Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified groups

In this paper, we prove a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation on the stratified Lie groups. Our proof is based on the concavity argument and the Poincaré inequality, established in Ruzhansky and Suragan (J Differ Eq 262:1799–1821, 2017) for stratified groups.


Introduction
The main purpose of this paper is to study the global existence and blow-up of the positive solutions to the initial-boundary problem of the nonlinear porous medium equation and the nonlinear pseudo-parabolic equation where m ≥ 1 and p ≥ 2, f is locally Lipschitz continuous on R, f (0) = 0, and such that f (u) > 0 for u > 0. Furthermore, we suppose that u 0 is a non-negative and non-trivial function in C 1 (D) with u 0 (x) = 0 on the boundary ∂D for p = 2 and in L ∞ (D) ∩ S1,p (D) for p > 2, respectively.
Definition 1.1.Let G be a stratified group.We say that an open set D ⊂ G is an admissible domain if it is bounded and if its boundary ∂D is piecewise smooth and simple, that is, it has no self-intersections.
Let G be a stratified group.Let D ⊂ G be an open set, then we define the functional spaces S 1,p (D) = {u : D → R; u, |∇ H u| ∈ L p (D)}. (1.3) We consider the following functional .
Thus, the functional class S1,p (D) can be defined as the completion of C 1 0 (D) in the norm generated by J p , see e.g.[7].
(b) Let N 1 be as in (a) and let X 1 , . . ., X N 1 be the left-invariant vector fields on G such that X k (0) = ∂ ∂x k | 0 for k = 1, . . ., N 1 .Then the Hörmander rank condition must be satisfied, that is, Then, we say that the triple Recall that the standard Lebesgue measure dx on R n is the Haar measure for G (see e.g.[14], [39]).The left-invariant vector field X j has an explicit form: see e.g.[39].The following notations are used throughout this paper: for the horizontal gradient, and for the p-sub-Laplacian.When p = 2, that is, the second order differential operator is called the sub-Laplacian on G.The sub-Laplacian L is a left-invariant homogeneous hypoelliptic differential operator and it is known that L is elliptic if and only if the step of G is equal to 1.
One of the important examples of the nonlinear parabolic equations is the porous medium equation, which describes widely processes involving fluid flow, heat transfer or diffusion, and its other applications in different fields such as mathematical biology, lubrication, boundary layer theory, and etc. Existence and nonexistence of solutions to problem (1.1) for the reaction term u m in the case m = 1 and m > 1 have been actively investigated by many authors, for example, [3,4,9,11,12,15,16,20,21,22,28,30,41,42,43], Grillo, Muratori and Punzo considered fractional porous medium equation [17,18], and it was also considered in the setting of Cartan-Hadamard manifolds [19].By using the concavity method, Schaefer [44] established a condition on the initial data of a Dirichlet type initial-boundary value problem for the porous medium equation with a power function reaction term when blow-up of the solution in finite time occurs and a global existence of the solution holds.We refer for more details to Vazquez's book [45] which provides a systematic presentation of the mathematical theory of the porous medium equation.
The energy for the isotropic material can be modeled by a pseudo-parabolic equation [10].Some wave processes [6], filtration of the two-phase flow in porous media with the dynamic capillary pressure [5] are also modeled by pseudo-parabolic equations.The global existence and finite-time blow-up for the solutions to pseudoparabolic equations in bounded and unbounded domains have been studied by many researchers, for example, see [26,27,33,34,37,47,48,49] and the references therein.
Also, blow-up of the solutions to the semi-linear diffusion and pseudo-parabolic equations on the Heisenberg groups was derived in [1,2,13,24,25].In addition, in [40] the authors found the Fujita exponent on general unimodular Lie groups.
In some of our considerations a crucial role is played by where introduced by Chung-Choi [8] for a parabolic equation.We will deal with several variants of such condition.
• The Poincaré inequality established by the first author and Suragan in [38] for stratified groups: where Note that it is possible to interpret the constant |N 1 −p| p (pR) p as a measure of the size of the domain D. Then β in (1.7) is dependent on the size of the domain D.
Our paper is organised so that we discuss the existence and nonexistence of positive solutions to the nonlinear porous medium equation in Section 2 and the nonlinear pseudo-parabolic equation in Section 3.

Nonlinear porous medium equation
In this section, we prove the global solutions and blow-up phenomena of the initialboundary value problem (1.1).

2.1.
Blow-up solutions of the nonlinear porous medium equation.We start with the blow-up properly.Assume that function f satisfies where for some Then any positive solution u of (1.1) blows up in finite time T * , i.e., there exists ) where M > 0 and σ = √ pmα m+1 − 1 > 0. In fact, in (2.3), we can take Remark 2.2.Note that condition on nonlinearity (2.1) includes the following cases: 1. Philippin and Proytcheva [35] used the condition where ǫ > 0. It is a special case of an abstract condition by Levine and Payne [31].2. Bandle and Brunner [4] relaxed this condition as follows where ǫ > 0 and γ > 0. These cases were established on the bounded domains of the Euclidean space, and it is a new result on the stratified groups.
Proof of Theorem 2.1.Assume that u(x, t) is a positive solution of (1.1).We use the concavity method for showing the blow-up phenomena.We introduce the functional and by (2.2) we have Moreover, J(t) can be written in the following form where Define with M > 0 to be chosen later.Then the first derivative with respect t of E(t) gives By applying (2.1), Lemma 1.2 and 0 < β m+1 , we estimate the second derivative of E(t) as follows By employing the Hölder and Cauchy-Schwarz inequalities, we obtain the estimate for [E ′ (t)] 2 as follows for arbitrary δ > 0. So we have (2.10) The previous estimates together with σ = δ = √ pmα m+1 − 1 > 0 where positivity comes from α > m + 1, imply By assumption J(0) > 0, thus if we select (2.11) We can see that the above expression for t ≥ 0 implies Then for σ = √ pmα m+1 − 1 > 0, we arrive at and some rearrangements with E(0) = M give Then the blow-up time That completes the proof.where for some where R = sup x∈D |x ′ | and x = (x ′ , x ′′ ) with x ′ being in the first stratum.Assume also that u 0 ∈ L ∞ (D) ∩ S1,p (D) satisfies inequality If u is a positive local solution of problem (1.1), then it is global and satisfies the following estimate: Proof of Theorem 2.3.Recall from the proof of Theorem 2.1, the functional Let us define By applying (2.12), Lemma 1.2 and m+1 , respectively, one finds We can rewrite E ′ (t) by using (2.9) and α ≤ 0 as follows That gives E(t) ≤ E(0).This completes the proof of Theorem 2.3.

Nonlinear pseudo-parabolic equation
In this section, we prove the global solutions and blow-up phenomena of the initialboundary value problem (1.2).
3.1.Blow-up phenomena for the pseudo-parabolic equation.We start with conditions ensuring the blow-up of solutions in finite time.Assume that αF (u) ≤ uf (u) + βu p + αγ, u > 0, ( where γ > 0 and R = sup Assume also that u 0 ∈ L ∞ (D) ∩ S1,p (D) satisfies Then any positive solution u of (1.2) blows up in finite time T * , i.e., there exists ) where σ = α 2 − 1 > 0 and Proof of Theorem 3.1.The proof is based on a concavity method.The main idea is to show that [E −σ p (t)] ′′ ≤ 0 which means that E −σ p (t) is a concave function, for E p (t) defined below.
Let us introduce some notations: We know that where Let us define with a positive constant M > 0 to be chosen later.Then Now we estimate E ′′ p (t) by using assumption (3.1) and integration by parts, that gives Next we apply Lemma 1.2, which gives with F (t) as in (3.6), then E ′′ p (t) can be rewritten in the following form Also we have for arbitrary δ > 0, in view of (3.7), Then by taking σ = δ = α 2 − 1 > 0, we arrive at Note that in the last line we have used the following inequality where making use of the Hölder inequality and Cauchy-Schawrz inequality we have .
By assumption F (0) > 0, thus we can select .9) We can see that the above expression for t ≥ 0 implies Then for σ = α 2 − 1 > 0, we arrive at This completes the proof.

3.2.
Global solution for the pseudo-parabolic equation.We now show that positive solutions, when they exist for some nonlinearities, can be controlled.

Theorem 2 . 1 .
Let G be a stratified group with N 1 being the dimension of the first stratum.Let D ⊂ G be an admissible domain.Let 2 ≤ p < ∞ with p = N 1 .

Theorem 3 . 1 .
Let G be a stratified group with N 1 being the dimension of the first stratum.Let D ⊂ G be an admissible domain.Let 2 ≤ p < ∞ with p = N 1 .