Quasilinear parabolic problems with nonlinear boundary conditions

One can then construct functions a j ∈ C(J × R n+1
+ ), 0 ≤ j ≤ Nb, as well as a j ∈
C(J × R n+1 ), Nb + 1 ≤ j ≤ Nb + N, which are subject to a 0 (t, x) = a(t, x), t ∈ J, x ∈
U0 ∩ R n+1
+ ; aj (t, x) = a(t, x), t ∈ J, x ∈ Uj ∩ R n+1
+ , j > 0; and
|a j (t, x)−a(Ti, xj)| B(R (n+1) 2 ) ≤η, t ∈ [Ti, Ti+1], x ∈ R n+1
+
resp. Rn+1 ,
for all i = 0, . . . , M − 1 and j = 0, . . . , Nb + N, where x0 = ∞. Similarly, one finds
functions b j , 0 ≤ j ≤ Nb, defined on J × R n enjoying the same regularity properties as
b, and which are such that b j equals b on J × {x ′ ∈ R n : (x ′ , 0) ∈ Uj} as well as
|b j (t, x ′ ) − b(Ti, x ′ j)| B(R n ,R) ≤ η, t ∈ [Ti, Ti+1], x ′ ∈ R n ,
for all i = 0, . . . , M − 1 and j = 0, . . . , Nb. With these functions we define spatially
local operators by A j
# (t, x, Dx) = −a j (t, x) : ∇ 2 x, 0 ≤ j ≤ Nb + N, and B j
# (t, x′ , Dx) =
−∂y + b j (t, x ′ ) · ∇x ′, 0 ≤ j ≤ Nb. Further, let A ij (Dx) = A j
# (Ti, xj, Dx) and B ij (Dx) =
B j
# (Ti, x ′ j , Dx).
We next choose a partition of unity {ϕj} Nb+N
⊂ C∞ (Rn+1 ) which has the properties
� Nb+N
j=0
⊂
C∞ (Rn+1 ) that satisfies ψj ≡ 1 on an open set Vj containing supp ϕj, and supp ψj ⊂ Uj.
To derive the local equations associated with {ϕj} Nb+N
j=0 we multiply both equations
in (4.30) by ϕj. For the Volterra equation, this again results in (4.5). Concerning
the boundary, for j = Nb + 1, . . . , Nb + N a condition does not appear, whereas for
j = 0, . . . , Nb we obtain
j=0
ϕj ≡ 1 on R n+1
+ , 0 ≤ ϕj(x) ≤ 1 and supp ϕj ⊂ Uj. Fix also a family {ψj} Nb+N
j=0
B ij (Dx)ϕjv = ϕjh − ϕjb0(·, x ′ )v + (B#(t, x ′ , Dx)ϕj)v
+(B j
# (Ti, x ′ j, Dx) − B j
# (t, x′ , Dx))ϕjv. (4.31)
Rephrasing, for j = Nb + 1, . . . , Nb + N we encounter the full space problems which were
already considered in the proof of Theorem 4.1.1 and which, with η being sufficiently
small, gave rise to the equations
ϕjv = ψj(I − S ij )| −1
Zi+1(ϕjv i ) hij (f, v). (4.32)
In case 0 ≤ j ≤ Nb we are led to half space problems of the form
�
w + k ∗ Aij (Dx)w = g, t ∈ [0, Tl], x ∈ R n+1
+ ,
−∂yw + b(Ti, x ′ j ) · ∇x ′w = φ, t ∈ [0, Tl], x ′ ∈ Rn , y = 0,
(4.33)
where 1 ≤ i+1, l ≤ M. By Theorem 4.2.2, (4.33) has a unique solution w =: L ij
H, l (g, φ)T
in the space Zl if and only if g and φ are subject to the conditions (i)-(v) enunciated
therein with J = [0, Tl]. Let us write for the latter (g, φ) ∈ ΞH, l. Moreover, an a priori
estimate for |w|Zl
in terms of the norms of the data g and φ holds, and the constant in
this estimate is independent of i, j, l, if the data belong to the spaces with vanishing
traces at t = 0. Note that for Zl, we use as in the proof of Theorem 4.1.1 the norm
|w|Zl
= |w|(k,1)
Hα p ([0,Tl];Lp(R n+1
+ )) + |∇2xw| (n+1)
X 2
l
Here Xl = Lp([0, Tl] × R n+1
+ ). The corresponding norm for
Yl := B
1 1
α( − 2 2p )
pp
([0, Tl]; Lp(R n )) ∩ Lp([0, Tl]; B
69
.
1
1− p
pp (R n )),