Quasilinear parabolic problems with nonlinear boundary conditions

(d) bD ∈ C(J0 × ΓD × U0), ∃Cb1 ∈ Lp(ΓD), Cb1 ≥ 0, ∃Cb2 ∈ Lp(J), Cb2 ≥ 0, and
∃σ2 > 1 − 1
p such that in case κ < 1: ∃σ1 > κ s.t.
|b D xΓ (t, x, ξ) − bD xΓ (t, ¯x, ξ)| ≤ Cb2 (t)|x − ¯x|σ2 , (6.12)
|b D ξ (t, x, ξ) − bD ξ (¯t, x, ξ)| ≤ Cb1 (x)|t − ¯t| σ1 ,
|b D xΓξ (t, x, ξ) − bD xΓξ (t, ¯x, ¯ ξ)| ≤ Cb2 (t)|x − ¯x|σ2 + C|ξ − ¯ ξ|, (6.13)
|b D ξξ (t, x, ξ) − bD ξξ (t, ¯x, ¯ ξ)| ≤ C(|x − ¯x| σ2 + |ξ − ¯ ξ|),
and in case κ > 1: ∃σ1 > κ − 1 s.t. (6.12), (6.13),
|b D t (t, x, ξ) − b D t (¯t, x, ξ)| ≤ Cb1 (x)|t − ¯t| σ1 ,
|b D tξ (t, x, ξ) − bD tξ (¯t, x, ¯ ξ)| ≤ Cb1 (x)|t − ¯t| σ1 + C|ξ − ¯ ξ|,
|b D ξξ (t, x, ξ) − bD ξξ (¯t, ¯x, ¯ ξ)| ≤ C(|t − ¯t| σ1 + |x − ¯x| σ2 + |ξ − ¯ ξ|),
all these inequalities being true for t, ¯t ∈ J0, x, ¯x ∈ ΓD, ξ, ¯ ξ ∈ U0; each of the
derivatives of b D occurring above is Carathéodory and essentially bounded on
J0 × ΓD × U0;
(e) bN ∈ C(J0×ΓN ×U), bN ζ ∈ L∞(J0×ΓN ×U; Rn+1 ) is Carathéodory, ∃Cb1 ∈ Lp(ΓN),
, s.t.
Cb1 ≥ 0, ∃Cb2 ∈ Lp(J), Cb2 ≥ 0, ∃σ1 > (1 + α)( 1 1
2 − 2p ), ∃σ2 > 1 − 1
p
|b N (t, x, ζ) − b N (¯t, ¯x, ζ)| ≤ Cb1 (x)|t − ¯t| σ1 + Cb2 (t)|x − ¯x|σ2 ,
|b N ζ (t, x, ζ) − bN ζ (¯t, ¯x, ¯ ζ)| ≤ Cb1 (x)|t − ¯t| σ1 + Cb2 (t)|x − ¯x|σ2 + C|ζ − ¯ ζ|,
t, ¯t ∈ J0, x, ¯x ∈ ΓN, ζ, ¯ ζ ∈ U;
(H4) (initial data): u0 ∈ Y0; u1 ∈ Y1, if α > 1
p (u1(x) := f(0, x, u0(x), ∇u0(x)), x ∈ Ω);
f(·, ·, u0(·), ∇u0(·)), g(·, ·, u0(·), ∇u0(·)) ∈ X T0 .
(H5) (compatibility): (u0(x), ∇u0(x)) ∈ U0 × U1, x ∈ Ω;
b D (0, x, u0(x)) = 0, x ∈ ΓD;
b N (0, x, u0(x), ∇u0(x)) = 0, x ∈ ΓN;
b D t (0, x, u0(x)) + b D ξ (0, x, u0(x))u1(x) = 0, x ∈ ΓD, if α > 3
2p−1 ;
(H6) (ellipticity): a(0, x, u0(x), ∇u0(x)) ∈ Sym{n}, x ∈ Ω; ∃c0 > 0 s.t.
(H7) (normality):
We have now the following result.
a(0, x, u0(x), ∇u0(x))ϱ · ϱ ≥ c0|ϱ| 2 , x ∈ Ω, ϱ ∈ R n ;
b D ξ (0, x, u0(x)) �= 0, x ∈ ΓD;
b N η (0, x, u0(x), ∇u0(x)) · ν(x) �= 0, x ∈ ΓN.
Theorem 6.1.1 Suppose that the assumptions (H1)-(H7) are satisfied. Let φ ∈ Z T0 be
as above, and assume that ρ ≤ ρ1. Then there exists T3 ∈ (0, T2] such that for each
T ∈ (0, T3] the following statements hold:
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