Global existence, exponential decay and finite time blow-up of solutions for a class of semilinear pseudo-parabolic equations with conical degeneration

In this paper, we study the semilinear pseudo-parabolic equations ut-▵Bu-▵But=up-1u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle u_{t} - \triangle _{{\mathbb {B}}}u - \triangle _{{\mathbb {B}}}u_{t} = \left| u\right| ^{p-1}u$$\end{document} on a manifold with conical singularity, where ▵B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\triangle _{{\mathbb {B}}}$$\end{document} is Fuchsian type Laplace operator investigated with totally characteristic degeneracy on the boundary x1=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{1} = 0$$\end{document}. Firstly, we discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence and nonexistence of global weak solution: for the low initial energy J(u0)0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(u_{0}) >0$$\end{document} or ‖∇Bu0‖L2n2(B)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \Vert \nabla _{{\mathbb {B}}}u_{0}\Vert _{L_{2}^{\frac{n}{2}}({\mathbb {B}})} = 0$$\end{document} and blows up in finite time with I(u0)<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(u_{0}) < 0$$\end{document}; for the critical initial energy J(u0)=d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J(u_{0}) = d$$\end{document}, the solution is global in time with I(u0)≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(u_{0}) \ge 0$$\end{document} and blows up in finite time with I(u0)<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I(u_{0}) < 0$$\end{document}. The decay estimate of the energy functional for the global solution and the estimates of the lifespan of local solution are also given.

Here B = [0, 1) × X , X is an (n − 1)-dimensional closed compact manifold, which is regarded as the local model near the conical points, and ∂B = {0} × X . Moreover, the operator B in (1.1) is defined by (x 1 ∂ x 1 ) 2 + ∂ 2 x 2 + · · · + ∂ 2 x n , which is an elliptic operator with conical degeneration on the boundary x 1 = 0 (we also called it Fuchsian type Laplace operator), and corresponding gradient operator is denoted by ∇ B = (x 1 ∂ x 1 , ∂ x 2 , . . . , ∂ x n ). Near ∂B we will use coordinates (x 1 The equation in (1.1) is a important physical model, appears in many applications to natural sciences, such as the unidirectional propagation of nonlinear, dispersive, long waves [2], the aggregation of population [21] and the nonstationary processes in crystalline semiconductors [11].
In the classical case, we have ⎧ ⎨ ⎩ u t − u t − u = |u| p−1 u, x ∈ , t > 0, u(0) = u 0 , x ∈ , u = 0, x ∈ ∂ , t ≥ 0, (1.2) where is an open bounded domain of R n with smooth boundary ∂ and is the standard Laplace operator. It's well known that problem (1.2) has been studied by many authors. A powerful technique for treating problem (1.2) is the so called "potential well method", which was established by Sattinger [23], Payne and Sattinger [22], and then improved by Liu and Zhao [18] by introducing a family of potential wells.
Recently, there are some interesting results about the global existence and blow-up of solutions for problem (1.2) in [28], in which Xu and Su proved the invariance of some sets, global existence, nonexistence and asymptotic behavior of solutions with initial energy J (u 0 ) ≤ d and obtained finite time blow-up with high initial energy J (u 0 ) > d by comparison principle. In [20], the author obtained a lower bound for blow-up time if p and the initial value satisfy some conditions. For other related works, we refer the readers to [3,4,9,15,16,[24][25][26][27] and the references therein.
In the conical degeneration case, Chen et al. established the corresponding Sobolev inequality and Poincaré inequality on the cone Sobolev spaces in [6]. Then in [5], Chen and Liu proved the existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions to the parabolic problem ⎧ ⎨ ⎩ u t − B u = |u| p−1 u, x ∈ intB, t > 0, u(0) = u 0 , x ∈ intB, u = 0, x ∈ ∂B, t ≥ 0, (1.3) where B is the same as above. In [8], Chen and Liu studied the initial boundary value problem for a class of semilinear edge-degenerate parabolic equations with singular potential term, and derived a threshold of the existence of global solutions with exponential decay, and the blow-up in finite time by introducing a family of potential wells. More works on equations with conical degeneration can be seen in the monograph [1,10] and references therein.
In this paper, we aim to use the improved potential well theory to prove the invariant sets, the vacuum isolating behavior, and the global existence, decay and finite time blow-up of solutions for problem (1.1) in weighted Sobolev space. For our purpose, we introduce a family of potential wells and its corresponding sets, and construct the relation between the existence of solution and the initial data u 0 via the method of the potential wells. Then, by the usage of Faedo-Galerkin method, the concavity argument and properties of a family of potential wells, we derive a threshold result of existence and nonexistence of global weak solution: for the low initial energy case (i.e., J (u 0 ) < d), the solution is global in time with I (u 0 ) > 0 or ∇ B u 0 The outline of this paper are as follows. In Sect. 2, we recall the cone Sobolev spaces and the corresponding properties. In Sect. 3, we give some preliminaries about the family of potential wells, after which we discuss the invariant sets and the vacuum isolating behavior of solutions for problem (1.1). In Sect. 4, we show the global existence, decay and finite time blow-up for problem (1.1) with low initial energy J (u 0 ) < d. In Sect. 5, we obtain the global existence, decay and finite time blow-up for problem (1.1) with critical initial energy J (u 0 ) = d.

Cone Sobolev spaces
In this section, we recall the manifold with conical singularities and the corresponding cone Sobolev spaces which are introduced in [6,7].
Let X be a closed, compact, C ∞ manifold. We set X =R + × X/({0} × X ) as a local model interpreted as a cone with the base X . Next, we denote X ∧ = R + × X as the corresponding open stretched cone with the base X .
An n−dimensional manifold B with conical singularities is a topological space with a finite subset B 0 = {b 1 , . . . , b M } ⊂ B of conical singularities, with the following two properties.
(2) Every b ∈ B 0 has an open neighbourhood U in B, such that there is a homeomorphism ϕ : U → X for some closed compact C ∞ manifold X = X (b), and ϕ restricts to a diffeomorphism ϕ : U \ {b} → X ∧ .
For simplicity, we assume that the manifold B has only one conical point on the boundary. Thus, near the conical point, we have a stretched manifold B, associated with B. Here B = [0, 1) × X , ∂B = {0} × X and X is a closed compact manifold of dimension n − 1. Also, near the conical point, we use the coordinates ( for any cut-off function ω, supported by a collar neighborhood of (0, 1) × ∂B. Moreover, the subspace H m,γ in Sobolev spaces W m, p (X ) whenX is a closed compact C ∞ manifold of dimension n that containing B as a submanifold with boundary.
with p, q ∈ (1, ∞) and 1 p + 1 q = 1, then we have the following Hölder's inequality In the sequel, for convenience we denote 3) Then J (u) and I (u) are well-defined and belong to C (H 1, n 2 2,0 (B)). We introduce the potential well and the outside sets of the corresponding potential well We define the potential well depth d as and the Nehari manifold Similar to the results in [28], one has 0 < d = inf u∈N J (u).

Invariant sets and vacuum isolating
In this section, we shall introduce a family of Nehari functionals I δ (u) in cone Sobolev spaces, the family of potential wells sets and give the corresponding lemmas, which will help us to demonstrate the invariant sets and the vacuum isolating behavior of solutions for problem (1.1).

Properties of potential wells
In this subsection, we shall introduce a family of potential wells W δ , its corresponding sets V δ and give a series of their properties which are useful in the proof of our main results. (2) On the interval 0 < λ < ∞, there exists a unique λ * = λ * (u), such that d dλ J (λu)| λ=λ * = 0.
which gives (2) An easy calculation shows that we have So, the conclusion of (3) holds.
which can be obtained from Propositions 1 and 2.
For δ > 0, we define in cone Sobolev spaces a set of Nehari functionals and denote (2) Notice that I δ (u) < 0, then we have ∇ B u 2 = 0. And we have .

So, we have
So, the conclusion of (4) holds.
Now, for δ > 0, we define the depth of a family of potential wells as follows Then, the depth d(δ) and its expression can be estimated. Additionally, we show that how d(δ) behaves with respect to δ in the following lemma.

Lemma 3.3 d(δ) satisfies the following properties:
Thus for λũ ∈ N δ , we get from the definition of d that Thus, for λū ∈ N δ , we get from the definition of d(δ) that .

Lemma 3.4 Let u
Therefore, we obtain = 0. If the sign of I δ (u) is changeable for δ 1 < δ < δ 2 , then we can chooseδ ∈ (δ 1 , δ 2 ) and Iδ(u) = 0. Therefore, we can have Now, we are in a position to introduce a series of potential wells. For 0 < δ < p+1 2 , we define From the definition of W δ , V δ and Lemma 3.3, we can obtain the following lemmas: where γ 0 (δ) is the unique real root of equation The remainder of this lemma follows from Lemmas 3.2 and 3.4.

Invariant sets and Vacuum isolating
In this subsection, we prove the invariance of some sets under the flow of (1.1) and the vacuum isolating behavior of problem (1.1).

Definition 3 (Maximal existence time)
Let u(t) be a weak solution of problem (1.1). We define the maximal existence time T max of u(t) as follows: 2,0 (B) and satisfies problem (1.1) in the distribution sense, i.e., (3.7) Now, we discuss the invariance of some sets corresponding to problem (1.1) inspired by the ideas in [19]. (2) All weak solutions u of problem (1.1) with 0 < J (u 0 ) ≤ e belong to V δ for where T max is the maximal existence time of u.
Proof ( = 0. T max is the maximal existence time of u(t).
To discuss about the invariant of the solutions with negative level energy, we introduce the following results.

Proposition 3 All nontrivial solutions of problem
where γ 0 is the unique real root of equation Proof Let u(t) be any solution of problem (1.1) with J (u 0 ) = 0, T max be the maximal existence time of u(t). From (3.8), we get J (u) ≤ 0 for 0 ≤ t < T max . Hence by we must have either ∇ B u  ≥ γ 0 for 0 ≤ t < T max . Again by (3.11) we get I δ (u) < 0 and Hence for above two cases we always have

Low initial energy J(u 0 ) < d
In this section, we prove a threshold result of global existence and nonexistence of solutions for problem (1.1) with the low initial energy J (u 0 ) < d.

Global existence with exponential decay
In this subsection, we establish the global existence of weak solutions for problem (1.  Proof We divide the proof into two steps. Step 1 Proof of global existence. Let {ω j (x)} be a system of base functions in H  Multiplying (4.1) by g sm (t), summing for s, and integrating with respect to t from 0 to t, we have By (4.2) we can get J (u m (0)) → J (u 0 ), then for sufficiently large m, we have From (4.3) and the proof of Theorem 3.1, we can get u m (t) ∈ W for 0 ≤ t < ∞ and sufficiently large m. Hence, by (4.3) and for sufficiently large m, which yields (4.7) Therefore, there exist a u and a subsequence {u v } such that In (4.1), we fixed s, letting m = v → ∞. Then, we get ∈ (0, T ).

Finite time blow-up of solution
In this subsection, we establish finite time blow-up of solution for problem (1.1) when J (u 0 ) < d and I (u 0 ) < 0 by using the concavity argument (see [13,14,17]) and properties of a family of potential wells. Furthermore, by making use of a differential inequality technique (see [20]) we determine a lower bound on blow-up time for certain solutions of problem (1.1) if blow-up occurs.
We need the following lemmas to prove finite time blow-up with J (u 0 ) < d.  for all t ∈ [0, T max ), where T max is the maximal existence time.
Proof Let u(t) be any weak solution of problem (1.1) with J (u 0 ) < d and I (u 0 ) < 0. By contradiction, we suppose that u(t) is global, then T max = ∞. For any T > 0 and for all t ∈ [0, T ], we define where b and T 0 are positive constants which will be specified later. Furthermore, and, consequently, Therefore, we get As a consequence, we read the following differential inequality for almost every t ∈ [0, T ], where ξ : [0, T ] → R + is the map defined by . Choosing b small enough we have . (4.19) Multiplying u(x, t) on two sides of Eq. (1.1), and integrating by part, we have Then by direct computation and (4.20), we have which implies If there exists t 0 ∈ [0, T max ) such that ϕ(t 0 ) = 0, then we can obtain ϕ(T max ) = 0, which contradicts with the fact that u(x, t) blows up at T max in H 1, n 2 2,0 (B)-norm. So we see Integrating the inequality (4.22) from 0 to t, we have So letting t → T max in (4.23), we can conclude that Remark 1 Noting that from follows from the definition of d that Thus, all possible cases already have been considered in Theorems 4.1 and 4.2.
From the discussion above, a threshold result of global existence and nonexistence of solutions for problem (1.1) has been obtained as follows.
By a direct computation we can see that Since I (u m ) ≥ 0, we can deduce (4.5), (4.6),  dτ is strictly increasing for 0 ≤ t < ∞.
Taking any t 1 > 0 and letting then by the energy inequality we get 0 Similar to the proof of Theorem 4.1, we can deduce the exponential decay (5.3) if we take t = t 1 as the initial time. (ii) Assume that there exists a t 1 > 0 such that I (u(t 1 )) = 0 and I (u) > 0 for 0 ≤ t < t 1 . We also have u t dτ ≤ J (u(t 1 )), t 1 ≤ t < ∞.
Hence from  and, consequently, for almost every t ∈ [0, T ]. Therefore, we get As a consequence, we read the following differential inequality for almost every t ∈ [0, T ], where ξ : [0, T ] → R + is the map defined by The reminder of the proof is the same as those of Theorem 4.2, therefore we omit it.