singular memory. SIAM J. Math. Anal. 24 (1993), 36 – 45.

[3] Besov, O. V., Il’jin, V. P. and S. M. Nikol’skij: Integral Representations of Functions and Embedding Theorems (in Russian). Moskva: Nauka 1975.

[4] Bochner, S. and K. Chandrasekharan: Fourier Transforms. Princeton: Univ. Press 1949.

[5] Di Perna, R.: Convergence of approximate solutions to conservation laws. Arch. Rat. Mech. Anal. 82 (1983), 27 – 70.

[6] Ditkin, V. A. and A. P. Prudnikov: Integral Transforms and Operation Research (in Russian). Moskva: FizMatGiz 1961.

[7] Gajewski, H., Gröger, K. and K. Zacharias: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Berlin: Akademie-Verlag 1974.

[8] Gripenberg, G., Londen, S.-O. and O. Staffans: Volterra Integral and Functional Equations. Cambridge: Univ. Press 1990.

[9] Hörmander, L.: The Analysis of Linear Partial Differential Operators. Part I. Berlin et al.: Springer-Verlag 1983.

[10] Kufner, A., John, O. and S. Fuçik: Function Spaces. Praha: Academia 1977.

[11] Miller, R. K.: Nonlinear Volterra Integral Equations. Menlo Park: Benjamin 1971.

[12] Milota, J., Nečas, J. and V. Šverák: On weak solutions to a viscoelasticity model. Comm. Math. Univ. Carolin. 31 (1990), 557 – 565.

[13] Nečas, J.: Introduction to the Theory of Nonlinear Eliptic Equations. Leipzig: Teubner Verlag 1983.

[14] Nečas, J. and I. Hlaváček: Mathematical Theory of Elastic and Elastico-Plastic Bodies: An Introduction. Amsterdam - Oxford - New York: Elsevier 1981.

[15] Nohel, J. A., Rogers, R. C. and A. E. Tzavaras: Weak solutions for a nonlinear system in viscoelasticity. Comm. Part. Diff. Eqs. 13 (1988), 97 – 127.

[16] Renardy, M., Hrusa, W.-J. and J. A. Nohel: Initial value problems in viscoelasticity. Appl. Mech. Rev. 41 (1988), 371 – 378.

[17] Renardy, M., Hrusa, W. J. and J. A. Nohel: Mathematical Problems in Viscoelasticity. New York: Longman 1987.

[18] Triebel, H.: Theory of Function Spaces. Leipzig: Akad. Verlagsges. Geest & Portig 1983.

Received 12.06.1997; in revised form 06.07.1999