Quasilinear parabolic problems with nonlinear boundary conditions
Quasilinear parabolic problems with nonlinear boundary conditions
Quasilinear parabolic problems with nonlinear boundary conditions
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for all t, τ ∈ J, and a.a. x ∈ Ω. Therefore,<br />
Finally,<br />
� T � T �<br />
(<br />
0 0 Ω<br />
|I5(t, τ, x)| p<br />
1<br />
dx dτ dt) p ≤ CT<br />
|t − τ| 1+s1p r−s1 [f − g]X m 1 .<br />
|I6(t, τ, x)| ≤ |bξξ| ∞, m 2|f − g|∞, m|ft(t, x)−ft(τ, x)| ≤ C[f − g]Y m|ft(t, x)−ft(τ, x)|.<br />
for all t, τ ∈ J, and a.a. x ∈ Ω. So we deduce<br />
� T � T �<br />
(<br />
0 0<br />
The assertion follows now from<br />
Ω<br />
[b(·, ·, f) − b(·, ·, g)] Y 1,s 1<br />
1<br />
|I6(t, τ, x)| p<br />
1<br />
dx dτ dt)<br />
|t − τ| 1+s1p<br />
p ≤ C|f − g|Y m[f] (Y 1,s1 1 ) m<br />
≤ C(ρ + [w] 1,s<br />
(Y 1<br />
1 ) m)|f − g|Y m ≤ C(ρ + µ(T ))|f − g|Y m.<br />
≤ I1 +<br />
6�<br />
(<br />
j=3<br />
� T � T<br />
0<br />
0<br />
�<br />
Ω<br />
|Ij(t, τ, x)| p<br />
1<br />
dx dτ dt) p . �<br />
|t − τ| 1+s1p<br />
Under the corresponding assumptions, cf. (H4) in Section 6.1, we can repeat the above<br />
steps <strong>with</strong> the roles of J and Ω being reversed to obtain the estimate<br />
�<br />
�<br />
≤ C ρ + µ(T ) + |bξ(·, ·, w)|∞, m<br />
[b(·, ·, f) − b(·, ·, g)] Y 1,s 2<br />
2<br />
|f − g| (Y 1,s 1<br />
1<br />
It is further not difficult to check that the same line of arguments also yields<br />
�<br />
�<br />
≤ C ρ + µ(T ) + |bξ(·, ·, w)|∞, m<br />
[b(·, ·, f) − b(·, ·, g)] Y 1,s 2<br />
2<br />
|f − g| (Y 0,s 1<br />
1<br />
∩Y 1,s2 2 ) m, f, g ∈ Σ.<br />
∩Y 1,s2 2 ) m, f, g ∈ Σ,<br />
in the case (k1, k2) = (0, 1). Here, the reader should recall that in the proof of Lemma<br />
6.2.2, we employed the embedding Y 1,s1<br />
1 ∩ Y 1,s2<br />
2 ↩→ Cr (J; C( ¯ Ω)) <strong>with</strong> some r ∈ (s1, 1).<br />
In the case (k1, k2) = (0, 1) the situation is more comfortable since, by assumption, we<br />
even have the embedding Y 0,s1<br />
1 ∩ Y 1,s2<br />
2 ↩→ C(J; C1 (Ω)).<br />
To conclude, we see that the estimate (6.19) holds true for F = Y k1,s1<br />
1 ∩ Y k2,s2<br />
2 . As<br />
already mentioned at the beginning of this section, this result can be transferred to the<br />
spaces Y T D and Y T N considered in Section 6.1 by means of the well-known method of local<br />
coordinates. If we apply the corresponding result to the function b defined by<br />
b(t, x, ξ) = b D (t, x, ξ)−b D (t, x, φ(t, x))−bξ(t, x, φ(t, x))(ξ−φ(t, x)), t ∈ J, x ∈ ΓD, ξ ∈ U0,<br />
then we get, owing to bξ(t, x, ξ) = bD ξ (t, x, ξ) − bD ξ (t, x, φ(t, x)), an estimate of the form<br />
|R φ<br />
D (u) − Rφ<br />
D (v)| Y T D ≤ M(µ(T ) + ρ + |bDξ (·, ·, w) − bDξ (·, ·, φ)|∞)|u − v| ZT for all u, v ∈ Σ(ρ, T, φ) ∪ {φ|J}. Since |b D ξ (·, ·, w) − bD ξ (·, ·, φ)|∞ → 0 as T → 0, the<br />
inequality (6.17) <strong>with</strong> K = D follows. In the same way one can also show the validity<br />
of (6.17) <strong>with</strong> K = N.<br />
We turn now to the spaces X m 1, s = Hs p(J; Lp(Ω)), 0 < s < 1.<br />
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