Quasilinear parabolic problems with nonlinear boundary conditions

The strategy for solving (4.1) is now as follows. First we determine the solution of
(4.1) on the interval [0, T1]. Let us denote it by v1 . Now suppose we already know the
solution vi of (4.1) on the interval [0, Ti], where 1 ≤ i ≤ M − 1. We then seek the
solution vi+1 of (4.1) on the larger interval [0, Ti+1] which equals vi on [0, Ti]. The last
step is repeated as long as i < M. Proceeding in this way we finally obtain the solution
of (4.1) on the entire interval [0, T ]. This is the basic idea concerning the localization
in time. Besides, we will also localize (4.1) with respect to the space variable. This will
be done in each single time step by means of a partition of unity {ϕj} N j=0 ⊂ C∞ (Rn )
which enjoys the properties �N j=0 ϕj ≡ 1, 0 ≤ ϕj(x) ≤ 1 and supp ϕj ⊂ Uj. We will
also make use of a fixed family {ψj} N j=0 ⊂ C∞ (Rn ) that satisfies ψj ≡ 1 on an open set
Vj containing supp ϕj, and supp ψj ⊂ Uj.
Suppose we are in the (i + 1)th time step of the above procedure. Set (0)Zi+1 =
(0)H α p ([0, Ti+1]; Lp(Rn )) ∩ Lp([0, Ti+1]; H2 p(Rn )). If i = 0 (initial time step), we have to
find v1 in the space
Z1(v 0 ) := {w ∈ Z1 : ∂ m t w|t=0 = ∂ m t f|t=0, if α > m + 1/p, m = 0, 1}.
If i > 0, we assume that v i =: V −1
i f lies in Zi. Here Vi refers to the operator I + k ∗
A(·, x, Dx) on [0, Ti]. Using the notation
Zi+1( ˜w) := {w ∈ Zi+1 : w| [0,Ti] = ˜w} for ˜w ∈ Zi, (4.3)
our aim is then to determine vi+1 in the space Zi+1(vi ). To achieve this, observe that
with d(w1, w2) = |w1 − w2|Zi+1 , (Zi+1(vi ), d) is a complete metric space. Our plan is
to transform (4.1) to an appropriate fixed point equation in Zi+1(vi ) and to apply the
contraction principle.
We first derive the local equations associated with {ϕj} N j=0 . Note that (4.1) is equivalent
to
v + k ∗ A#(·, x, Dx)v = f − k ∗ AR(·, x, Dx)v,
which when multiplied by ϕj becomes
ϕjv + k ∗ A j
# (·, x, Dx)ϕjv = ϕjf − k ∗ ϕjAR(·, x, Dx)v
+k ∗ [A#(·, x, Dx), ϕj]v. (4.4)
We freeze the coefficients of the local operator A j
# (t, x, Dx) at the point (Ti, xj) to get the
homogeneous differential operator with constant coefficients Aij (Dx) := A j
# (Ti, xj, Dx).
Then (4.4) can be written as
ϕjv + k ∗ A ij (Dx)ϕjv = ϕjf − k ∗ ϕjAR(·, x, Dx)v + k ∗ [A#(·, x, Dx), ϕj]v
+k ∗ (A j
# (Ti, xj, Dx) − A j
# (·, x, Dx))ϕjv. (4.5)
Let A ij be the Lp -realization of the differential operator A ij (Dx). For l ∈ {1, . . . , M},
we put Xl = Lp([0, Tl]; Lp(R n )) and define the space Ξl as the set of all functions g ∈
H α p ([0, Tl]; Lp(R n )) satisfying g|t=0 ∈ γ0Z in case α > 1/p, and ∂tg|t=0 ∈ γ1Z in case
α > 1 + 1/p. Ξl endowed with the norm
|g|Ξl
= |g|(k,1)
H α p ([0,Tl];Lp(R n )) + χ ( 1
p ,2)(α)|g|t=0|γ0Z + χ (1+ 1
p ,2)(α)|∂tg|t=0|γ1Z
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