The next lemma is a prerequisite for applying the A-caloric approximation technique.
Lemma 4.1 Let be a weak solution to (1.9) under the assumptions (H1)-(H6). Then for any , we have
for any and with and any affine function independent of time, satisfying . Here and we write
Proof Without loss of generality, we can assume that . From (1.12) and the fact that and , we deduce
In turn, we split the first integral as follows:
and , .
We proceed estimating the two resulting pieces. As for , using (H6), the fact that is concave and Jensen’s inequality (note that ), we get
To estimate , we preliminarily observe that, using Hölder inequality,
Similarly, we also have
Using (H1), (H2) and the previous inequality, we then conclude the estimate of as follows:
Combining the estimates found for and , we have
For the remaining pieces, using (H4′), we deduce
Here we have used that and the assumption that . Using again (H4′) and Young’s inequality, we estimate
Noting the definition of H and combining the estimates just found for I, II, III and IV, we obtain
A simple scaling argument yields the result for general φ. □
The next lemma is a standard estimate for weak solutions to linear parabolic systems with constant coefficients , Lemma 5.1.
Lemma 4.2 Let be a weak solution in of the following linear parabolic system with constant coefficients:
where the coefficients satisfy , for any . Then h is smooth in and there exists a constant such that
Here we write
In the following we consider a weak solution u of the nonlinear parabolic system (1.9) on a fixed sub-cylinder and .
Lemma 4.3 Given and , there exist and depending only on n, N, λ, L, β, α and m such that if
and such if
Proof Given . And we shall always consider . We first want to apply Lemma 4.1 on to , where is an affine function independent of t satisfying . We observe that Ψ has the following property:
From Caccioppoli’s second inequality, we infer
From Lemma 4.1 we therefore get, for any , that
For given to be specified later, we let to be constant from Lemma 2.3. Define and .
Then from (4.3) we deduce that, for all , the following holds:
Moreover, we estimate, using Caccioppoli’s second inequality, (4.1) and (4.2),
provided we have chosen large enough.
Assuming the smallness condition,
satisfied. Then (4.4) and (4.5) allow us to apply Lemma 2.4, i.e., they yield the existence of solving the -heat equation on and satisfying
From Lemma 4.2 we recall that h satisfies, for any , the a priori estimate (note that )
Here we have used that , and and (4.7). Combining the previous estimate with (4.8), we deduce
Recalling back via , we arrive at
Next we use the minimizing property of
At the same time, from (4.11), we can see that: For (), we have , where
with . Therefore we can find such that .
Using Sobolev’s, Caccioppoli’s and Young’s inequalities together with (4.11), we have
Using Lemma 2.5, Caccioppoli’s inequality, (4.4), (4.6), (4.12) and Young’s inequality, we obtain
From (4.12) and (4.13), we conclude
provided and we fixed . That it is to say,
Combining (4.11) and (4.15) yields the desired estimate
for . Given , we choose such that with . This also fixes the constants and . Thus we have shown Lemma 4.3. □
In the following, we want to iterate Lemma 4.3. That is,
Lemma 4.4 For and , suppose that the conditions
are satisfied. Then, for every , we have
Moreover, the limit
exists, and the estimate
is valid for a constant .
Proof For fixed we shall denote . For given (and ), we determine , and according to Lemma 4.3. Then we can find sufficiently small such that
Given this, we can also find so small that, writing
Now, suppose that the conditions (i), (ii) and (iii) are satisfied on . Then, for , we shall show
Note first that combined with (ii), (iii) and (4.19) yields
Moreover, we have and . There we can apply Lemma 4.3 to conclude that holds. Furthermore, using Lemma 2.5, (iii) and (4.18), we deduce
i.e., holds. We now assume that and for hold. We can apply Lemma 4.3 to calculate
showing . To show we estimate
Here we have used in turn Lemma 2.5, the definition of and for .
Since . We are in a position to apply Theorem 3.1. We obtain
We now consider . We fix with . Then the previous estimate implies
Next, we show that is a Cauchy sequence in . For we deduce
This proves the claim. Therefore the limit exists and from the previous estimate, we infer (taking the limit )
Combining this with (4.20), we arrive at
For , we find with . Then the previous estimate implies
This proves the assertion of the lemma. □
An immediate consequence of the previous lemma and of isomorphism theorem of Campanato-Da Prato  is the following result.
Theorem 4.1 (Description of regularity points)
Let be a weak solution to the system (1.9) under the assumptions (H1)-(H3) and (H4′), (H5), and denote by Σ the singular set of u. Then , where
At last, we have the following.
Theorem 4.2 (Almost everywhere regularity)
Let be a weak solution to the system (1.9) under the assumptions (H1)-(H3) and (H4), (H5), and denote by Σ the singular set of u. Then , where is as in Theorem 4.1 and
Proof We start taking a point such that
The proof is complete if we show that such points are regularity points.
Step 1: a comparison estimate. Consider the unique weak solution of the initial boundary value problem
Then the difference satisfies
for every . We now choose with for , on , and for , where . Then
Letting , we easily obtain that for a.e.
The second term of the left-hand side of the previous equation can be estimated by the use of monotonicity, i.e., (H3). We therefore obtain
To estimate the right-hand side, we use (H4) which easily yields
Using the previous estimate, Young’s inequality and the fact that , we have
Having combined the previous estimate with (4.23), we arrive at
We shall provide on estimate for III. We denote , .
If we let , then
We now split III
and estimate IV and V. We have, using that , (4.25) and (4.22)
From the definition of θ, we have
Noting that , we have
We now choose the parameter t carefully, i.e., and let ε suitably small. Then connecting the previous estimates for II, III, IV and V to (4.24), we easily have the estimate we were interested in, that is,
In particular, we see that
We observe that, as a consequence of (4.21) and (4.22), we have that
Step 2: A Poincare-type inequality. Let us define
where for every . From , Theorem 3.1, we conclude that and that
In view of the previous estimate, using the Poincare inequality for v and (4.26), we find
Finally, by comparison, we get the Poincare inequality for u via (4.26) and the previous estimate
for a constant .
Step 3: Conclusion. From the previous estimate and (4.28), the assertion readily follows. Indeed if satisfies (4.21) and (4.22), then we have
therefore is a regular point in view of Theorem 4.1. □