Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity \begin{document}$u_t-\triangle_{X} u_t-\triangle_{X} u=u\log|u|$\end{document} , where \begin{document}$X=(X_1, X_2, ··· , X_m)$\end{document} is an infinitely degenerate system of vector fields, and \begin{document}$\triangle_{X}:=\sum^{m}_{j=1}X^{2}_{j}$\end{document} is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique, the logarithmic Sobolev inequality and Poincare inequality, we obtain the global existence and blow-up at \begin{document}$+∞$\end{document} of solutions with low initial energy or critical initial energy, and discuss the asymptotic behavior of the solutions.


Introduction.
Let Ω be an open domain in R n (n ≥ 2), and X = (X 1 , · · · , X m ) be a system of real smooth vector fields defined in Ω . We assume that each X i is formally skew-adjoint operator (i.e. X i = −X * i ). Denote that I = (i 1 , i 2 , · · · , i k ), and X I = [X i1 , [X i2 , · · · [X i k−1 , X i k ] · · · ]] as the k-th repeated commutators of (X 1 , X 2 , . . . , X m ), here 1 ≤ i j ≤ m, and |I| = k is called the length of the commutators. We say that X satisfies Hörmander's condition in Ω with Hörmander's index Q < +∞, if there is an integer Q ≥ 1 such that the vector fields {X 1 , X 2 , . . . , X m } together with their commutators of length at most Q can span the tangent space at each point x ∈ Ω . If X satisfies the Hörmander's condition then we say that X is finite degenerate, otherwise X is infinitely degenerate and the operator X := m j=1 X 2 j is called as infinitely degenerate elliptic operator.
In this paper, we consider the following initial-boundary value problem of infinitely degenerate semilinear pseudo-parabolic equation with logarithmic nonlinear term where Ω is a bounded open domain and Ω ⊂⊂ Ω . Here, we pose the following hypotheses: (H-1) ∂Ω is C ∞ and non-characteristic for the system of vector fields X.
(H-2) X satisfies the finite type of Hörmander's condition with Hörmander index Q ≥ 1 on Ω except an union of smooth surfaces Γ which are also non-characteristic for X.
Pseudo-parabolic equations describe a variety of important physical processes, such as the seepage of homogeneous fluids through a fissured rock [2], the unidirectional propagation of nonlinear, dispersive, long waves [4,37], and the aggregation of populations [31]. The pseudo-parabolic equation where k is a positive constant, Ω is a bounded domain of R n with smooth boundary ∂Ω and is the standard Laplace operator, can be used in the analysis of nonstationary processes in semiconductors in the presence of sources [21,22]. Furthermore, Eq. (3) can be regarded as a Sobolev type equation or a Sobolev-Galpern type equation [36].
Ting, Showalter and Gopala Rao [14,35,38] investigated the initial-boundary value problem and the Cauchy problem for the linear pseudo-parabolic equation, and established the existence and uniqueness of solutions. From then on, considerable attentions have been paid to the study of nonlinear pseudo-parabolic equations, even including singular pseudo-parabolic equations and degenerate pseudo-parabolic equations, see [5,13,33] and the references therein for details. For the nonlinear case, besides the existence and uniqueness results, the properties of solutions, such as asymptotic behavior and regularity, were also investigated.
On the one hand, if f (u) in (3) is a polynomial such as u p with p > 0, Benedetto and Pierre [3] established the maximum principle of Eq. (3). Cao, Yin, and Wang [6] studied the Cauchy problem of Eq. (3). In fact, by the integral representation and the contraction-mapping principle, they not only proved the existence, uniqueness and comparison principle for mild solutions, but also got large time behavior of solutions, the critical global existence exponent and the critical Fujita exponent for Eq. (3). Later, Xu and Su [39] considered the initial-boundary value problem (3) with p > 1 and showed the invariance of some sets, global existence, blow-up in finite time and asymptotic behavior of solutions. Here, they used the so-called potential well method, which was found and developed by Sattinger [34] in 1968 and Payne and Sattinger [32], respectively. Later, the method was greatly improved by Liu and Zhao [24].
On the other hand, in case of f (u) = u log |u| in (3), Chen et al. [8,11] first studied this kind of initial boundary value problem (3). They used the generalized potential well method and the following logarithmic Sobolev inequality for any u ∈ H 1 0 (Ω) and a > 0, to prove that the global existence, blow-up at +∞ and asymptotic behaviour for the solutions. However, for the problem (1), such kind of "good" logarithmic Sobolev inequality (4) is not satisfied, and we have only the logarithmic regularity estimate (2). Thus it is more difficult to deal with the nonlinear term f (u) = u log |u| for the infinitely degenerate problem (1). Ji, Yin and Cao [16] recently study the periodic problem for semilinear heat equation and pseudo-parabolic equation (3) with logarithmic source, and show the existence and instability of positive periodic solutions. Associated with the system of vector fields X = (X 1 , X 2 , · · · , X m ), we introduced the following Sobolev space (cf. [30]): which is a Hilbert space with norm where Xu 2 L 2 (Ω ) = m j=1 X j u 2 L 2 (Ω ) . The space H 1 X,0 (Ω) is defined by the closure of C ∞ 0 (Ω) in H 1 X (Ω ), which is also a Hilbert space. In this paper, under the hypotheses (H-1), (H-2) and (H-3), our goal is to prove the global existence and blow-up at +∞ of the solutions for the problem (1) in H 1 X,0 (Ω). Now, for our purpose, we introduce the following definitions.  (1) on Ω × [0, T ), if u ∈ L ∞ (0, T ; H 1 X,0 (Ω)) with u t ∈ L 2 (0, T ; H 1 X,0 (Ω)) satisfies the problem (1) in the distribution sense, i.e.
Define two functionals on H 1 X,0 (Ω) as follows: Then, it is obvious that The mountain pass level d, also known as potential well depth, is defined as where λ 1 is the first eigenvalue of the operator − X , C X can be got by the hypothesis (H-3) and the logarithmic Sobolev inequality (13). (in Lemma 3.3 below we can show that d ≥ L).
Our main results can be stated as follows. where • if L ≤ J(u 0 ) ≤ d and I(u 0 ) > 0, then for any given sufficiently small positive number ξ ∈ (0, L), there exists t ξ > 0 such that where where the positive constant C 1 is determined by C X and u 0 ; • for any η ∈ (0, 1), there exists t η > 0 such that where the positive constant C 2 is dependent on η and t η .
We organize this paper as follows. After recalling the basic properties for the infinitely degenerate system of vector fields X in Section 2, we introduce a family of potential wells relative to the logarithmic nonlinear term u log |u| and discuss the invariance of some sets under the flow of (1) and vacuum isolating behavior of solutions for problem (1) in Section 3. Finally, we pose the proof of Theorem 1.3 and 1.4 in Section 4, respectively.
2. The basic properties for the infinitely degenerate system of vector fields. Proposition 1. (Logarithmic Sobolev inequality, cf. [30]) Suppose that the system of vector fields X = (X 1 , · · · , X m ) verifies the estimate (2) for some s > 1 2 . Then there exists C 0 > 0 such that Proposition 2. (Poincaré inequality, cf. [30]) Assume that the system of vector fields X verifies the logarithmic regularity estimate (2) for s > 1, ∂Ω is C ∞ and non-characteristic. Then the first eigenvalue λ 1 of the operator − X is strictly positive and there holds By Proposition 2, we can use as an equivalent norm of the space H 1 X,0 (Ω).
3. Potential wells. In this section, under the hypotheses (H-1), (H-2), and (H-3), we first introduce a family of potential wells for problem (1) and give a series of their properties.
and takes the maximum at λ = λ X ; where λ X = exp .
By introducing the so-called Nehari manifold we see from Lemma 3.1 that d > 0, and the potential well depth d is also characterized by Thus, we can define the potential well For δ > 0, we introduce where λ 1 is the first eigenvalue of the operator − X . Moreover, for δ ∈ (0, 1 + 1 2λ1 ), we define }. Now, we are ready to prove the following Lemma 3.2. For u ∈ H 1 X,0 (Ω) and r(δ) as defined by (17), there hold (1) Hypothesis (H-3) and the logarithmic Sobolev inequality (13) give that where C X is defined in (9). By Young's inequality, (18) and Poincaré inequality (14), we see that By the definitions of r(δ) and I δ (u), Lemma 3.2 (1) follows.
Proof. First, the inequality J(u) > d 0 > 0 implies Xu L 2 (Ω) = 0. If the sign of I δ (u) is changed when δ ∈ (δ 1 , δ 2 ), then there exists a δ ∈ (δ 1 , δ 2 ) such that I δ (u) = 0. Thus by the definition of d(δ) we have J(u) ≥ d(δ ), which is contradictive with Note that by Definition 1.1 the weak solution u satisfies the following energy equality Next, by using above potential wells, we discuss the invariance of some sets under the flow of (1) and vacuum isolating behavior of solutions for problem (1).
From Corollary 1 and Lemma 3.3 we get the following result.
Corollary 1 show that for the set of all solutions of the problem (1) with d 0 < J(u 0 ) ≤ µ < d, there exists a vacuum region such that no solution of the problem (1) belongs to U µ . The vacuum region U µ becomes bigger and bigger when µ is decreasing. As the limit case we obtain Here, by Lemma 3.3 we have chosen δ 0 ∈ (1, 1 + 1 2λ1 ) such that d(δ 0 ) = d 0 . Proposition 5. Assume that u 0 ∈ H 1 X,0 (Ω), J(u 0 ) ≤ 0, I(u 0 ) < 0, and u is a weak solution of the problem (1), then I(u) < 0 for all 0 ≤ t < T .

4.1.
Global existence with exponential decay. In this subsection, under the hypotheses (H-1), (H-2) and (H-3), we shall use the Galerkin approximation technique and the potential well method to prove the global existence of weak solutions for problem (1). Meanwhile, we shall obtain the asymptotic stability of the global solutions.
In order to get the asymptotic behavior of the global solutions, we need the following well-known estimate Lemma 4.1 (See [20]). Let y(t) : R + → R + be a nonincreasing function. Assume that there is a constant A > 0 such that +∞ s y(t)dt ≤ Ay(s), 0 ≤ s < +∞.
Proof of Theorem 1.3. We divide the proof into four steps.
Step 1. Global existence in the case of J(u 0 ) < d.
First, we can excluse some special cases as follows.
Step 2. Global existence in the critical case of J(u 0 ) = d.
Step 3. Decay estimate in the case of J(u 0 ) < L.

4.2.
Blow-up at +∞ of solution. In this subsection, by using the properties of a family of potential wells, we can prove blow-up at +∞ of solutions for problem (1) when J(u 0 ) ≤ d and I(u 0 ) < 0.
Proof of Theorem 1.4. We divide the proof into three steps.