Quasilinear parabolic problems with nonlinear boundary conditions

to assume that m, c are bounded functions of the form m(t) = m0 + (1 ∗ m1)(t), t > 0,
m(0) = 0, and c(t) = c0 + (1 ∗ c1)(t), t > 0, c(0) = 0, respectively, with m0 > 0, c0 ≥ 0,
and m1, c1 ∈ L1(R+). Here and in the sequel, f1 ∗ f2 denotes the convolution of two
functions defined by (f1 ∗ f2)(t) = � t
0 f1(t − τ)f2(τ) dτ, t ≥ 0.
Without loss of generality we may assume that the material is at zero temperature
up to time t = 0, and is then exposed to a sudden change of temperature u(0, x) = u0(x),
x ∈ Ω; otherwise one has to add a known forcing term in both (1.8) and (1.10) below
that incorporates the history of the temperature up to time t = 0. Then (1.3)-(1.7) yield
∂t(dm ∗ u) − dc ∗ (div σ(∇u)) = h, t > 0, x ∈ Ω, (1.8)
u = ub, t > 0, x ∈ Γb, (1.9)
dc ∗ σ(∇u) = qf , t > 0, x ∈ Γf , (1.10)
u|t=0 = u0, x ∈ Ω. (1.11)
We show now that (1.8)-(1.11) can be transformed to a problem of the form (1.1),
see also [23]. Note first that without restriction of generality we may assume m0 = 1.
By integrating (1.8) with respect to time we obtain
u + m1 ∗ u − c ∗ (div σ(∇u)) = 1 ∗ h + u0, t ≥ 0, x ∈ Ω. (1.12)
Define the resolvent kernel r ∈ L1, loc(R+) associated with m1 as the unique solution of
the convolution equation
r + m1 ∗ r = m1, t ≥ 0.
Application of the operator (I − r∗) to (1.12) then results in
u − (c − r ∗ c) ∗ (div σ(∇u)) = 1 ∗ (h − r ∗ h − ru0) + u0. (1.13)
Using (formally) the chain rule yields
div σ(∇u(t, x)) = Dσ(∇u(t, x)) : ∇ 2 u(t, x), t ≥ 0, x ∈ Ω,
Dσ denoting the Jacobian of σ. Hence, with k = c − r ∗ c, f = h − r ∗ h − ru0, and
a = Dσ, it follows by differentiation of (1.13) that
∂tu − dk ∗ (a(∇u) : ∇ 2 u) = f, t ≥ 0, x ∈ Ω,
which is a special form of the integrodifferential equation in (1.1). Lastly, if c belongs to
a certain class of ’nice’ kernels, one can invert the convolution with the measure dc and
thus rewrite (1.10) as a nonlinear boundary condition of non-memory type as in (1.1).
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Another important application is the theory of viscoelasticity; here problems of the form
(1.1) naturally occur when balance of momentum is combined with nonlinear stress-strain
relations of memory type. General treatises on this field are, for example, Antman [3],
Christensen [12], and Renardy, Hrusa, and Nohel [71], but we also refer the reader to
Chow [11], Engler [33], and Prüss [63]. A short account of the basic equations in the
linear vector-valued case is given in Chapter 5.
Having motivated the investigation of (1.1) by examples from mathematical physics,
we describe next the main result to (1.1), which is stated in Theorem 6.1.2. For T > 0
and 1 < p < ∞, set J = [0, T ] and define the space Z T by
Z T = H 1+α
p (J; Lp(Ω)) ∩ Lp(J; H 2 p(Ω)).
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