Quasilinear parabolic problems with nonlinear boundary conditions

which allows us to write the first two equations in (5.21) as
� ∂tv − da ∗ (∆xv + ∂ 2 yv) + db ∗ ∇x(p − p|t=0) + 1
3 da ∗ ∇xp = fv − b(∇xp|t=0),
∂tw − da ∗ (∆xw + ∂ 2 yw) + db ∗ ∂y(p − p|t=0) + 1
3 da ∗ ∂yp = fw − b(∂yp|t=0).
(5.56)
Owing to (v, w) ∈ Z and
(∇x(p − p|t=0), ∂y(p − p|t=0)) ∈ (0H κ p (J; Lp(R n+1
+ )))n+1 ,
it is clear that all terms on the left-hand side of (5.56) are functions in the space
Hδa−1 p (J; Lp(R n+1
+ )). Thus
with
and furthermore
fv = hv + b(∇xp|t=0), fw = hw + b(∂yp|t=0), (5.57)
(hv, hw) ∈ (H δa−1
p (J; Lp(R n+1
+ )))n+1 , (5.58)
(hv, hw)|t=0 ∈ (B
1 1
2(1− − δa pδa )
pp (R n+1
+ ))n+1 . (5.59)
As in the two cases before one can see that (5.14),(5.16),(5.17) are necessary.
We now consider (5.23). Note that in view of (5.57) and (5.58) we have in the
distributional sense
∇x · fv + ∂yfw = ∇x · hv + ∂yhw + b(∆x + ∂ 2 y)p|t=0.
So it follows from (v, w) ∈ Z and (5.49), cp. Theorem 3.3.1, that
and
(∇x · hv + ∂yhw)|t=0 ∈ B
2κ
1+ − δb 2
− δb 2
pδb pp
Turning to gw, observe first that (v, w) ∈ Z and (5.49) entail
Therefore
δb2 +κ−1
∂tp ∈ Hp
(J; Lp(R n+1
δb2 (1−
γ∂tp ∈ B
1
p )+κ−1
pp
as well as
δb2 (1−
γp ∈ B
1
p )+κ
pp
+ )) ∩ Lp(J; H
(J; Lp(R n )) ∩ Lp(J; B
2κ
1+ − δb 2
δb p
(J; Lp(R n )) ∩ H κ p (J; B
(γ∂tp)|t=0 ∈ B η pp(R n ), if η := 1 + 2κ
δb
(R n+1
+ ). (5.60)
(R n+1
+ )), if δb
2 + κ − 1 > 0,
1
1− p
pp (R n )).
2κ
1+ − δb 2
− δb 1
p
pp (R n )), if δb
2
2 2 1
− − − δb pδb p > 0,
1 (1 − p ) + κ > 1,
the latter also being a consequence of (5.60). So we conclude from (5.22) that gw is of
the structure
gw = da ∗ ψ1 + db ∗ ψ2, with ψ1 ∈ Y, ψ2 ∈ Yκ, (5.61)
where Yκ is defined as in the previous case, that is
δb2 (1−
Yκ = B
1
p )+κ
pp
(J; Lp(R n )) ∩ H κ p (J; B
93
1
1− p
pp (R n )),