Quasilinear parabolic problems with nonlinear boundary conditions

where δij denotes Kronecker’s symbol. The function a describes how the material responds
to shear, while b determines its behaviour under compression. Therefore, a is
called shear modulus and b compression modulus. The constitutive law (5.7) becomes
� ∞
S(t, x) = 2
0
0
da(τ, x) ˙
E(t − τ, x) + 1
3 I
0
� ∞
(3db(τ, x) − 2da(τ, x))tr ˙ E(t − τ, x).
Besides, we want to assume that the material is homogeneous, i.e. ρ0 as well as a and
b do not depend on the material points x ∈ Ω. For simplicity, let us put ρ0(x) ≡ 1, x ∈ Ω.
To summarize, we obtain the following integro-differential equation for homogeneous
and isotropic materials:
� ∞
� ∞
ü(t, x) = da(τ)∆ ˙u(t − τ, x) + (db(τ) + 1
da(τ))∇∇ · ˙u(t − τ, x) + f(t, x), (5.10)
3
for all t ∈ R and x ∈ Ω. This equation has to be supplemented by the boundary
conditions (5.3),(5.4).
Let us consider a material which is at rest up to time t = 0, but is then suddenly
moved with the velocity v0(x), x ∈ Ω. More precisely, we want to assume that v(t, x) =
˙u(t, x) = 0, t < 0, x ∈ Ω, and v(0, x) = v0(x), x ∈ Ω. Then the problem (5.10),(5.3),(5.4)
amounts to
⎧
⎪⎨
⎪⎩
∂tv − da ∗ ∆v − (db + 1
3da) ∗ ∇∇ · v = f, t > 0, x ∈ Ω
(2da ∗ ˙
E + 1
3
I(3db − 2da) ∗ tr ˙
0
v = vd, t > 0, x ∈ Γd
E) n = gs, t > 0, x ∈ Γs
v|t=0 = v0, x ∈ Ω,
where we use the notation (dk ∗ φ)(t) = � t
0 dk(τ)φ(t − τ), t > 0.
(5.11)
5.2 Assumptions on the kernels and formulation of the goal
Given f, vd, gs, v0, our goal is to solve (5.11) for v. For this to be possible it is necessary
that the convolution terms in (5.11) do not produce terms involving the displacement u,
which would be the case, if for example a(t) = t, t ≥ 0. That means the problem must
not be hyperbolic. Further we need certain regularity assumptions on a and b so that
we can apply the results from Section 3.5. It turns out that the following class of kernels
is appropriate for our problem.
Definition 5.2.1 A function k : [0, ∞) → R is said to be of type (E) if
(E1) k(0) = 0, and k is of the form k(t) = k0 + � t
0 k1(τ) dτ, t > 0, where k0 ≥ 0 and
k1 ∈ L1, loc(R+);
(E2) k1 is completely monotonic, i.e. k1 ∈ C∞ (0, ∞) and (−1) lk (l)
1 (t) ≥ 0 for all
t > 0, l ∈ N0;
(E3) if k1 �= 0, then k1 ∈ K∞ π
(α, θk1 ), for some α ∈ [0, 1) and θk1 < 2 .
Observe that (E1) and (E2) imply that the function k in Definition 5.2.1, restricted to
the interval (0, ∞), is a Bernstein function, which by definition means that k is R+valued,
infinitely differentiable on (0, ∞), and that k ′ is completely monotonic. We
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