Quasilinear parabolic problems with nonlinear boundary conditions

to write
where
h2(t, τ, x) = h21(t, τ, x) · gt(τ, x) +
+(a − c)(A − B) + (a − b)(A − C)
+(−bξ(t, x, f(t, x)) + bξ(t, x, g(t, x)) + bξ(τ, x, f(τ, x))) · h22(t, τ, x)
+(bξ(t, x, f(t, x)) − bξ(τ, x, f(τ, x)) · (ft(t, x) − gt(t, x)) +
+(bξ(t, x, f(t, x)) − bξ(t, x, g(t, x))) · (ft(t, x) − ft(τ, x))
=: I3(t, τ, x) + I4(t, τ, x) + I5(t, τ, x) + I6(t, τ, x),
h21(t, τ, x) = bξ(t, x, f(t, x)) − bξ(t, x, g(t, x)) − bξ(τ, x, f(τ, x)) + bξ(τ, x, g(τ, x)),
h22(t, τ, x) = ft(t, x) − gt(t, x) − ft(τ, x) + gt(τ, x), t, τ ∈ J, a.a. x ∈ Ω.
The summand I3 can be estimated by mimicking the middle part of the proof of Lemma
6.2.1. Letting
h23(t, τ, x) = f(t, x) − g(t, x) − f(τ, x) + g(τ, x)
and r0 = min{r, r1} we get
|h21(t,τ, x)| ≤
≤ C(|h23(t, τ, x)| + |f − g|∞, m(|f(t, x) − f(τ, x)| + |g(t, x) − g(τ, x)| + |t − τ| r1 ))
≤ C(|t − τ| r [f − g] (C r 1 ) m + |f − g|∞, m(|t − τ| r ([f] (C r 1 ) m + [g] (C r 1 )m) + |t − τ|r1 ))
≤ C(|t − τ| r [f − g]Y m(1 + ρ + [w] (C r 1 )m) + [f − g]Y m|t − τ|r1 )
≤ C|t − τ| r0 [f − g]Y m
for all t, τ ∈ J, and a.a. x ∈ Ω. Thus,
� T � T �
|I3(t, τ, x)|
(
0 0 Ω
p
1
dx dτ dt) p ≤
|t − τ| 1+s1p
≤ C[f − g]Y m(
� T �
|gt(τ, x)|
0 Ω
p � T
(
0
dt
) dx dτ)1 p
|t − τ| 1+(s1−r0)p
|f − g|Y m ≤ Cµ(T )|f − g|Y m.
≤ CT r0−s1 [g]X m 1
Turning to I4, we immediately see that
� T � T �
(
0 0 Ω
As for I5, we estimate
|I4(t, τ, x)| p
1
dx dτ dt) p ≤ M(ρ + |bξ(·, ·, w)|∞,
|t − τ| 1+s1p m)[f − g] 1,s
(Y1 ) m.
|I5(t, τ, x)| ≤ |bξ(t, x, f(t, x)) − bξ(τ, x, f(t, x))||ft(t, x) − gt(t, x)|+
+ |bξ(τ, x, f(t, x)) − bξ(τ, x, f(τ, x))||ft(t, x) − gt(t, x)|
≤ (|btξ|∞, m|t − τ| + |bξξ| ∞, m 2[f] (C r 1 ) m|t − τ|r )|ft(t, x) − gt(t, x)|
≤ C|t − τ| r |ft(t, x) − gt(t, x)|
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