Let us set m, N positive integers, α = (α1, ..., α
n
) a multi-index and |α| = α1 + · · · + α
n
the order of α. We denote by the Cartesian product
and , pα ∈ ℝN, the generic point of . If , we set p = (p', p") where , , and
We consider, as usual,
Let us consider the following differential nonlinear variational parabolic system of order 2m :
(3.1)
where aα(X, p) = aα(X, p', p") are functions of in ℝN, satisfying the following conditions:
(3.2) for every α : |α| < m and every , the function X → aα(X, p), defined in Q having values in ℝN, is measurable in X;
(3.3) for every α : |α| < m and every X ∈ Q, the function p → aα(X, p), defined in having values in ℝN, is continuous in p;
(3.4) for every α : |α| < m and every (X, p) ∈ Λ, such that ||p'|| ≤ K, we have
where fα ∈ L2(Q);
(3.5) for every α : |α| = m, the function aα(X, p', p"), defined in having values in ℝN, are of class C1 in and, for every with ||p'|| ≤ K, we have
(3.6) ∃ ν = ν(K) > 0 such that:
for every ξ = (ξα) ∈ R" and for every , with ||p'|| ≤ K. If the coefficients aα satisfy condition (3.6) we say that the system (3.1) is strictly elliptic in Ω.
Theorem 3.1. If u ∈ L2(-T, 0, Hm(Ω, ℝN)) ∩ Cm-1,λ(Q, ℝN), 0 < λ < 1, is a weak solution of the system (3.1) and if the assumptions (3.2)-(3.6) hold, then ∀B(3σ) = B(x0, 3σ) ⊂⊂ Ω, ∀a, b ∈ (0, T), a < b, it results
(3.7)
and the following estimate holds
(3.8)
where K = sup
Q
||D' u|| and .
Proof Let us observe that, using Theorem 2.III in [21], for every 0 < ϑ < 1 and , we have
and
(3.9)
Hence, we remark that , then, it results, for a. e. t ∈ (-b*, 0),
Then, from Theorem 2.4 with , 1 - λ < θ < 1, for , and for a.e. t ∈ (-b*, 0):
and there exists a constant c = c(θ, λ, σ, m, n) such that
where .
The choice ensures that for a. e. t ∈ (-b*, 0) we have
(3.10)
and
(3.11)
where .
Then we have, for a. e. t ∈ (-b*, 0), the following inclusion between Sobolev spaces
(3.12)
then, using (3.9), written with , and (3.10)-(3.12), we have
(3.13)
then it follows the requested inequality (3.8). ■
Theorem 3.2 (main result). If u ∈ L2(-T, 0, Hm(Ω, ℝN)) ∩ Cm-1,λ(Q, ℝN), 0< λ < 1, is a weak solution of the system (3.1) and if the assumptions (3.2)-(3.6) hold, then ∀B(3σ) = B(x0, 3σ) ⊂⊂ Ω, ∀a, b ∈ (0, T), a < b it results
(3.14)
and the following estimate holds
(3.15)
where K = sup
Q
||D' u|| and .
Proof Let us fix B(3σ) = B(x0, 3σ) ⊂⊂ Ω, a, b ∈ (0, T) with a < b and h ∈ ℝ such that , set , and let a real function satisfying the following properties 0 ≤ ψ ≤ 1 in ℝn, ψ = 1 in B(σ), ψ = 0 in ℝn\B(2σ), in ℝn.
Let us also define the function ρ
μ
(t), for , μ integer, the following real function
(3.16)
Moreover set {g
s
(t)} the sequence of symmetric regularizing functions such that
Let i be a positive integer, i ≤ n, and h a real number such that . For every and for every , let us define the following "test function"
(3.17)
Substituting in (2.2) the above defined function φ, we have
(3.18)
For every α : |α| = m and a. e. X = (x, t) ∈ Q, we have
where, if b = b(X, p), for simplicity of notation, we set
Then, equality (3.18) becomes
Taking into account, for α : |α| = m, that
where
we obtain
For s → +∞, using ellipticity condition (3.6), symmetry hypothesis, convolution property of g
s
and that
we have
(3.19)
where
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
We observe that, for every ε > 0, we have
(3.25)
The term B can be estimated, for every ε > 0, as follows
(3.26)
Let us consider the term C, for every ε > 0, we have
To estimate the term D, we firstly observe that
(3.27)
then, using Theorem 2.2, we obtain
(3.28)
Finally, using (3.4) condition, the term E can be expressed as follows
(3.29)
Then, from (3.19) estimating the terms A, B, C, D, and E, for every ε > 0, we have
(3.30)
We observe that the function
is continuous in the origin, then ∃h0(ν, K, U, λ, σ, m, n), , such that for every |h| < h0, we have
For each integer i = 1, ..., n, for and every h such that |h| < h0(< 1), it follows
(3.31)
Let us focus our attention on the last term, taking into account that from (3.12), for a. e. t ∈ (-b*, 0), we have
then using Hölder and Young inequalities, for every α such that |α| < m, for every ε > 0, it follows
Furthermore, for every α such that |α| < m, from Theorem 2.2 for every h ∈ ℝ with |h| < h0 and for every ε > 0, we have
the last inequality follows, as before, applying Theorem 2.2 for p = 2.
Let us now choose , it ensures
Multiplying each term for and integrating respect to and applying (3.13), we achieve
Taking into consideration the last inequality and the properties of the function ψ, from (3.31) we deduce
From which, passing the limit μ → ∞, we get
(3.32)
Let us now estimate the last term in (3.32). Using Hölder inequality, applying Theorem 2.2 (for p = 4, instead of B(σ) and ) and formula (3.13), for every |h| < h0, it follows
Integrating in (-b*, 0), from (3.32), it follows
(3.33)
If , for every i = 1, 2, ..., n we easily obtain
It is then proved, for every and every i ∈ {1, 2, ..., n}, that