Supercaloric functions for the parabolic $p$-Laplace equation in the fast diffusion case

We study a generalized class of supersolutions, so-called $p$-supercaloric functions, to the parabolic $p$-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for $p\ge 2$, but little is known in the fast diffusion case $1<p<2$. Every bounded $p$-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic $p$-Laplace equation for the entire range $1<p<\infty$. Our main result shows that unbounded $p$-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case $\frac{2n}{n+1}<p<2$. The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case $1<p\le \frac{2n}{n+1}$ and the theory is not yet well understood.


Introduction
This paper studies classes of supersolutions to the parabolic p-Laplace equation ∂ t u − div |∇u| p−2 ∇u = 0. (1.1) The general theory covers the entire parameter range 1 < p < ∞, but different phenomena occur in the slow diffusion case p > 2 and in the fast diffusion case 1 < p < 2. For p = 2 we have the heat equation. We do not only consider weak solutions, but also weak supersolutions and, more generally, p-supercaloric functions to (1.1). They are pointwise defined lower semicontinuous functions, finite in a dense subset, and are required to satisfy the comparison principle with respect to the solutions of (1.1), see Definition 2.6 below. The definition of supercaloric functions is the same as in classical potential theory for the heat equation when p = 2, see Watson [22]. By Juutinen et al. [12], the class of p-supercaloric functions is the same as the viscosity supersolutions to (1.1) for 1 < p < ∞. Our results can be extended to more general quasilinear equations ∂ t u − div A(x, t, u, ∇u) = 0, with the p-growth, which are discussed in DiBenedetto [7], DiBenedetto et al. [8] and Wu et al. [23]. For simplicity, we discuss only the prototype case in (1.1). A p-supercaloric function does not, in general, belong to the natural Sobolev space for (1.1). The only connection to the equation is through the comparison principle. However, Kinnunen and Lindqvist [14] proved that bounded p-supercaloric functions belong to the appropriate Sobolev space and are weak supersolutions to (1.1) for p ≥ 2. Korte et al. [16] extended the study for a more general class of parabolic equations with p-growth. In this paper we show that bounded p-supercaloric functions are weak solutions to (1.1) for the entire range 1 < p < ∞.
We are mainly interested in unbounded p-supercaloric functions. Assume that u is a psupercaloric function in Ω T = Ω × (0, T ), where Ω is an open set in R n and T > 0. One of the main results of Kuusi et al. [21] asserts that for p > 2 there are two mutually exclusive alternatives: Either u ∈ L q loc (Ω T ) for every 0 < q < p − 1 + p n or u / ∈ L p−2 loc (Ω T ). In particular, if u ∈ L p−2 loc (Ω T ), then u ∈ L q loc (Ω T ) for every 0 < q < p − 1 + p n . For the corresponding theory for the porous medium equation, see Kinnunen et al. [18].
Examples based on the Barenblatt solution (see Barenblatt [3]) and the friendly giant show that both alternatives occur. In the first alternative the upper bound for the exponent is given by the Barenblatt  , (x, t) ∈ R n × (0, ∞), (1.2) where 2 < p < ∞, λ = n(p − 2) + p and the constant c is a positive number, which can be chosen such thatˆR n U (x, t) dx = 1 (1.3) for every t > 0. The Barenblatt solution is a weak solution to (1.1) in R n × (0, ∞) and the zero extension is p-supercaloric in R n × R with u ∈ L q loc (R n × R) for every 0 < q < p − 1 + p n . This function solves (1.1) with a finite point source, but it fails to belong to the natural Sobolev space, since |∇u| / ∈ L p loc (R n × R). For the second alternative, we consider a bounded open set Ω in R n with a smooth boundary. By separation of variables, we obtain the friendly giant where u ∈ C(Ω) ∩ W 1,p 0 (Ω) is a weak solution to the elliptic equation div |∇u| p−2 ∇u + 1 p−2 u = 0 in Ω with u(x) > 0 for every x ∈ Ω. The function U is a weak solution to (1.1) in Ω × (0, ∞) and the zero extension as in (1.4) is p-supercaloric in Ω × R with u / ∈ L p−2 loc (Ω × R). We prove the corresponding result in the supercritical range 2n n+1 < p < 2 for a p-supercaloric function u in Ω T . It asserts that either u ∈ L q loc (Ω T ) for every 0 < q < p−1+ p n or u / ∈ L n p (2−p) loc (Ω T ).
In particular, if u ∈ L n p (2−p) loc (Ω T ), then u ∈ L q loc (Ω T ) for every 0 < q < p − 1 + p n . Again, both alternatives occur. For 2n n+1 < p < 2, the Barenblatt solution (see Wu et al. [23, 2.7.2] and Bidaut-Véron [1]) of (1.1) is given by formula where λ = n(p − 2) + p and the constant c is a positive number so that (1.3) holds for every t > 0. Observe that p > 2n n+1 is equivalent with λ > 0. Barenblatt solutions do not exist in the subcritical case 1 < p ≤ 2n n+1 and, to our knowledge, the theory is not yet well understood. The Barenblatt solution is compactly supported for every t > 0 for 2 < p < ∞, but it is positive everywhere for 2n n+1 < p < 2. As above, the zero extension to the negative times is p-supercaloric in R n × R with u ∈ L q loc (R n × R) for every 0 < q < p − 1 + p n . A prime example of a p-supercaloric function for 2n n+1 < p < 2 that does not belong to the Barenblatt class is the infinite point source solution see Chasseigne and Vázquez [6]. This function is a solution to (1.1) in (R n \ {0}) × (0, ∞) and has singularity at x = 0 for every t > 0. Observe that (1.6) is obtained by setting c = 0 in the Barenblatt solution (1.5) and thus solves (1.1) with an infinite point source. The zero extension as in (1.4) is a p-supercaloric function u in R n ×R with u / ∈ L n p (2−p) loc (R n ×R). Our main result asserts, roughly speaking, that a p-supercaloric function and its gradient have similar local integrability properties than the Barenblatt solution or the corresponding properties are at least as bad as for the infinite point source solution. A Moser type iteration scheme and Harnack estimates are applied in the argument.
Acknowledgments. The authors would like to thank the Academy of Finland for support. K. Moring has been supported by the Magnus Ehrnrooth Foundation.
Weak solutions to (1.1) are assumed to belong to a parabolic Sobolev space, which guarantees a priori local integrability for the function and its weak gradient. We state the definition and results in a space-time cylinder Ω T , with T > 0, but extensions to arbitrary cylinders Ω t1,t2 , with t 1 < t 2 , are obvious. Definition 2.1. Let 1 < p < ∞ and let Ω be an open set in R n . A function u ∈ L p loc (0, T ; W 1,p loc (Ω)) is called a weak solution to (1.1), if ΩT −u∂ t ϕ + |∇u| p−2 ∇u · ∇ϕ dx dt = 0 for every ϕ ∈ C ∞ 0 (Ω T ). Furthermore, we say that u is a weak supersolution if the integral above is nonnegative for all nonnegative test functions ϕ ∈ C ∞ 0 (Ω T ). If the integral is non-positive for such test functions, we call u a weak subsolution.
Theorem 2.2. Let 1 < p < ∞ and let Ω be a bounded open set in R n with a Lipschitz boundary and g ∈ C(∂ p Ω T ). Then there exists a unique weak solution u ∈ C(Ω T ) to (1.1) with u = g on ∂ p Ω T . Moreover, if g belongs to C(0, T ; L 2 (Ω)) ∩ L p (0, T ; W 1,p (Ω)), so does u. Theorem 2.4. Let 1 < p < ∞ and let Ω be an open set in R n . Assume that u i , i = 1, 2, . . . , are weak supersolutions to (1.1) in Ω T such that u i L ∞ (ΩT ) ≤ L < ∞ for every i = 1, 2 . . . and u i → u almost everywhere in Ω T as i → ∞. Then u is a weak supersolution to (1.1) in Ω T .
The class of weak supersolutions is closed under taking minimum of two functions, see Korte el al. [16,Lemma 3.2]. See also Kinnunen and Lindqvist [14].
Lemma 2.5. Let 1 < p < ∞ and let Ω be an open set in R n . If u and v are weak supersolutions to (1.1) in Ω T , then min{u, v} is a weak supersolution in Ω T .
The local boundedness assumption in Theorem 2.4 can be replaced with a uniform Sobolev space bound, see Kinnunen and Lindqvist [14] and Korte et al. [16,Remark 5.6]. This implies that if u ∈ L p loc (0, T ; W 1,p loc (Ω)) and min{u, k} ∈ L p loc (0, T ; W 1,p loc (Ω)) is a weak supersolution for every k = 1, 2, . . . , then u is a weak supersolution. In general, we have to consider more general class of solutions than weak supersolutions. Thus we define p-supercaloric functions as in Kilpeläinen and Lindqvist [13, Lemma 3.1], see also Kinnunen and Lindqvist [14].
Since p-supercaloric functions are lower semicontinuous, they are locally bounded from below. Thus by adding a constant, we may assume that a p-supercaloric function is nonnegative, when we discuss local properties. If u is a nonnegative p-supercaloric function in Ω T , the zero extension in the past as in (1.4) is a p-supercaloric function in Ω × (−∞, T ). The following assertion is a direct consequence of the comparison principle in Definition 2.6.
For the following comparison principle, we refer to Björn We will apply an existence result for the obstacle problem. In order to guarantee continuity of the solution up to the boundary, we assume that the domain where the obstacle problem is considered has Lipschitz boundary. This condition can be relaxed, see [17, Theorem 3.1].
Theorem 2.11. Let 1 < p < ∞. Assume that Ω is an open and bounded set in R n with Lipschitz boundary and let ψ ∈ C(Ω T ). There exists u ∈ C(Ω T ) that is a weak supersolution to (1.1) in Ω T with the following properties: Next we show that every bounded p-supercaloric function u is a weak supersolution in Ω T . In this case u ∈ L p loc (0, T ; W 1,p loc (Ω)) and  [16,Theorem 5.8], but we repeat it here to show that it applies in the full range 1 < p < ∞.
Proof. Since u is lower semicontinuous, there exists a sequence of continuous functions ψ i , i = 1, 2, . . . , such that ψ 1 ≤ ψ 2 ≤ · · · ≤ u and for every (x, t) ∈ Ω T . We claim that u is a weak supersolution in Ω T . To conclude this it is sufficient to show that u is a weak supersolution in every space-time box Q t1,t2 ⋐ Ω T . By Theorem 2.11, for every i = 1, 2, . . . , there exists a solution v i ∈ C(Q t1,t2 ) to the obstacle problem in Q t1,t2 with the obstacle ψ i . The function v i is a continuous weak solution in the open set Since v i = ψ i on ∂ p Q t1,t2 , v i ∈ C(Q t1,t2 ) and ψ i ∈ C(Q t1,t2 ), it follows that v i = ψ i on the boundary ∂U , except possibly when t = t 2 . By Corollary 2.10, we conclude that u ≥ v i in U for every i = 1, 2, . . . . Consequently, ψ i ≤ v i ≤ u in Q t1,t2 for every i = 1, 2, . . . and thus for every (x, t) ∈ Q t1,t2 . According to the Theorem 2.4, the function u is a weak supersolution in Q t1,t2 . Since this holds true for every Q t1,t2 ⋐ Ω T we conclude that u is a weak supersolution in Ω T .

Barenblatt solutions
This section discusses p-supercaloric functions with a Barenblatt type behaviour in the supercritical range. in the weak sense, where δ is Dirac's delta. It follows that the weak gradient ∇u(·, t) exists, in the sense of (3.2), for almost every t and it is locally integrable to any power 0 < q < p − 1 + 1 n+1 . However, the function u is not a weak supersolution to (1.1) in R n × R, sincê t2 t1ˆB(0,r) for every r > 0, t 1 ≤ 0 and t 2 > 0. This implies that u / ∈ L p loc (R; W 1,p loc (R n )). Observe, that the truncations min{u, k}, k = 1, 2, . . . belong to L p loc (R; W 1,p loc (R n )) and are weak solutions to (1.1) in R n+1 by Theorem 2.12.
Lemma 3.2. Let 1 < p < ∞, 0 < ε < 1 and let Ω be an open set in R n . Assume that u is a nonnegative weak supersolution in Ω T . There exists a constant c = c(p, ε) such thaẗ for every nonnegative test function ϕ ∈ C ∞ 0 (Ω T ). The following version of Sobolev's inequality will be useful for us, see DiBenedetto where q = p + pm n . We prove a general local integrability result for unbounded p-supercaloric functions. The obtained integrability exponent is sharp as shown by the Barenblatt solution. We apply a Moser type iteration scheme.
Theorem 3.4. Let 2n n+1 < p < 2 and let Ω be an open set in R n . Assume that u is a psupercaloric function in Ω T . If u ∈ L s loc (Ω T ) for some s > n p (2 − p), then u ∈ L q loc (Ω T ) whenever 0 < q < p − 1 + p n . Proof. Since u is locally bounded from below, by adding a constant, we may assume that u ≥ 1. We may assume 0 < s < 1, otherwise we have u ∈ L 1 loc (Ω T ) and we can directly proceed to the last step of the proof.
Since p > 2n n+1 , we have n p (2−p) < 1. We consider the truncations u k = min{u, k}, k = 2, 3, . . . . By Theorem 2.12, the function u k is a weak supersolution to (1.1) and it satisfies the Caccioppoli estimate in Lemma 3.2. Let ϕ ∈ C ∞ 0 (Ω T ), 0 ≤ ϕ ≤ 1 and ϕ = 1 in a compact subset of Ω T . For the first step of iteration, we choose 0 < ε < 1 in Lemma 3.2 so that 1 − ε = s. Since u k ≥ 1, we obtain  (Ω T ). Denote s 0 = s and s 1 = s 0 (1+ p n )−(2−p). After the first step of iteration we have u ∈ L s1 loc (Ω T ). If s 1 < 1, then in the second step of iteration we choose 1 − ε = s 1 and again combine the Caccioppoli inequality and the Sobolev inequality with m = ps1 s1−(2−p) . We continue in this way and obtain an increasing sequence of numbers s i , satisfying which can be written in terms of s 0 as After the ith step of iteration we have u ∈ L si loc (Ω T ). After a finite number of iterations we have s i ≥ 1 and thus u ∈ L 1 loc (Ω T ). In order to pass from u ∈ L 1 loc (Ω T ) to u ∈ L p−1+ p n −σ loc (Ω T ) for every σ > 0 we apply a similar argument once more. Let ε = σ By Lemma 3.2, with u k ≥ 1, this implies Let u be a nonnegative p-supercaloric function in Ω T . It may happen that u does not have a locally integrable weak derivative. In this case we consider the truncations u k = min{u, k} ∈ L p loc (Ω T ), k = 1, 2, . . . and define the weak gradient by ∇u = lim k→∞ ∇u k , which is a well defined measurable function but does not necessarily belong to L 1 loc (Ω T ). If ∇u ∈ L 1 loc (Ω T ), then ∇u(·, t) is the Sobolev gradient of u(·, t) for almost every t with 0 < t < T . Theorem 3.5. Let 2n n+1 < p < 2 and let Ω be an open set in R n . Assume that u is a psupercaloric function in Ω T with u ∈ L s loc (Ω T ) for some s > n p (2 − p), Then ∇u ∈ L q loc (Ω T ) whenever 0 < q < p − 1 + 1 n+1 . Proof. Since u is locally bounded from below, by adding a constant, we may assume that u ≥ 1. Let 0 < t 1 < t 2 < T , Ω ′ ⋐ Ω and ε ∈ (0, 1). We consider the truncations u k = min{u, k}, k = 1, 2, . . . . By Hölder's inequality, we havê where the first integral is uniformly bounded with respect to k as in (3.1) and the second integral is finite whenever q 1+ε p−q < p − 1 + p n by Theorem 3.4. From this we can conclude that the right hand side is finite for any 0 < q < p − 1 + 1 n+1 . Now we have that ∇u k is uniformly bounded in L q loc (Ω T ).
Remark 3.7. Let 2n n+1 < p < 2 and let Ω be an open set in R n . Assume that u is a p-supercaloric function in Ω T with u ∈ L s loc (Ω T ) for some s > n p (2−p). By Theorem 3.5 we have ∇u ∈ L p−1 loc (Ω T ). Theorem 2.12 implies for every nonnegative ϕ ∈ C ∞ 0 (Ω T ). By the Riesz representation theorem there exists a nonnegative Radon measure µ on R n+1 such thaẗ ΩT −u∂ t ϕ + |∇u| p−2 ∇u · ∇ϕ dx dt =¨Ω T ϕ dµ for every ϕ ∈ C ∞ 0 (Ω T ). This means that u is a solution to the measure data problem ∂ t u − div |∇u| p−2 ∇u = µ.
Note that ∇u / ∈ L p loc (Ω T ), in general.
Remark 3.8. Let 2n n+1 < p < 2 and let Ω be an open set in R n . Assume that u is a nonnegative p-supercaloric function in Ω T with u ∈ L 2−p loc (Ω T ). Then ∇ log u ∈ L p loc (Ω T ), where ∇ stands for the usual Sobolev gradient. This property holds for u satisfying the assumptions in Theorem 3.4, since 2 − p < n p (2 − p). The logarithmic estimate can be obtained as in [15] by taking ε = p − 1 in Lemma 3.2.
Remark 3.9. By the following dichotomy it follows that p-supercaloric function is either locally integrable to any power smaller than p − 1 + p n > 1, or it is not integrable to a power n p (2 − p) < 1. Observe that as p ց 2n n+1 we have n p (2−p) ր 1 and p−1+ p n ց 1. The gap between the exponents becomes smaller and shrinks to a point as p approaches the critical value from above.
We have the following characterization for Barenblatt type p-supercaloric functions.

Infinite point source solutions
This section discusses p-supercaloric functions in the supercritical case that are not covered by Theorem 3.10. The leading example is the infinite point source solution in (1.6). In contrast with the friendly giant in the slow diffusion case p > 2, it is the singularity in space, not in time, that fails to be locally integrable to an appropriate power. Roughly speaking, at every time slice the singularity of the infinite point source solution is worse power type singularity than u(x, t) = |x| − n−p p−1 , 1 < p < 2, based on the fundamental solution to the elliptic p-Laplace equation. for every r > 0, t 1 > −∞ and t 2 > 0. This implies that u / ∈ L p loc (R; W 1,p loc (R n )). Observe, that the truncations min{u, k}, k = 1, 2, . . . belong to L p loc (R; W 1,p loc (R n )) and are weak solutions to (1.1) by Theorem 2.12. where q ≥ p and the constant c will be determined later. We show that u is a supersolution to (1.1) in (B(0, 1) \ {0}) × (0, ∞). A direct computation shows that It is easy to see that q 2−p − q + p − n > 0 for any q ≥ p, and q 2−p ≥ −q + p + q 2−p , so that |x| q−p− q 2−p ≤ |x| − q 2−p for every x ∈ B(0, 1). By choosing we have ∂ t u(x, t) ≥ div |∇u(x, t)| p−2 ∇u(x, t) for every (x, t) ∈ B(0, 1) × (0, ∞). Lemma 2.8 implies that u is a p-supercaloric function in B(0, 1) × (0, ∞). For any ε > 0, by choosing q > n ε (2 − p), we obtain a p-supercaloric function u / ∈ L ε loc (B(0, 1) × (0, ∞)). This shows that, in general, a p-supercaloric function is not locally integrable to any positive power. The same example with the constant in (4.1) is a supersolution to (1.1) also for 1 < p ≤ 2n n+1 , with the additional requirement q > (n−p)(2−p) The function u is p-supercaloric for any c > 0 with 0 < q ≤ (n−p)(2−p) p−1 and 1 < p < 2, since then for (x, t) ∈ R n × (0, ∞).
The following intrinsic weak Harnack inequality for supersolutions can be found in [10, Proposition 3.1]. Lemma 4.3. Let 2n n+1 < p < 2 and let Ω be an open set in R n . Assume that u is a nonnegative lower semicontinuous weak supersolution to (1.1) in Ω T . There exist constants c 1 = c 1 (n, p) ∈ (0, 1) and c 2 = c 2 (n, p) ∈ (0, 1) such that, for almost every s ∈ (0, T ), we have We shall also apply the following Harnack inequality for weak solutions, see [ Next lemma will be a useful building block in the characterization of the complementary class of the Barenblatt type supercaloric functions.
Lemma 4.5. Let 2n n+1 < p < 2 and let Ω be an open set in R n . Assume that u is a nonnegative supercaloric function in Ω T and let (x 0 , t 0 ) ∈ Ω T . Let 0 < t j < T , j = 1, 2, . . . , with t j → t 0 as j → ∞. If for every r 0 > 0 there exists 0 < r ≤ r 0 such that for every τ ∈ (t 0 , t). Notice that the construction above can be done for any r ∈ (0, r 0 ], which implies that (4.4) holds for every τ ∈ (t 0 , t) and r ∈ (0, r 0 ]. Let (t j ) j∈N be a sequence for which (4.2) holds and let r j → 0 as j → ∞. It follows that This implies that Let ε ∈ [0, c 2 (t − t 0 )) and, for every j large enough, choose the truncation levels k j so that The constant c 2 = c 2 (n, p) ∈ (0, 1) is from Lemma 4.3. Since u kj is a nonnegative weak supersolution to (1.1) in Ω T , the weak Harnack inequality in Lemma 4.3 implies for every t ∈ [t j + 3 4 ε, t j + ε] and j large enough. Since sequence r j → 0 is arbitrary, the claim follows.
(iii) =⇒ (iv): This is the implication which requires a bit more machinery, especially use of Harnack inequalities. From (iii) we have that there exists a point t 0 ∈ (0, T ) and a sequence t j → t 0 as j → ∞ such that lim j→∞ˆΩ′ u(x, t j ) dx = ∞.
The collection {B(x, r x ) : x ∈ Ω ′ , r x > 0} is an open cover of Ω ′ . Since Ω ′ is compact, there exists a finite subcover {B(x i , r i ) : i = 1, . . . , N } and u(x, t j ) dx for every j = 1, 2, . . . . By taking the limit j → ∞ it follows that lim j→∞ˆB (xi,ri) u(x, t j ) dx = ∞ u(x, t j ) dx, we obtain a contradiction by taking the limit j → ∞ as the left hand side goes to infinity and right hand side does not. Thus (4.7) holds. Now the assumption for Lemma 4.5 holds which completes the proof. (Ω T ). Also the assertion u ∈ L ∞ loc (0, T ; L 1 loc (Ω)) in Theorem 3.10 follows similarly from Theorem 4.6.