Quasilinear parabolic problems with nonlinear boundary conditions

Apply now V#, i+1 := I + k ∗ A#(·, x, Dx) to QH, i+1(g, gb); by (4.36) this gives
whence
Nb+N �
V#, i+1QH, i+1(g, gb) =
Similarly,
j=0
ψj(ϕjg+k ∗ Cj(·, x, Dx)QH, i+1(g, gb))+K (1)
H, i+1 (g, gb)
= g − k ∗ AR(·, x, Dx)QH, i+1(g, gb)) + K (1)
H, i+1 (g, gb),
Vi+1QH, i+1(g, gb) = g + K (1)
H, i+1 (g, gb) = f.
γyB#(t, x ′ Nb �
, Dx)QH, i+1(g, gb) = ψj(ϕjgb+γyCH, j(t, x)QH, i+1(g, gb))+K (2)
H, i+1 (g, gb)
j=0
= gb − b0γyQH, i+1(g, gb) + K (2)
H, i+1 (g, gb),
which entails
γyB(t, x ′ , Dx)QH, i+1(g, gb) = gb + K (2)
H, i+1 (g, gb) = h.
This proves surjectivity of VH, i+1, provided δ is sufficiently small.
All in all, we have shown that there exists a unique solution v of (4.30) in the space
Z. �
4.3 Problems in domains
In this section let Ω ⊂ R n+1 be a domain with compact C 2 -boundary Γ which decomposes
according to Γ = ΓD ∪ ΓN and dist(ΓD, ΓN) > 0. Let further 1 < p < ∞ and J = [0, T ].
We consider the problem
⎧
⎨
⎩
v + k ∗ A(·, x, Dx)v = f, t ∈ J, x ∈ Ω,
v = g, t ∈ J, x ∈ ΓD,
B(t, x, Dx)v = h, t ∈ J, x ∈ ΓN.
Here the differential operators A(t, x, Dx) and B(t, x, Dx) are of the form
respectively
(4.40)
A(t, x, Dx) = −a(t, x) : ∇ 2 x + a1(t, x) · ∇x + a0(t, x), t ∈ J, x ∈ Ω, (4.41)
B(t, x, Dx) = b(t, x) · ∇x + b0(t, x), t ∈ J, x ∈ ΓN.
Put Y = Bs1 pp(J; Lp(ΓN)) ∩ Lp(J; Bs2 pp(ΓN)) with s1 = α( 1 1
2 − 2p ), s2 = 1 − 1
p , as well as
M = �
ri> si Cr1 (J; C(ΓN)) ∩ C(J; Cr2 (ΓN)). Let further ν(x) denote the outer unit
normal of Ω at x ∈ Γ. Then our assumptions read as follows.
(H1) (kernel): k ∈ K1 �
1
(α, θ) with θ < π and α ∈ (0, 2)\ p , 2
2p−1 , �
2 1 3
p−1 , 1 + p , 1 + 2p−1 ;
(H2) (smoothness of coefficients): a ∈ Cul(J×Ω, Sym{n+1}), a1 ∈ L∞(J×Ω, Rn+1 ), a0 ∈
L∞(J × Ω), as well as (b, b0) ∈ Y n+2 in case p > n + 1 + 2/α, and (b, b0) ∈ M n+2 , otherwise;
(H3) (uniform ellipticity): ∃c0 > 0 s.t. a(t, x)ξ · ξ ≥ c0|ξ| 2 , t ∈ J, x ∈ Ω, ξ ∈ Rn+1 ;
(H4) (normality): b(t, x) · ν(x) �= 0, t ∈ J, x ∈ ΓN.
The aim of this section is to prove the subsequent result.
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