On the local integrability and boundedness of solutions to quasilinear parabolic systems ∗

We introduce a structure condition of parabolic type, which allows for the generalization to quasilinear parabolic systems of the known results of integrability, and boundedness of local solutions to singular and degenerate quasilinear parabolic equations.


Introduction
In this note, we investigate under which conditions it is possible to extend to systems the results of local integrability and local boundedness known to hold for solutions to a general class of degenerate and singular quasilinear parabolic equations.In particular, we show that the results presented by DiBenedetto in [1, Chp.VIII] are true for a larger class of problems, by providing conditions under which one can recover for weak solutions of quasilinear parabolic systems the work contained in [5,6].Fundamental to our approach is a new condition for the parabolicity of systems, which can be viewed as the extension of an analogous notion for parabolic equations, introduced in [1, Lemma 1.1 pg 19].
Generalizations of the results in [1, Chp.VIII] to initial-boundary value problems for systems have been proven in [7].

Remark
We would like to point out that the parabolicity condition (H5) is a quite natural one to consider.In fact, for the case of a single equation it reduces to the condition A j (x, t, u, v)u 2 v j ≥ 0, which, for u = 0, is equivalent to the weak parabolicity condition presented in [1,Lemma 1.1,p.19].Further, in the simple case where our requirement is satisfied if the matrix a jm (x, t, u, ∇u) is for example positive definite.Indeed, since for the above system one has the identity (H5) can be rewritten as where we set w h = l u l v lh .Finally, we note that (H5) is not so restrictive that the equation must have one of these simple forms.For example, consider the perturbation where the matrix a jm is positive definite.Define λ(x, t, u, v) = min |w|=1 a jm (x, t, u, v)w j w m > 0; this exists and is obtained because w → a jm (x, t, u, v)w j w m is positive and continuous for each (x, t, u, v) on the compact set {w ∈ R N : |w| = 1}.Then for any vector w ∈ R N , w = 0 Condition (H5) will be verified if the perturbation α ij satisfies the smallness condition EJQTDE, 2004 No. 14, p. 3 Indeed, we have We follow the approach of [1,5,6] and start with the derivation, presented in Section 2, of a local energy estimates for weak solutions to (1).We then outline, in Section 3 and Section 4 how the methods in [5] can be applied to obtain local integrability and boundedness.
We also remark that the techniques presented can be modified to handle doubly degenerate problems, where for some Φ, following the same lines as the proof in [6].

Notation & Preliminaries
Let (x 0 , t 0 ) ∈ Ω T , without loss of generality we can assume (x 0 , t 0 ) = (0, 0).For R > 0 we set We also require that While, for η > 0, we let J η be a smooth, symmetric, mollifying kernel in space-time, and for a given function f we use the notation f η ≡ J η * f to represent its convolution with J η .
Finally, for fixed > 0, and κ > 0, we consider the function In the following, we will use the fact that 0 ≤ f (s) ≤ 1, and that provided 0 < < 1 2 .We are now ready to start the derivation of our energy estimate.Fix η > 0, κ > 0 and consider the test function function for η sufficiently small, we can substitute it into the definition of weak solution to obtain For convenience of notation, we rewrite (5) in compact form as I 1 + I 2 = I 3 , and discuss each of these terms in turn.

Estimate of I 1
We begin by using the symmetry of the mollifying kernel, and integration by parts to rewrite I 1 as We then notice that summing over the index i implies and we derive If we now let k → ∞, thanks to the uniform convergence of ζ k → ζ, and the smoothness of the mollified functions we obtain EJQTDE, 2004 No. 14, p. 5 Proceeding in a standard fashion, we rewrite the integral on the right hand side as and applying integration by parts, since ζ = 0 on t = −R p , we gather We would like to take the limit for η ↓ 0 in ( 7), and we are able to do so, since from where γ 2 is a constant that depends on σ, R and p. (Note that the above limits are zero due to the fact that u ∈ L ∞,loc (0, T ; L 2,loc (Ω)).)In conclusion, we have the following estimate EJQTDE, 2004 No. 14, p. 6

Estimate of I 2
We start as in Section 2.2, and use the symmetry of the mollifying kernel to rewrite I 2 : We then take the limit for k → ∞, and by the smoothness of the mollified functions we obtain As done while deriving the estimate for I 1 , we would like to consider the limit for η ↓ 0 as well.To do so, we notice that the structure condition (H2) implies the inequality From which, we have that ), since δ < m and since by the classical embedding theorems for parabolic spaces we know Therefore, we obtain A ij,η (x, t, u, ∇u) hence from u i,η → u i and ∇u i,η → ∇u i almost everywhere [3, Appendix C, Theorem 6] we conclude that If next we use our estimates for f and f , we have the upper bound which, applying a slight generalization of Lebesgue's Dominated Convergence Theorem [4, §1.8], gives EJQTDE, 2004 No. 14, p. 7 We then have that equation (9) yields The first integral above can be estimated with the help of (H1) as follows: To handle the second integral, we use the parabolicity assumption (H5), and the equal- For the last integral, we need (H2) to derive Finally, we combine (11), ( 12), (13), and (14) so to obtain the inequality: EJQTDE, 2004 No. 14, p. 8

Estimate of I 3
Once again, our first step is to rewrite I 3 in the form and to consider the limit for k → ∞: To justify taking the limit for η ↓ 0 in this case, we proceed by noticing that (H3) implies , in view of the embedding (10), and the relations δ < m, p Moreover, since we know that we can apply the same generalization of Lebesgue's Dominated Convergence Theorem to see that and we can use (H3) once more to conclude

The Energy Estimate
To derive our energy estimate (presented in Proposition 3 below), we use the intermediate result stated as Lemma 2. This is a direct consequence of equations ( 8), ( 15) and ( 17): starting from (5), one can use the bounds given in the previous sections, and then apply Young's inequality to treat the terms involving |∇u| p−1 , |∇u| p(1− 1 δ ) , and EJQTDE, 2004 No. 14, p. 9 Lemma 2 Let p > 1, let f be defined by (3), and let u ∈ L ∞,loc (0, T ; L 2,loc (Ω)) ∩ L p,loc (0, T ; W 1 p,loc (Ω)) be a weak solution of (1).If the assumptions (H1)-(H5) are verified, then for any To extract useful information from Lemma 2, we need to substitute our choice of f (s), and then let ↓ 0. We first note that By remarking that if < κ < s then EJQTDE, 2004 No. 14, p. 10 Moreover, since we can use the Monotone Convergence Theorem to pass to the limit as ↓ 0 in the remaining terms of (18), and gather the bound

Higher Integrability of u
Owing to Proposition 3, we can proceed as in [5] to show higher integrability properties for u, that is the first part of Theorem 1.In fact, thanks to the Sobolev embedding for EJQTDE, 2004 No. 14, p. 11 parabolic spaces [1, Chap.1], and hypotheses (H6) for the functions φ 0 , φ 1 , and φ 2 , we have Inequality ( 21) is the key link needed to obtain for our systems exactly the same higher integrability result proven in [5, Proposition 3] for single equations: Proposition 4 Under the hypotheses and notation of Theorem 1, we have that if s, µ ≥ (N +p) p , then u ∈ L q,loc (Ω T ) for any q < ∞; if s, µ < (N +p) p , then u ∈ L q,loc (Ω T ) for any q < q * .Indeed, suppose u ∈ L β,loc .Then we can use (21) to see that then u ∈ L weak α(β),loc .Thus u ∈ L q,loc (Q R ) for all q < α(β), and we can iterate this process starting from β 0 = max{2, N + 2 N p, r} to obtain the result.The details can be found in [5].

Boundedness of u
The L ∞ local estimate part of Theorem 1 is a straightforward application of DeGiorgi's technique; again the details can be found in [5].In particular, we fix ρ > 0, σ > 0, so EJQTDE, 2004 No. 14, p. 12 that Q ρ ⊂⊂ Ω T .For each integer n, we define and set Q n = Q ρn .Next we fix κ > 0 to be chosen later, and set For N +2 N p > 2, we consider N p ≤ 2, we take for λ sufficiently large.This is well defined thanks to the local integrability proven in Section 3. We then apply the local energy estimate (21)in a standard way to obtain an estimate of the form for positive constants γ, B 1 , B 2 , B 3 , 1 , 2 and 3 .As final step, we choose κ sufficiently large so to have Y n → 0 as n → ∞ which implies |u| < κ in Q σρ .It should be clear from the above presentation how the crucial roles in the generalization of the results in [5] to system of the form (1) are played by the local energy estimate of Proposition 3, and by the fact that the techniques in [5] really depend just on |u|.In a similar fashion, it is an easy exercise to check that the same ingredients (Proposition 3 and replacement of u by |u|) lead to the more general results of [6] .
t)|u|χ[|u| > κ] dx dt .In turn, the above inequality leads to the classical local energy estimate stated in Proposition 3 below, if one takes in account the relation ∇|u| p ≤ |∇u| p .