Pointwise estimates for solutions of singular quasi-linear parabolic equations

For a class of singular divergence type quasi-linear parabolic equations with a Radon measure on the right hand side we derive pointwise estimates for solutions via the nonlinear Wolff potentials.


Introduction and main results
In this note we give a parabolic version of a by now classical result by Kilpeläinen-Malý [7], who proved pointwise estimates for solutions to quasi-linear p-Laplace type elliptic equations with measure in the right hand side. The estimates are expressed in terms of the nonlinear Wolff potential of the right hand side. These estimates were subsequently extended to fully nonlinear equations by Labutin [8] and fully nonlinear and subelliptic quasi-linear equations by Trudinger and Wang [16]. The pointwise estimates proved to be extremely useful in various regularity and solvability problems for quasilinear and fully nonlinear equations [7,8,13,14,16]. An immediate consequence of these estimates is the sufficient condition of local boundedness of weak solutions which, as examples show, is optimal.
For the heat equations the corresponding result was recently given in [5]. The degenerate case p > 2 was studied recently in [10]. Here we provide the pointwise estimates for the singular supercritical case 2n n+1 < p < 2.
Let Ω be a domain in R n , T > 0. Let µ be a Radon measure on Ω. Let 2n n+1 < p < 2. We are concerned with pointwise estimates for a class of non-homogeneous divergence type quasi-linear parabolic equations of the type (1.1) u t − div A(x, t, u, ∇u) = µ in Ω T = Ω × (0, T ), Ω ⊂ R n .
Before formulating the main results, let us remind the reader of the definition of a weak solution to equation (1.1).
We say that u is a weak solution to (1.1) if u ∈ V (Ω T ) := C([0, T ]; L 2 loc (Ω)) ∩ L p loc (0, T ; W 1,p loc (Ω)) and for any compact subset K of Ω and any interval [t 1 , t 2 ] ⊂ (0, T ) the integral identity for any ϕ ∈ W 1,2 loc (0, T ; C(K))∩L p loc (0, T ; • W 1,p (K)). Note that ϕ is required to be continuous with respect to the spatial variable so that the right hand side of (1.5) is well defined.
The crucial role in our results is played by the truncated version of the Wolff potential defined by In the sequel, γ stands for a constant which depends only on n, p, c 1 , c 2 which may vary from line to line.
Theorem 1.1. Let u be a weak solution to equation (1.1). Let β = p + n(p − 2) > 0. There exists γ > 0 depending on n, c 1 , c 2 and p, such that for almost all (x 0 , t 0 ) ∈ Ω T there exists R 0 > 0 satisfying the Estimate (i) is not homogeneous in u which is usual for such type of equations [2,4]. The proof of Theorem 1.1 is based on a suitable modifications of De Giorgi's iteration technique [1] following the adaptation of Kilpeläinen-Malý technique [7] to parabolic equations with ideas from [11,15].
The following test for local boundedness of solutions to (1.1) is an immediate consequence of Theorem 1.
Then u ∈ L ∞ loc (Ω T ). Remark 1.3. 1. The range of p in the above results is optimal for the validity of the Harnack inequality (cf. [2]), however for weak solutions of the considered class it is plausible to conjecture that the results are valid for 2n n+2 < p < 2, although it may require some additional global information as in [2, Chapter V]. 2. Stationary solutions of (1.1) solve the corresponding elliptic equation of p-Laplace type with measure on the right hand side, and the Kilpeläinen-Malý upper bound is valid for them [7,12]. The difference with our result in Theorem 1.1(ii) is in the additional term R 2 on the right hand side of the estimate. It is not clear yet whether this is a result of the employed technique or it lies in the essence of the problem, as even for the homogeneous structure (µ = 0) a similar term is present in the estimates (cf. [2, Chapter V]).
The rest of the paper is organized as follows. In Section 2 we give auxiliary energy type estimates. Section 3 contains the proof of Theorem 1.1. In Section 4 we provide an application giving the global supremum estimate for the solution to a simple initial boundary problem.

Integral estimates of solutions
We start with some auxiliary integral estimates for the solutions of (1.1) which are formulated in the next lemma. Define ρ/2 (y, s) and |∇ξ| ≤ 2 ρ , |ξ t | ≤ cδ p−2 ρ −p for some c > 0. Then there exists a constant γ > 0 depending only on n, p, c 1 , c 2 such that for any l, δ > 0, any cylinder Q Proof. Further on, we assume that u t ∈ L 2 loc (Ω T ), since otherwise we can pass to Steklov averages. First, note that Let η be the standard mollifier in R n and as usual η Using first (2.3) on the right hand side, then passing to the limit σ → 0 on the left, after applying the Schwarz inequality we obtain for any t > 0 Passing to the limit ε → 0 on the right hand side of the above inequality and using the Fatou lemma on the left, by (2.4) we obtain the required (2.2). Now set The next lemma is a direct consequence of Lemma 2.1.
Proof. Follows directly from the previous lemma and the condition on λ.

3
Proof of Theorem 1.1

Construction of the iteration sequences
where here and below 1 S stands for the characteristic function of the set S.
The sequences of positive numbers (l j ) j∈N and (δ j ) j∈N are defined inductively as follows.
which will be fixed later. Let j ≥ 1. Suppose we have already chosen the values l i , i = 1, 2, . . . , j and Let us divide I j into the equal parts by the points τ * j,m , m = 1, 2, . . . , M * (j), in such a way that For a fixed j ≥ 1 and every m = 1, 2, . . . M * (j) we define where G is defined in (2.1) and k will be fixed later. It follows from (3.3) that Fix a number κ ∈ (0, 1) depending on n, p, c 1 , c 2 , which will be specified later. Choose B such that This implies that A * j (l j + Bρ −n j ) ≤ κ 2 . Note that there exists a small enough R 0 such that (3.1) and (3.5) are consistent (with some B ≥ 1). If In this case we set l j+1 =l and δ j = l j+1 − l j in both cases.

Main lemma
The following lemma is a key in the Kilpeläinen-Malý technique [7]. Lemma 3.1. Let the conditions of Theorem 1.1 be fulfilled. There exists γ > 0 depending on the data, such that for all j ≥ 1 we have Proof. Fix j ≥ 1. Without loss assume that since otherwise (3.16) is evident. The second inequality in (3.17) guarantees that A j (l j+1 ) = κ. Next we claim that under conditions (3.17) there is a γ > 0 such that Indeed, for (x, t) ∈ L j one has which proves the claim.
Recall that (3.20) Let us estimate the terms in the right hand side of (3.20). For this we decompose L j as L j = L ′ j ∪ L ′′ j , where ε ∈ (0, 1) depending on n, p, c 1 , c 2 is small enough to be determined later. By (3.18) we have The following inequalities are easy to verify The integral in the second terms of the right hand side of (3.26) is estimated by using the Gagliardo-Nirenberg inequality in the form [9, Chapter II,Theorem 2.2] as follows Let us estimate separately the first factor in the right hand side of (3.27).
For the last term in the above inequality we estimate by (3.18) and (3.24) Using the decomposition (3.21) and the first inequality in (3.17) we have Thus we obtain the following estimate for the first term of A j (l j+1 ): Let us estimate the second term in the right hand side of (3.20). By (3.31) we have (by using the decomposition (3.21) and (3.33)) Combining (3.32) and (3.34) and choosing ε appropriately we can find γ 1 and γ such that It remains to estimate δ 0 . There are two cases to consider. Either δ 0 = 1 2 ρ 2 0 = 1 2 R 2 , or l 1 and δ 0 are defined by A * 0 (l 1 ) = κ with κ fixed in the proof of Lemma 3.1 by (3.36). It follows that there exists m such that A * 0,m (l 1 ) = κ. Using the decomposition (3.21) with ε chosen via κ, and Lemma 2.2 together with (3.2) one can see that Then by (3.20) we conclude that either In case of (3.39) we obtain Combining this with the first case and chosing R 0 such that Hence the sequence (l j ) j∈N is convergent, and δ j → 0 (j → ∞), and we can pass to the limit J → ∞ in (3.38). Let l = lim j→∞ l j . From (3.8) we conclude that Choosing (x 0 , t 0 ) as a Lebesgue point of the function (u − l) + we conclude that u(x 0 , t 0 ) ≤ l and hence u(x 0 , t 0 ) is estimated from above by Applicability of the Lebesgue differentiation theorem follows from [6, Chap. II, Sec. 3]. This completes the proof of the first assertion of Theorem 1.1. Estimate (ii) is immediate consequence of (3.44).

Example of application
In this section we give an application of our main result, Theorem 1.1, to the weak solution of the following model initial boundary value problem.
where µ is a (positive) Radon measure on B R .
Before formulating the result we need to clarify what we understand by the weak solution to the initial boundary value problem (4.1). We assume that Otherwise, Proof. We start with a proof of the following inequality As in the proof of Lemma 2.1 we need the test function for (4.1) to be continuous with respect to the spatial variable to make it µ-measurable. It is clear that u can be approximated by the functions u n which are step functions with respect to t on (0, T ) with values in C ∞ 0 (B R ). Without loss u n → u pointwise almost everywhere and in L p (0, T ) ; • W 1,p (B R ) . Now, by v h we will denote the symmetric Steklov average v h (x, t) = 1 2h t+h t−h v(x, τ )dτ . Taking t 1 > 0 and t 2 = t ≤ T in the integral identity (1.5), testing it with ϕ = (u n ) h |(u n ) h | + ε h , ε > 0, and noting that |ϕ| ≤ 1 we obtain t t1 BR In the above inequality we first pass to the limit n → ∞. Next passing to the limit h → 0 we obtain BR |u(x, t)| − |u(x, t 1 )| dx − ε BR ln |u(x, t)| + ε |u(x, t 1 )| + ε dx + ε t t1 BR |∇u| p (|u| + ε) 2 dxdτ ≤ (t − t 1 )µ(B R ).
Passing to the limit ε → 0 in the above inequality and then t 1 → 0 we prove (4.4).
We are going to apply estimate (ii) of Theorem 1.1 for the points (x,t) in the form (4.5) u(x,t) ≤ γ R 2 + 1 R n ess sup 0<t<T BR u + dx + W µ p (x, 2R) .