Quasilinear parabolic problems with nonlinear boundary conditions

We remark that the constant C2 stems from finding an upper bound for the integral
� T2
T1 |t − τ|−β dt, τ ∈ [0, T2], where β < 1 is a fixed number. Thus if T1 and T2 vary
within the set [0, T ], C2 can be chosen to be independent of those numbers.
Set
and
|m| M T 1 = |m| L∞([0,T1]×R n ) + [m] C r 1 ([0,T1]; C(R n )) + [m] C([0,T1]; C r 2 (R n ))
[m] M T 1 , T 2 = [m] C r 1 ([T1,T2]; C(R n )) + [m] C([T1,T2]; C r 2 (R n )) .
Then our observations can be summarized as follows.
Lemma 4.2.2 Suppose that 1 < p < ∞, 0 < T , 0 ≤ T1 ≤ T2 ≤ T , and 0 <
si < ri < 1 for i = 1, 2. Let Y T2 s1 = Bpp([0; T2]; Lp(Rn )) ∩ Lp([0, T2]; Bs2 pp(Rn )), and
M T2 r1 = C ([0, T2]; C(Rn )) ∩ C([0, T2]; Cr2 (Rn )). Then there exists a constant C > 0 not
depending on T1 and T2, such that
�
|mf| Y T2 ≤ C
(4.27)
for all m ∈ M T2 and f ∈ Y T2 .
4.2.3 Variable coefficients
�
|m| M T 1 |f| Y T 1 + |m| L∞([T1,T2]×R n )(1 + [m] M T 1 , T 2 )|f| Y T 2
The goal of this subparagraph is to extend the results proven in Subsection 4.2.1 to variable
coefficients. Besides we no longer stick to differential operators consisting only of the
principal part but consider general operators of second (Volterra equation) respectively
first order (boundary condition).
To start with, recall the notation R n+1
+ = {(x′ , y) ∈ Rn+1 : x ′ ∈ Rn , y > 0}. Define
the operators A(t, x, Dx) and B(t, x ′ , Dx) by
and
A(t, x, Dx) = −a(t, x) : ∇ 2 x + a1(t, x) · ∇x + a0(t, x), t ∈ J, x ∈ R n+1
+ ,
B(t, x ′ , Dx) = −∂y + b(t, x ′ ) · ∇x ′ + b0(t, x ′ ), t ∈ J, x ′ ∈ R n , (4.28)
respectively. We are concerned with the two separate problems
� v + k ∗ A(·, x, Dx)v = f, t ∈ J, x ∈ R n+1
+ ,
v = g, t ∈ J, x ′ ∈ R n , y = 0
� v + k ∗ A(·, x, Dx)v = f, t ∈ J, x ∈ R n+1
+ ,
B(t, x ′ , Dx)v = h, t ∈ J, x ′ ∈ R n , y = 0
and seek, as in Subsection 4.2.1, unique solutions in the regularity class
Z = H α p (J; Lp(R n+1
+ )) ∩ Lp(J; H 2 p(R n+1
+ )).
Concerning (4.29), we have the following result.
, (4.29)
, (4.30)
Theorem 4.2.3 Let 1 < p < ∞, J = [0, T ], n ∈ N, and k ∈ K 1 (α, θ), where θ < π, and
α ∈ (0, 2) \
� 1
p , 2
2p−1
1 3
, 1 + p , 1 + 2p−1
�
. Suppose a ∈ Cul(J × R n+1
+ , Sym{n + 1}), a1 ∈
L∞(J × R n+1
+ , Rn+1 ), a0 ∈ L∞(J × R n+1
+ ), and assume further that there exists c0 > 0
such that a(t, x)ξ · ξ ≥ c0|ξ| 2 , t ∈ J, x ∈ Rn+1 , ξ ∈ Rn+1 .
Then (4.29) has a unique solution in the space Z if and only if the data f and g are
subject to the conditions (i)-(vi) stated in Theorem 4.2.1.
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