A comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups

In this paper we present a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups. Moreover, using the comparison principle we obtain blow-up type results and global in $t$-boundedness of solutions of nonlinear equations for the heat $p$-sub-Laplacian on stratified Lie groups. In particular, this paper generalises and extends previous results obtained by the first author and Suragan in [RS18].


Introduction
A connected simply connected Lie group G is called a graded Lie group if its Lie algebra admits a gradation. The graded Lie groups form the subclass of homogeneous nilpotent Lie groups admitting homogeneous hypoelliptic left-invariant differential operators ( [Mil80], [tER97], see also a discussion in [FR16,Section 4.1]). These operators are called Rockland operators from the Rockland conjecture, solved by Helffer and Nourrigat [HN79]. So, we understand by a Rockland operator any leftinvariant homogeneous hypoelliptic differential operator on G. Thus, the considered setting includes the higher order operators on R n as well as higher order hypoelliptic invariant differential operators on the Heisenberg group, on general stratified Lie groups, and on general graded Lie groups.
Let us also recall that the standard Lebesgue measure is the Haar measure for G. Let Ω ⊂ G be a bounded set with smooth boundary. We denote the Sobolev space by where ν is the homogeneous order of the Rockland operator R. We have allowed ourselves to write · L ∞ (G) = · L ∞o (G) for the supremum norm, in the notation of [FR16,Chapter 4]. Let us also define the functional class S a,p 0 (Ω) to be the completion of C ∞ 0 (Ω) in the norm (1.1). For a general discussion of Sobolev spaces on graded Lie groups we refer to [FR16,Chapter 4] and [FR17].
In this paper we study the higher order nonlinear hypoelliptic heat equation for u = u(t, x), |R a 2 ν 2 2 u| r j + γ n 3 k=1 |u| s k −1 u (1.2) for x ∈ Ω and t > 0, with the initial-boundary conditions where a 1 , a 2 ≥ 0, and α, β, γ ∈ R, and n 1 , n 2 , n 3 ∈ N, p j > 1 and Here, ν 1 and ν 2 are the homogeneous orders of the Rockland operators R 1 and R 2 , respectively. We also assume that the initial data satisfies where a = max{a 1 , a 2 }.
(Ω)) such that φ ≥ 0, φ| S T = 0. Then u is called a weak solution if it is a super-solution and a sub-solution. Here and after, we use T max to denote the maximal existence time.
Our goal in this paper is to give a simple proof of a comparison principle for the initial boundary value problem for higher order nonlinear hypoelliptic heat operators on graded Lie groups using pure algebraic relations, inspired by the works [Att12] and [ZL13].
The structure of this paper is as follows. Section 2 establishes a comparison principle for the problem (1.2)-(1.4). Then, in Section 3, using the comparison principle, we investigate the blow-up or the boundedness of solution of (1.2)-(1.4) depending on the signs of α, β, γ, and relations between parameters p j , q i , r j , s k , and on u 0 .

A comparison principle on graded Lie groups
In this section we state a comparison principle for the problem (1.2)-(1.4).
The proof of the comparison principle mostly based on the following algebraic lemma (see e.g. [Att12, Lemma 2.1]).
Proof of Theorem 2.1. First, let us consider the case α, β and γ > 0. Denote φ := max{u − v, 0}, hence φ(0, x) = 0 and φ(t, x)| x∈∂Ω = 0. By the definitions of sub-and super-solutions, using φ as the test function, for any τ ∈ (0, T ), we have By Lemma 2.3, for I 1 we have Let us now estimate the term I 2 . We put h(s) = s A combination of (2.3) and (2.4) leads to For I 3 , by the mean value theorem we obtain Similarly, for I 4 we have Choosing 0 < ǫ j < 4/(βCp 2 j ) and combining the estimates (2.1), (2.2), (2.5), (2.6) and (2.7), we obtain for any τ ∈ (0, T ) that Then by Gronwall's lemma we conclude that φ ≡ 0 almost everywhere. The case α = β = γ = 0 is trivial. Now, we discuss the case, when not all, but at least one of the parameters α, β, γ is positive. Note that I 3 is positive, since for q i > 0 we have Similarly, one can verify that I 4 is positive for s k > 0. Therefore, in the case when α < 0 or β < 0 (or γ < 0) by dropping I 3 or I 2 (or I 4 ), respectively, we can always get (2.8).
3. Some applications to nonlinear equations for the heat p-sub-Laplacian In this section, we give some applications of Theorem 2.1 to nonlinear equations for the heat p-sub-Laplacian on stratified Lie groups. These groups are an important class of graded Lie groups, investigated thoroughly by Folland [Fol75]. There are many different, equivalent ways to define a stratified Lie group (see, for example, [BLU07,FS82] or [FR16,RS19] for the Lie group and Lie algebra points of view, respectively). A Lie group G = R N , • is called a stratified Lie group if it satisfies the following two conditions: • for every λ > 0 the dilation δ λ : . . , r with N 1 + · · · + N r = N and R N = R N 1 × · · · × R Nr . • let X 1 , . . . , X N 1 be the left invariant vector fields on G such that X j (0) = ∂ ∂x j | 0 for j = 1, . . . , N 1 . Then, for every x ∈ R N the Hörmander condition rank (Lie {X 1 , . . . , X N 1 }) = N holds, that is, X 1 , . . . , X N 1 with their iterated commutators span the whole Lie algebra of the group G. Let us also recall that the left invariant vector field X j has an explicit form given by (see, e.g. [FR16, Section 3.1.5]) Throughout this section, we will also use the following notations for the horizontal gradient, for the p-sub-Laplacian, and for the Euclidean norm on R N 1 .
Using (3.1) one can observe that (see, e.g. [RS17]) and Let us first consider the following initial boundary value problem for the p-sub- (Ω), and the parameters α, β, q and r will be determined later. By Definition 1.1 let us recall that T max is the maximal existence time of a weak solution of (3.5).
Then a weak solution of (3.5) is globally in t-bounded, that is, there exists a constant M depending only on p, q, r, α, β, N 1 , Ω and u 0 such that for every T > 0 we have 0 ≤ u ≤ M on (0, T ).
Proof of Theorem 3.1. Part (i). For convenience, we assume that β = −α = 1. Set We also introduce the following notations Let us now find suitable positive K 1 and σ 1 such that V 1 (t, x) is a super-solution of (3.5). By using the identities (3.3) and (3.4), we observe that Thus, we have . Now, we need to find σ 1 and K 1 such that M p V 1 0, that is, Multiplying both sides of the inequality by K −p+1 1 e −(p−1)σ 1 R , we derive that Taking into account ε R < R ′ + 1, we see that in order to prove (3.6) it is sufficient to show Thus, to have M p V 1 0 we can choose when r + 1 > p, and when r + 1 = p. We also need that K 1 u 0 L ∞ (Ω) such that V 1 (0, x) = K 1 e σ 1 R u 0 (x). Obviously, we also have V 1 (t, x) ≥ 0 = u(t, x) on ∂Ω. Therefore, V 1 (t, x) is a super-solution of (3.5). Then, Theorem 2.1 concludes that (3.7) Note that the right-hand side of (3.7) is independent of t, hence u(t, x) is globally in t-bounded. Part (ii). In this case, we may assume that α = −β = 1. We recall from Part (i) that R ′ = max 0 , x ′′ 0 ) ∈ G\Ω and ε ∈ (0, 1).
First, let us consider the case r > q. Here, we will use the following notations Now, we need to find a suitable positive K 2 such that V 2 (t, x) is a super-solution of (3.5). By using the identities (3.3) and (3.4), we observe that Then we have From this, we have Thus, it is sufficient to choose K 2 such that Note that the inequality (3.8) is satisfied if we take provided that r > p − 1. We divide inequality (3.9) by K q 2 R r p−1 to derive For qp ≥ r, we can set while for qp < r, we can set We also need that K 2 ≥ σ 2 u 0 L ∞ ε σ 2 to have V 2 (0, x) ≥ u 0 . Thus, taking K 2 as follows x) is a super-solution of (3.5). Then, Theorem 2.1 concludes that In the case when r = q, we can take such that the function V 3 (t, x) = K 3 R σ 3 is a super-solution of (3.5). By the same procedure, one can obtain the uniform boundedness of u(t, x).
There exists M > 0 such that if Ω u 2r−p r−p 0 dx > M, then T max < ∞.
In the case r = q, the proof above is still valid for β ≫ |α|.
As another application of the comparison principle, we now investigate the following initial boundary value problem for the p-sub-Laplacian, 1 < p < ∞, (Ω), and the parameters α, γ, q i and s i will be determined later.
Theorem 3.4. Let Ω ⊂ G be a bounded open set in a stratified Lie group with N 1 being the dimension of the first stratum. Let s = min{s i } and q = min{q i }. Assume that α, γ, q i and s i in (3.15) satisfy one of the following conditions: (i) α > 0, γ < 0, and q i ≥ 1 with 1 < p < s + 1 and s i < q i ; (ii) α < 0, γ > 0, and s i ≥ 1 with 1 < p < q + 1 and s i > q i . Then a weak solution of (3.15) is globally in t-bounded, that is, there exists a constant M depending only on p, q i , s i , α, γ, N 1 , Ω and u 0 such that for every T > 0 we have 0 ≤ u ≤ M on (0, T ).
Proof of Theorem 3.4. We only prove Part (i), since Part (ii) is actually the same, but only α and q i are swapped by β and s i , respectively. For convenience, we assume that α = −γ = 1. We recall that R ′ = max 0 , x ′′ 0 ) ∈ G\Ω and ε ∈ (0, 1). We also employ the following notations Now, we look for a suitable positive K 4 such that V 4 (t, x) is a super-solution of (3.15). Then we have From this, we note that So, it is sufficient to choose K 4 such that , that is, K 4 ≥ 2σ 4 ε − p p−1 . We also need that K 4 ≥ σ 4 u 0 L ∞ ε σ 4 to ensure V 4 (0, x) ≥ u 0 . Thus, choosing K 4 as follows we obtain K p V 4 ≥ 0 and V 4 (0, x) ≥ u 0 . Clearly, we also have V 4 (t, x) ≥ 0 = u(t, x) on ∂Ω. Therefore, we can conclude that V 4 (t, x) is a super-solution of (3.15). Then, the comparison principle yields that 0 ≤ u(t, x) ≤ K 4 (R ′ + 1) Since the right-hand side of (3.18) is independent of t, we can conclude that u(t, x) is globally in t-bounded.
By the same procedure as in the proof of Theorem 3.2, one can obtain the following result for the problem (3.15) when n 1 = 1: Theorem 3.6. Let α > 0, γ < 0, p > 1 and s > 0. If q > max{s, p − 1, 1}, then the solution of the problem (3.15) blows up in finite time for some large u 0 > 0.
Proof of Theorem 3.6. As in the proof of Theorem 3.2, one can show that the same function v from (3.10) is a sub-solution of the problem (3.15). Then, the comparison principle (Theorem 2.1) concludes the proof.