Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth

In this paper we deal with uniqueness of solutions to the following problem \[ \begin{cases} \begin{split}&u_t-\Delta_p u=H(t,x,\nabla u)&\quad \text{in}\quad Q_T,\\&u (t,x) =0&\quad \text{on}\quad(0,T)\times \partial \Omega,\\&u(0,x)=u_0(x)&\quad \displaystyle\text{in }\quad \Omega \end{split} \end{cases} \] where $Q_T=(0,T)\times \Omega$ is the parabolic cylinder, $\Omega$ is an open subset of $\mathbb{R}^N$, $N\ge2$, $1<p<N$, and the right hand side $\displaystyle H(t,x,\xi):(0,T)\times\Omega \times \mathbb{R}^N\to \mathbb{R}$ exhibits a superlinear growth with respect to the gradient term.


Introduction
The present paper is devoted to the study of the uniqueness and, more in general, to the comparison principle between sub and supersolutions of nonlinear parabolic problems with lower order terms that have at most a power growth with respect to the gradient. More specifically, we set Ω a bounded open subset of R N , with N ≥ 3, and T > 0. We consider a Cauchy-Dirichlet problem of the type in Ω, (1.1) where Q T = (0, T )×Ω denotes the parabolic cylinder, −∆ p is the usual p−Laplacian with p > 1, the functions u 0 and f belong to suitable Lebesgue spaces and h(t, x, ξ) is a Cartheodory function that has (at most) q−growth with respect to the last variable, being q "superlinear" and smaller than p.
The literature about comparison principles for weak sub/super solutions of (1.2) is mainly devoted to cases in which solutions are smooth (say for instance continuous), the equation is exactly the one in (1.2) or the growth of the nonlinear term is "sublinear".
Our aim is to generalize this kind of results to the case of unbounded solutions and non regular data (both the initial datum and the forcing term), dealing with sub/supersolutions in a suitable class.
Let us mention that in the elliptic framework such a kind of results have been studied in several papers using different techniques. Let us recall the papers [3], [4] [10], [11], [12], [24] (and references cited therein) where unbounded solutions for quasilinear equations have been treated. We want also to highlight the results of [28] (see also [6], [22] and [21]) that have inspired our work, where the comparison principle among unbounded sub/supersolutions has been proved, for sub/supersolutions that have a suitable power that belongs to the energy space.
Let us also mention that, as well explained in [1] (see also [2] for the parabolic counterpart) things change drastically when one deals with the so called natural growth (i.e. q = p in (1.2)), since in this case the right class in which looking for uniqueness involves a suitable exponential of the solution (one can convince himself just by performing the Hopf-Cole transformation to the equation in (1.2)).
The literature is much poorer in the parabolic case, especially when unbounded solutions are considered. Let us mention the results in [18], [17] where nonlinear problem of the type (1.1) are considered where h(t, x, ξ) has a sublinear (in the sense of the p-Laplacian type operators, see [23] for more details about such a threshold) growth with respect to the last variable.
In order to prove the comparison principle (that has the uniqueness has byproduct), several techniques have been developed. Let us mention, among the others, the results that have been proved by using the monotone rearrangement technique (see for instance [10] and references cited therein) and by means of viscosity solutions (see for instance [16], [5] and references cited therein).
Our choice, that has been mainly inspired by [28], uses both an argument via linearization and a method that exploits a sort of convexity of the hamiltonian term with respect to the gradient. These two approaches are, in some sense, complementary since the first one (the linearization) works in the case 1 < p ≤ 2 while the second one (the " convex"one) deals with p ≥ 2. Of course, the only case in which both of them are in force is when p = 2.
Since we want to deal with unbounded solutions and irregular data, the way of defining properly the sub/supersolutions is through the renormalized formulation (see [13], [25], [14] and [26]).
The renormalized formulation, that is the most natural one in this framework, is helpful in order to face the first difficulty of our problem, that is the unboundedness of the sub/supersolutions. Indeed we can decompose the sub/supersolutions into their bounded part plus a reminder that can be estimated, using the uniqueness class we are working in.
According with the results in the stationary case, we prove that the uniqueness class (i.e. the class of functions for which we can prove the comparison principle, and uniqueness as a byproduct) is the set of functions whose a suitable power γ (that depends only on q, p and N ) belongs to the energy space. Let us recall (see [23]) that such a class is also the right one in order to have existence of solutions.
Even more, we show, through a counterexample, that at least for p = 2, the class of uniqueness is the right one, adapting an argument of [7]- [8] to our case.
We first consider the case with 1 < p ≤ 2, and we look for an inequality solved by the difference between the bounded parts of the sub and supersolutions, using the linearization of the lower order term. Let us recall that this is the typical approach for singular (i.e. p ≤ 2) operators, that has extensively used in several previous papers (see for instance [18] and [3] and references cited therein). In this case we are allowed to deal with general Leray-Lions operators, even if, due to a lack of regularity of the the sub/supersolutions, we cannot cover all the superlinear and subnatural growths.
The second part of the paper is devoted to the case p ≥ 2 that is, in some way, more complicated, due to the degenerate nature of the operator. In fact, we need to straight the hypotheses on the differential operator considering a perturbation (through a matrix with bounded coefficients) of the standard p-Laplacian.
Here the idea is to perturb the difference between the bounded parts of the sub and supersolutions and to exploit the convexity of the lower order term with respect to the gradient (at least in the case p = 2, otherwise the general hypothesis is more involved).
The plan of the paper is the following: in Section 2 we collect all the statement of our results, while Section 3 is devoted to some technical results. The proofs of the main results are set in Section 4, if 1 < p ≤ 2 and in Section 5 if p ≥ 2.
Finally in the Appendix there is an example that shows that the uniqueness class is the right one, at least for p = 2 and 1 < q < 2.

Assumptions and statements of the results
As already explained in the Introduction, we deal with the following Cauchy-Dirichlet problem: in Ω, The main assumptions on the functions involved in (2.1) are the following: the vector valued function a(t, x, ξ) : (0, T ) × Ω × R N → R N is a Carathéodory function such that a(t, x, 0) = 0 .
As far as the lower order term is concerned, we suppose that H(t, x, ξ) : (0, T ) × Ω × R N → R N is a Carathéodory function that satisfies the following growth condition: for a.e. (t, x) ∈ Q T , ∀ξ ∈ R N and with f = f (t, x) belonging to some Lebesgue space.
First of all we need to determine the meaning of sub/supersolutions we want to deal with. Since we are interested in possibly irregular data and, in general, in unbounded solutions, the most natural way to mean sub/supersolutions is trough the renormalized formulation. In order to introduce such an issue, we first need to define a natural space where such sub/supersolutions are defined: taking inspiration from [9], we set where T k (s) = max{−k, min{k, s}}, for k ≥ 0 and s ∈ R.
Now we are ready to define the renormalized sub/super solutions to (2.1).
Definition 2.1. We say that a function u ∈ T 1,p 0 (Q T ) is a renormalized subsolution (respectively a supersolution) of (2.1) if H(t, x, ∇u) ∈ L 1 (Q T ), u ∈ C([0, T ]; L 1 (Ω)) and it satisfies: for every S ∈ W 2,∞ (R) such that S ′ (·) is nonnegative, compactly supported and for every Some remarks about the above definition are in order to be given. Such a condition is required in order to guarantee that renormalized solution are, in fact, distributional ones. In our case we do not have to ask, in general, (2.7) to hold since it is a consequence of the class of uniqueness that we consider (see Lemma 3.4). More specifically, we have to impose such a condition only in the case in which we deal with L 1 -data and with "low" values of q.

Remark 2.3.
i) Note that a subsolution (a supersolution) on Q T turns out to be a subsolution (a supersolution) on Q t for any 0 < t ≤ T . Thus, with an abuse of notation, we refer to Definition 2.1 even if we take into account (2.6) evaluated over Q t , with 0 < t ≤ T . ii) For renormalized solutions of an equation of the type (1.2), the regularity u ∈ C([0, T ]; L 1 (Ω)) is deduced directly by the renormalized formulation, via a trace result (see [27]). However, since we are dealing with sub/supersolutions, we need to add it to the definition.

2.1.
Assumptions for p = 2. As already announced in the Introduction, for problem (2.1) with p = 2 we can use both the approach by linearization and the one by "convexity".
The first approach we want to deal with is the one by linearization. Hence we assume that a(t, x, ξ) : (2.4), that in this particular case read as: Moreover we assume the growth assumption a.e. in (t, x) ∈ Q T , ∀ξ ∈ R N . In addition we suppose the following locally (weighted) Lipschitz condition is in force, for some function g(t, x) belonging to a suitable Lebesgue space we specify later.
Before stating our comparison results, we need to introduce the class of uniqueness. As for the elliptic case (see [28,6] and also [19]- [20]), the right framework is the set of sub/supersolutions u, v whose power γ = γ(q) belongs to the energy space for a suitable choice of γ. Moreover we consider the initial data u 0 , v 0 belonging to L σ (Ω) for some σ ≥ 1. More precisely, we consider sub/supersolutions satisfying and (1 + |u|) (2.14) Such a class of uniqueness makes sense whenever 1 ≤ σ, i.e. if q ≥ 2 − N N +1 .
Remark 2.4. One can convince himself that the uniqueness class is the right one just by constructing a counterexample of a problem of the type (2.1) that admits (at least) two solutions, whose just one belongs to the right class. The construction of such a pair of solutions is a bit involved and we left it to the Appendix A.
The assumptions about the data are strictly related to the value of the superlinearity q. For this reason, we split the superlinear growth of the gradient term into two subintervals for which, in turn, we require two different compatibility conditions on the data and two different class of uniqueness.
We first consider superlinear rates belonging to the range that correspond to the case 1 < σ ≤ 2, and that allows us to consider f (t, x) ∈ L r (0, T ; L m (Ω)) in (2.11) that verifies Our first result is the following.
Theorem 2.5. Assume that a(t, x, ξ) satisfies (2.8)-(2.10), H(t, x, ξ) satisfies (2.11) (2.12) and that (2.15)-(2.17) hold true. Let u and v be a renormalized subsolution and a supersolution of (2.1), respectively, satisfying (2.13), (2.14) and let u 0 , v 0 ∈ L σ (Ω) such that u 0 ≤ v 0 . Then u ≤ v in Q T . Remark 2.6. As far as the limit the case q = 2 − N N −1 is concerned, we observe that the result of Theorem 2.5 still holds true assuming the data for sub/supersolutions u, v that belong to the class ). The proof follows as the one of Theorem 2.5 with minor changes, so we omit it.
Secondly consider the range given by that correspond to σ < 1, and we require that the functions f and g satisfy Let us observe that, in fact, our results do not cover all the interval 1 ≤ q < 2. This is due to a lack of regularity of the sub/super solutions (see Remark 2.12 below).
The second approach to the comparison principle deals with a trick that uses the convexity of the lower order order term. Such a method is not as robust as the linearization one, so we need to strength the hypotheses on the differential operator.
We consider here the following problem in Ω . (2.21) We assume that A : (0, T ) × Ω → R N ×N is a bounded and uniformly elliptic matrix with measurable coefficients, i.e.
such that ∃α, β : 0 < α ≤ β and α|ξ| 2 ≤ A(t, x)ξ · ξ ≤ β|ξ| 2 , for almost every (t, x) ∈ Q T and for every ξ ∈ R N , As far as the lower order term is concerned, we suppose that the nonlinear term H(t, x, ξ) satisfies (2.11) with 1 < q < 2 and it can be decomposed as where, for a.e. (t, x) ∈ Q T and for every ξ, η in R N , the functions H 1 (t, x, ξ) and H 2 (t, x, ξ) verify: that satisfies the following inequality for sufficiently small ε > 0 for almost every (t, x) ∈ Q T and for all ξ, η ∈ R N .
As for the approach by linearization, we have two types of results, depending on the regularity of the sub/supersolutions under consideration.
First we deal with solutions in the class (2.13)-(2.14): in this case we consider lower order terms whose growth with respect to the gradient is at most a power q in the range and we assume that f in (2.11) belongs to L r (0, T ; L m (Ω)) with (m, r) such that (2.16) holds true.
Hence we have the following result.  16), and let u and v be, respectively, a renormalized subsolution and a supersolution of (2.21) satisfying (2.13)-(2.14). and let u 0 , v 0 ∈ L σ (Ω) be such that u 0 ≤ v 0 . Then we have that u ≤ v in Q T .
As far as the low values of q are considered, we deal with the same range considered in (2.18) and L 1 data. . Let u and v be a renormalized subsolution and a supersolution of (2.21), respectively, satisfying (2.7) and let u 0 , v 0 ∈ L 1 (Ω) with u 0 ≤ v 0 . Then we have that u ≤ v in Q T .
As far as the lower order term is concerned, we suppose that it satisfies the growth condition (2.5) and we assume that a suitable weighted Lipschitz assumption with respect to the last variable is in force, i.e.
for a.e. (t, x) ∈ Q T , for all ξ, η ∈ R N and for some measurable function g = g(t, x) belonging to L d (Q T ), for a suitable choice of d ≥ 1.
In this setting, we determine two ranges of q each of them giving a different type of result in function of the required class of uniqueness (and the regularity) of the solutions.
We start by considering (2.31) When we deal with this range, we assume the continuity regularity and we deal with the uniqueness class In this ranges of values of q we have that σ ∈ (1, 2). The next range corresponds to the case of lower values of q. Namely, we consider and that g fulfils As far as the source term f is concerned, we suppose Thus the result is the following.
Remark 2.12. Let us mention some peculiarity of our results, for 1 < p ≤ 2.
i) It is worth pointing out that the case p− N N +2 < q < p is not considered here. Indeed, as already observed in the elliptic case (see [28,Remark 3.5]), we would need to require more regularity on the gradient of the sub/supersolutions, in order to apply the linearization technique, which turns out to be unnatural in our framework. Indeed we should require the sub/supersolution to belong to the space Lemma 3.4) and µ > p if p − N N +2 < q. Consequently we would have such a result under the additional assumption that u ∈ L µ (0, T ; W 1,µ 0 (Ω)). ii) Note that the critical growth q = p − N N +1 , that corresponds to the case σ = 1 and m = r = 1, has been excluded in (2.29) ((2.15) if p = 2). For such a value we have a slightly different result whose proof follows from the one of Theorem 2.11, with the following hypotheses on the data: u 0 ∈ L 1+ω (Ω) and f ∈ L 1+ω (Q T ) with ω > 0.
2.3. Assumptions for 2 < p < N . In this case we change (2.21) into the following problem: in Ω, where the matrix A(x) is bounded, coercive and with measurable coefficients, while the right hand side satisfies a superlinear growth condition with respect to the gradient. More precisely, we assume that A : Ω → R N ×N is a bounded and uniformly elliptic matrix with measurable coefficients, i.e.
As far as the Hamiltonian term is concerned, we suppose, in addition to (2.5), that ∃M > 0, such that for any ε ∈ (0, 1) The above hypothesis seems to be quite technical, since it combines several properties of the nonlinear lower order term. It is not so hard to see that, for example, the model Hamiltonian satisfies (2.39) for p − 1 < q < p and for a function f 0 bounded above and not increasing with respect to the t variable. Let us underline that also some perturbations, through locally Lipschitz function, weighted with the (p − 2)/2 power of the gradient of such a model Hamiltonian still fulfill hypothesis (2.39).
As for the previous results, we have two regimes depending on the values of q.
We first deal with the range Next, we consider the last case, that is the range and we have the following result.

Notation and basic tools
With the purpose of dealing with the bounded part of the sub/supersolutions considered during the paper, we here introduce a smooth approximation of the classical truncation function T n (s) = max{−n, min{n, s}}.
We define the smoothed truncation function S n (·) and θ n (·) as follows: Moreover, here and in all the paper, we denote by G k (z) the function for every z ∈ R and for any k ≥ 0.
Here we recall a classical parabolic regularity result that we use systematically in the following.
where the pair (j, y) fulfils Moreover there exists a positive constant c(N, η, h) such that the following inequality holds true: Next we state two useful Lemmata that we will use in the sequel in order to conclude the proofs of our results.
Proof. The proof follows direclty from the one of Lemma 2.1 in [28].
The next Lemma is a sort of parabolic version of the above one.
) be a function satisfying the following inequality: sup Proof. Without loss of generality we take into account the case m = p. Then, since w(0) L σ (Ω) = 0 and by the continuity assumption we define from (3.3). Now, let us suppose by contradiction that T * < T . Then, if t = T * and by definition of T * we would find w(T * ) σ L σ (Ω) = 1 (2c0) σ η ≤ 0 which is in contrast with the assumption c 0 > 0. We thus deduce that (3.4) holds for all t ≤ T and, in particular, we conclude that w(t) L γ (Ω) ≡ 0 for every t ∈ [0, T ].
During the proof of our main results, we need some regularity results, as the next two Lemmata. Lemma 3.4. Let u ∈ C([0, T ]; L σ (Ω)) be such that (2.33) holds true. Then Proof. Let us start with the proof of the regularity in (3.5); using (2.33) and Theorem 3.1 with η = p and h = σ γ we deduce that u ∈ L p N γ+σ N (Q T ). Then, by Young's inequality, we get and (3.5) follows by definitions of σ and γ.
Lemma 3.5. Let u, v be respectively a renormalized subsolution and a renormalized supersolution of (2.1) such that (2.7), (2.5) with (2.34) hold. Then Proof. We only deal with the case of subsolution u, since having v be a supersolution implies that −v is a subsolution. In fact, (3.7) is a consequence of Corollary 4.7 (see also Remark 4.2) applied to the positive part of u, that yields to the following inequality By the standard results on the regularity, we deduce that u + ∈ L r (0, T ; W 1,r (Ω)) for every r < p − N N +1 (see [29] and [15]).
We conclude this Section with another useful result. where L, M, µ, ν > 0 and 0 < γ < ρ. Then ∃c = c(ρ, γ, N, µ, m) : Proof. The above assumptions on w imply that we can apply Theorem 3.1 to the function (Ω)) and we get the regularity estimate Then, since the inequality (a + b) α ≤ a α + b α holds for a, b > 0, 0 < α < 1 and ρ−γ ρ < 1, we have by (3.10), combined with (3.8), and Fatou's Lemma Minimizing the right hand side above with respect to µ, we get that the minimum is achieved at µ = c L ρ N M N N (ρ−γ+ν)+mρ for a constant c that depends only on ρ, γ, N, ν, m. We are now ready to prove (3.9): we use the Hölder's inequality with ( ρ b , ρ ρ−b ) in order to get and we use that b satisfies γb ρ−b = N (ρ−γ)+mρ N , so that (3.9) follows from the choice of µ.

4.
Proofs in the case 1 < p ≤ 2 We start by proving Theorem 2.10 and Theorem 2.5.
Proof of Theorem 2.10 and of Theorem 2.5. As anticipated, we need to rewrite the inequalities satisfied by sub/supersolutions in terms of their bounded parts plus some (quantified) reminder. To this aim, we set u n = S n (u) and v n = S n (v) (4.1) where S n (·) has been defined in (3.1). We consider the renormalized formulation in (2.6) with S(u) = θ n (u) so we obtain Reasoning in the same way on the supersolution v -of course, with S(v) = θ n (v) -and considering the difference between the above inequalities, we get ∀0 < t ≤ T , We now define z n = u n − v n (4.2) and rewrite the above inequality as In virtue of the density result [26, Proposition 4.2], we are allowed to take in the inequality in (4.3) and then, thanks also to (2.2) and (2.28), we obtain We want to prove now that lim n→∞ R n = 0. From now on, we denote by ω n any quantity that vanishes as n diverges, and we set R n = I 1 + I 2 + I 3 + I 4 (4.5) where (s, x, ∇v)] · ∇z n ϕ ′ (z n ) dx ds, (4.6) Let us start by studying I 1 . The definition of θ n (·) and (2.4) imply that being a(s, x, ∇u)θ n (u) − a(s, x, ∇u n ) ≡ 0 when |u| ≤ n and since ∇u n ≡ 0 when |u| ≥ 2n (and the same hols for v n ). We just prove that the first integral in (4.7) behaves as ω n since the second one can be dealt in the same way. The definition of ϕ(·) and (2.3) allow us to deduce that and thanks to (2.33) we estimate the first integral in the right hand side above, since using that |{(t, x) ∈ (0, T ) × Ω : n < |u| < 2n}| → 0 as n → ∞. As far as the third integral is concerned, Hölder's inequality with (p, p ′ ) implies thanks again to (2.33). Finally, we deal with the second term in (4.8) (the fourth is treated in the same way) applying again Hölder's inequality with (p, p ′ ) and so obtaining A similar argument can be done for the last term in (4.7), so that we have that I 1 = ω n .
As far as I 3 is concerned, we have that by definition of θ n (·): indeed, θ n (u)H(s, x, ∇u) − H(s, x, ∇u n ) = 0 if |u| ≤ n and |∇u n | = 0 if |u| ≥ 2n. We only consider the first term in the inequality above since the second one can be dealt with in the same way. Thanks to the growth assumption (2.11), the desired convergence of the first term follows once we prove that An application of Hölder's inequality with indices p q , p * p−q , N p−q , Sobolev's embedding and Theorem 3.1 (see (3.2)) lead us to for the same reasons given above. As far as the integral involving the forcing term is concerned, we observe that, applying the Hölder inequality with indices (m, m ′ ) and (r, r ′ ) as in (2.31), we get We observe that since the couple (m, r) satisfies (2.31), we have that the pair (j, y) fulfills We thus proceed through Lebesgue spaces inclusion and we deduce = ω n , that implies I 3 = ω n . Finally, recalling that u 0 ≤ v 0 and the definition of θ n (·), we conclude that also I 4 ≤ ω n . so that that R n = ω n .
We now get into the main step of the proof. Let E n,k be the subset of Q t defined by E n,k = {(t, x) ∈ Q t : z n > k and |∇z n | > 0} , (4.11) and we set where the parameter a ≤ p ≤ 2 has to be fixed. An application of Hölder's inequality with indices ( 2 a , 2 2−a ) and the inequality in (4.4) provide with the following estimate: (4.13) We recall (2.30) (i.e. we know that g ∈ L d (Ω) for d = N (q−(p−1))−p+2q q−1 ) and we set a such that a(2−p) Then, the gradient regularity (3.5) contained in Lemma 3.4 applied on both |∇u n | and |∇v n | (we recall also (2.33)) implies that the first integral in the right hand side of (4.13) is finite.
Now, let us focus on the second one. Hölder's inequality with indices a, d, 2 and then we take advantage of (4.14) in (4.13), we obtain (4. 15) We observe that the first two integrals in the right hand side above are bounded thanks to (2.30), the definition of a and Lemma 3.4. Moreover, using (4.15) in (4.14) leads to + ω n (4.16) and the uniform estimate of the right hand side in (4.4) is closed.
Next, we observe that the definitions of a, γ and σ imply that N is the Gagliardo-Nirenberg regularity exponent applied with spaces L ∞ (0, T ; L 2 (Ω)) ∩ L a (0, T ; W 1,a 0 (Ω)). In particular, the inequality in (3.2) applied to the the function (4.17) So far, we know that the second integral in (4.17) is bounded thanks to (4.15). Furthermore, it holds from (4.4) and (4.16) that + ω n and then, using (4.15) and (4.17), it follows that Finally, passing to the limit first with respect to n (we apply Lebesgue Theorem on the right hand side) and subsequently as µ → 0, we obtain: We conclude the proof applying Lemma 3.2 with ρ = a, ν = 2 to the function G k (u − v).
Using the same ideas, adapted to the case of L 1 data and low values of q, we prove now Theorem 2.11 and Theorem 2.7.
Proof of Theorem 2.11 and Theorem 2.7. Before getting into the real proof, we recall (2.35) and observe that Indeed, Lemma 3.5 provides us that |∇u + |, |∇v − | ∈ L q (Q T ) for every q < 2 − N N +1 . So |∇u + | q−1 and |∇v − | q−1 belong to L r (Q T ) for every r > N + 2. Furthermore, being (u − v) + a subsolution itself and reasoning as in the just mentioned Lemma with T k ((z n ) + ), then (u − v) + inherits the L q (0, T ; W 1,q (Ω)) regularity for every q < 2 − N N +1 . We underline that we use the fact that ω n is assumed to be uniformly bounded in n -it will soon proved -and also converging to 0. Then, the regularity in (4.19) follows since We use the same notation of Theorem 2.10 (see (4.1) and (4.2)); in particular wee take again S(u) = S n (u) in (4.3) and S(v) = S n (v) in the formulation of the supersolution, and we set where B n,1 and R n have been defined in (4.12) and (4.6), respectively. Once again, we have that lim n→∞ R n = 0; the proof of this fact is quite similar to the one contained in Theorem 2.10. We just recall the decompositions (4.5)-(4.6) and that, using the current choice of ϕ(·) and the asymptotic energy condition (2.7), we deduce that I 1 + I 2 = ω n . The terms I 3 and I 4 follow as in Theorem 2.10, just observing that now f only belong to L 1 (Q T ), using again (2.7).
The uniform boundedness of the right hand side in (4.20) follows from the above remark by (4.19). Now, let the parameter a ≤ p be such that a(2−p) 2−a = q, i.e. a = 2q q−p+2 . Then, recalling the inequality in (4.20), we obtain for B n,1 , B n,2 as in (4.12), E n,k as in (4.11). In particular, (4.20) provides us with with c independent from µ, so that We use (4.23) in (4.22), so that Ω G k (z n (t)) dx ≤ c E n,k |∇G k (z n )| B n,1 dx ds + ω n + cµ λ that becomes, taking the limits as µ → 0, and then as n → +∞, where the last convergence follows thanks to Lemma 3.5. Furthermore, letting n → ∞ also in (4.21), we get with E k as in (4.18) and We now apply Lemma 3.6 with ρ = a, ν = λa 2 , m = 1, γ = a 1 + λ 2 and In particular, the estimate in (3.9) holds with b = a N (1 − λ) + 2 2(N + 1) = q by the definitions of a, λ and .
Observe that, since q q−1 ց p(N +1)−N p(N +1)−(2N +1) as q ր p − N N +1 , the integral involving B 1 is bounded (up to choose q closer to the threshold). Then we conclude by applying Lemma 3.2 with ρ = q.

Proofs in the case 2 ≤ p < N
We start by proving the results for p = 2, since their proofs are different from those of the case p > 2.
Proof of Theorem 2.8. We follow the same notation that we have used for the proof of Theorem 2.10, by defining u n , v n as in (4.1). Thus we consider the inequalities in (2.6) satisfied by the sub/supersolutions with S(u) = u n and S(v) = v n respectively so that, we have where the inequality related to the supersolution has been multiplied by (1 − ε), for ε ∈ (0, 1). Then, taking into account the difference between the inequalities above, we get We use the hypothesis (2.23) in order to we rewrite the above inequality as Let us observe that the conditions (2.25) and (2.26) imply that byYoung's inequality. On the other hand, since H 1 (t, x, ξ) satisfies (2.24) (i.e. the convexity assumption with respect to ξ), we have Finally, the growth assumption on H 1 (s, x, ξ) contained in (2.11) allows us to improve (5.1) as In particular, the inequality above can be written in terms of the function getting Observe that since ϕ is supported where z ε n ≥ k, then and we are reduced to study We just note that, by definitions of θ n (·), ϕ(·) and thanks to (2.22), (2.33), the proof of lim n→∞ R n = 0 follows reasoning as in Theorem 2.10.
We observe that the definition of Ψ k (·) combined with Hölder's inequality with indices 1 2−q , 1 q−1 and also an application Young inequality with 2 q , 2 2−q leads to We now focus on the term involving the source f . We recall the estimate in Theorem 2.10 and apply Hölder's inequality with (m, m ′ ) and (r, r ′ ) getting where r, m verifies (2.31). Finally, we gather together the estimates above and find that ) and thanks to (2.33), the energy integral in the left hand side is uniformly bounded.
Moreover by Theorem 3.1 we have that and, reasoning as in (4.10), we get so that the inequality (5.4) reads as for any µ > 0, where we have used that z ε n (0) ≤ v 0 . Now, reasoning as in the proof of the a priori estimates contained in [23]. We fix a value δ 0 such that max δ L r (0,T ;L m (Ω)) < δ 0 for any k ≥ k 0 . We also define T * = sup τ > 0 : G k (z ε n (s)) σ L σ (Ω) ≤ δ 0 ∀s ≤ τ which is strictly positive by (2.13) and since u 0 ≤ v 0 in Ω. Note also that T * continuously depends on n by (2.13).
Then, for k ≥ k 0 and for any t ≤ T * we have sup t∈[0,T * ] Ω Φ k (z ε n (t)) dx < δ 0 and, letting µ → 0, we deduce Now, if T * < T , then (5.5) would be in contrast with both the continuity regularity in (2.13) and the definition of T * , so (5.5) holds up to T . We thus deduce a bound, uniform in ε, for the function z n ε (t) in L σ (Ω). Indeed, we have and then, letting n → ∞ and recalling the definition of z n in (5.2), leads to which, letting ε → 0, implies u ≤ v in Q T and thus the assertion is proved.
Next we prove Theorem 2.9.
Proof of Theorem 2.9. We start recalling the inequality in (5.3), with z ε n defined in (5.2), and we set where Φ(w) = w 0 ϕ(y)dy. Again, we observe that so we drop it, and we just deal with (5.6) Our purpose is to recover an a priori estimate for z ε n . We begin applying Young's inequality with 2 q , 2 2−q to the first integral in the right hand side of (5.6) obtaining We observe that, since µ < 1, it holds that 1 − 1 (1+G k (z ε n )) µ ≤ G k (z ε n ) 1+G k (z ε n ) , and since 2 2−q > 1, it follows 2−q ≤ G k (z ε n ) and thus we deduce the uniform boundedness since, as already observed, the asymptotic condition (2.7) takes the place of (2.33) in the proof that lim n→∞ R n = 0. In particular, this means that the energy term above is uniformly bounded in n.
The above choice of δ 0 and (5.8) imply Therefore, by definition of C 1 and C 0 and thanks to (5.7), we obtain By the continuity regularity u, v ∈ C([0, T ]; L 1 (Ω)) and (5.10) we deduce that T * = T , since if T * < T then (5.10) would be in contrast with the definition of T * and since u, v ∈ C([0, T ]; L 1 (Ω)). Once we have (5.10) for T = T * then we have which, letting n → ∞ and recalling the definition of z n in (5.2), leads to and the proof follows once we let ε → 0.
5.2. The case 2 < p < N . Here we prove our results via the "convexity" method.
Proof of Theorem 2.13. We want to follow the first part of Theorem 2.8. In order to do it, we recall the definitions of u n , v n in (4.1) and consider the renormalized formulations in (2.6). We focus on the one related to the supersolution v: we consider S(v) = v n and multiply its inequality by (1 − ε) p−1 , we get by definition of Ψ p,µ (·) and Hölder's inequality with indices An application of Young's inequality with p q , p The uniform boundedness of the right hand side above is due to the fact that pγ + p(q−(p−1)) p−q = p N β+σ N and that (u − v) ∈ L p N +σ N (Q T ) by (2.33). We continue applying once more the Hölder inequality with indices p q , p * p−q , N p−q and also Sobolev's embedding, so we finally get where c = c(γ, N, q, T ) and finally deduce that Then, the above uniform boundedness in n on the difference between sub/supersolutions allow us to let n → ∞ getting We now reason as in Theorem 2.8 and, being Φ(w) −→ |w| σ σ(σ−1) as µ → 0 and thanks to Lemma 3.3, we have that G k (z ε ) ≡ 0 in Q t . In particular, this means that e −λt u t, x − (1 − ε)v (1 − ε) p−2 t, x − εv 0 − εM t ≤ εk and letting ε vanishes we deduce that u(t, x) ≤ v(t, x) in Q T , as desired.
Appendix A.
Our current goal is proving that one needs to consider sub/supersolutions belonging to the regularity class (2.13)-(2.14) in order to have a uniqueness result for problems of (1.2) type.
Here we use a result contained in [7, Section 3] (see also [8]), where it is proved that the Cauchy problem Consequently it is easy to see that both u 1 ≡ 0 and u 2 (t, x) = t − N 2σ U (|x|/ √ t) solve (A.1).
We use such a result in order to show that the class of uniqueness (2.13)-(2.14) in the right one in order to have comparison (and thus uniqueness). Indeed we construct, for the following problem for a suitable choice of f smooth, a pair of solutions, whose only one belong to the class (2.13)-(2.14), while the other one is not regular enough.
In order to prove that v 2 (t) L µ (BR(0)) → 0, as t → 0 + , for µ < σ, we compute where the last inequality follows from (A.5). Then, if µ < σ, we have N 2 1 − µ σ > 0 which implies that the right hand side of the above inequality vanishes as t → 0 + . If, on the contrary, we set µ = σ then the integral above becomes which is bounded from below, thanks to (A.5), by a positive constant. In order to prove (A.8), we observe that