Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

In this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

The inequality f (u)u ≥ 0 means that the direction of external force is the same as the direction of displacement. Problem (1.1)-(1.3) describes some important physical issues (see [2]) and has attracted the attention of many scholars. For example, Benedetto, Pierre, Gopala Rao, Slowalter, and Ting [1,5,20,22] considered the initial boundary value problem for the equation u tk u tu = 0, got the maximum principle and established the existence and uniqueness of solution. Cao, Yin, and Wang in [3] studied the Cauchy problem of semilinear pseudo-parabolic equations. With the initial data being appropriately smooth, they got the existence and uniqueness of mild solution. Moreover, they proved that there exist the critical global existence exponent and the critical Fujita exponent for pseudo-parabolic equations.
For the equation Xu and Su in [26] considered the initial boundary value problem and proved the global existence, nonexistence, and asymptotic behavior of solutions when J(u 0 ) ≤ d. Moreover, they proved finite time blow-up when J(u 0 ) > d by the comparison principle. For the background of the problem in [26], one can refer to [7,8,14,21]. Up to now, there has been no paper to study the nonlinear evolution equations of arbitrary higher order by the method of potential well, and the study for nonlinear evolution equations of arbitrary higher order by the method of potential well is more difficult. If M = 1, there have been some results about the initial boundary value problem (1.1)-(1.3) (see [26]). If M ≥ 2, there have not been any results about the initial boundary value problem (1.1)-(1.3). In this paper, for M ∈ N + = {1, 2, 3, . . .}, the initial boundary value problem (1.1)-(1.3) is studied by the method of potential well. In the growth condition of nonlinear term f (u), the growth exponent p depends not only on space dimension n, but also on M (see condition (H)). The result of this paper is applicable for a big class of higher-order n-dimensional nonlinear pseudo-parabolic equations. Moreover, the study for the higherorder nonlinear evolution equation will be related to the properties about the higher-order eigenfunctions and the higher-order eigenvalues of Laplace operator.
The method of potential well was proposed by Sattinger in [19] and was used to prove the existence of global weak solutions to nonlinear hyperbolic equations which do not have positive energy. Payne and Sattinger in [18] have shown the theory of potential well in detail and made the further study of potential well. Under some conditions they have proved that the solutions to the equation u ttu = f (u) (x ∈ , t > 0) will blow up in finite time.
Recently, the potential well method has been well applied in some important physical problems. Liu and Zhao in [13] generalized the family of potential wells to the initial boundary value problems of semilinear hyperbolic equations and parabolic equations. They not only gave a threshold result of global existence and nonexistence of solutions, but also obtained the vacuum isolating of solutions. On the basis of Liu and Zhao's work, Xu in [24] continued to study the initial boundary value problem with critical initial data E(0) = d (or J(u 0 ) = d), I(u 0 ) < 0 and proved that there exist non-global solutions under the classical conditions on f . By the method of potential well, Liu, Xu, and Yu in [12] studied the Cauchy problem of semilinear heat equations and obtained a threshold result for the global existence and nonexistence of solutions. Moreover, they got the asymptotic behavior of the solution when J(u 0 ) ≤ d. Xu in [23] discussed the initial boundary value problem of semilinear parabolic equations with semilinear term f (u). By introducing a family of potential wells, not only the solutions decay to zero with f (u) satisfying some conditions when J(u 0 ) ≤ d and I(u 0 ) > 0 was proved, but also finite time blows up when J(u 0 ) ≤ d and I(u 0 ) < 0 was obtained. In addition, based on the potential well method, for the case of high initial energy, finite time blow-up was proved (see [9,25,27]); for dealing with problems with variational methods, one can also refer to [15,16]. In fact, there are many well-known results, which affect both global existence and finite time blow-up of solutions to the nonlinear damping wave equations, if we focus on the initial data (see [10,17,28,29]). This paper is organized as follows.
(i) In Sect. 2: we study the static problem which is closely related to the evolution problem and is the important basis of the evolution problem. (ii) In Sect. 3: some useful lemmas and their proofs are given. (iii) In Sect. 4: by using the method of potential well, we prove the existence of global weak solutions to problem (1.1)-(1.3) with subcritical energy, i.e., E(0) < d, which is shown in Theorem 4.1. Throughout the paper, we denote Let us begin with defining the following total energy functional: where ∇ = ( ∂ ∂x 1 , ∂ ∂x 2 , . . . , ∂ ∂x n ) is an n-dimensional gradient operator and F(u) = u 0 f (s) ds. Then the potential energy functional 5) and the Nehari functional By I(u) we define the Nehari manifold the potential well and the set outside the potential well where d is the depth of the potential well which is defined as follows: (1.7)

Static problem
In this section, the existence of nonzero weak solution to the following boundary value problem is considered. This is the important basis of evolution problem, where , M, , γ , and D γ are the same as in Sect. 1.
Assume that the function f (·) satisfies: where a > b > 0, a 1 > 0, a, b, and a 1 are positive real constants. In addition, the positive real number p satisfies that when 2M < n, 1 < p < n+2M n-2M ; when 2M ≥ n, 1 < p < +∞. Assume that there exists 0 < γ < 1 2 such that where a and b are positive real constants (a > b ). Take f (u) = 1 4 u 2m+1 , where m is a positive integer. Then the above conditions can be satisfied.
Obviously, u = 0 is the trivial weak solution to problem (2.1). In what follows we will prove that there exists a nontrivial weak solution to problem (2.1).

Make the functional
First, we will prove that if I(u 0 (x)) = min u∈H M 0 ( ) I(u), the function u 0 (x) must be the weak solution to problem (2.1).

Lemma 2.1
If I(u 0 (x)) = min u∈H M 0 ( ) I(u), then the function u 0 (x) must be the weak solution to problem (2.1).
Proof Assume that the functional I(u) takes its minimum at u 0 (x) ∈ H M 0 ( ), it concludes that ∀h(x) ∈ H M 0 ( ), the function ϕ(t) = I(u 0 (x) + th(x)) takes its minimum at t = 0, that is, Make the following computations: therefore, By ϕ (0) = 0, we conclude that Hence, for each h(x) ∈ H M 0 ( ), we get The function u 0 (x) ∈ H M 0 ( ) is the weak solution to problem (2.1). The proof of Lemma 2.1 is completed.
From the above discussion we know that if we hope to find the nontrivial weak solution to problem (2.1), we need to find the nonzero minimum point of the functional I(u) in H M 0 ( ). We will prove the conclusion by the mountain pass theorem. Notice that ∇ M u(x) L 2 ( ) is an equivalent norm in H M 0 ( ), we can define the inner product in By the above inner product we can induce the following norm: which is an equivalent norm in H M 0 ( ). The following Theorem 2.1 shows that the functional I(u) has a nonzero minimum point in H M 0 ( ). Proof We first consider the case when 2M < n.
Step 1: We will verify that For u(x), w(x) ∈ H M 0 ( ), by some computations, Therefore, the functional I 1 is differential at u(x) and I 1 (u) = u. It concludes that I 1 (u) is a C 1 functional in H M 0 ( ). In what follows we will prove that the functional we can define a bounded linear functional w * ∈ H -M ( ) = (H M 0 ( )) * .
By 2M < n and 1 < p < n+2M n-2M , it concludes that 2np n+2M < 2n From the above discussion it shows that by f (u) we can define a bounded linear func- ) is a weak solution to the following problem: In what follows we prove that, for each u(x) ∈ H M 0 ( ), we can get For a, b ∈ R, by some computations, we derive By equality (2.5), for each w(x) ∈ H M 0 ( ), In what follows we make some estimates about R.
To save notations, in the following discussion C represents different positive constants in different places.
Since w(x), u(x) ∈ H M 0 ( ), by the Sobolev imbedding theorem, there exists a positive constant C such that By Hölder's inequality, we have Notice that p + 1 < n+2M n-2M + 1 = 2 * M , by u(x), w(x) ∈ H M 0 ( ), and the Sobolev imbedding theorem, we can get that Therefore, By the above conclusions we can get that . By the definition of Fréchet derivative, In what follows we will prove that I 2 : where L is a positive constant. Then we have , and the Sobolev imbedding theorem, we can get that Therefore, Summarizing the above discussion, there exists a positive constant C such that Therefore, mapping I 2 : H M 0 ( ) → H M 0 ( ) is Lipschitz continuous on a bounded subset of H M 0 ( ), it means that I 2 is a C 1 functional. Because I 1 is also a C 1 functional, I(u) = I 1 (u) -I 2 (u) is a C 1 functional.
The proof of Step 1 is completed.
Step 2: In what follows we verify that functional I(u) satisfies the Palais-Smale condition. Assume that {u k (x)} +∞ k=1 ⊂ H M 0 ( ) satisfies: |I(u k )| ≤ C (k = 1, 2, 3, . . .) and when k → +∞, Under the above conditions we will prove that there exists a subse- is strongly convergent in H M 0 ( ). By the known conditions, we get When k → +∞, I (u k ) H M 0 ( ) → 0, it concludes that I (u k ) M → 0. By 0 < γ < 1 2 , the I (u k ) M is bounded with respect to k and the inequality is strongly convergent in L s ( ). That is, there exists a function u * (x) ∈ L s ( ) such that when j → +∞, we conclude that Since u k j (x) ∈ L s ( ), from the above inequality we can get that f (u k j (x)) ∈ L 2n n+2M ( ). Therefore, f is a mapping from L s ( ) to L 2n n+2M ( ). The function f (u) is continuous with respect to u, by Theorem 1.1 in [6] the mapping f : Hence, we have Then we can get the following estimates: Therefore, When k → +∞, I (u k ) → 0 in H M 0 ( ), it concludes that when i, j → +∞, By the above discussion we know that f (u k i )f (u k j ) The proof of Step 2 is completed.
Step 3: In what follows we will verify other conditions of mountain pass theorem. The functional by F(0) = 0 it concludes that I(0) = 0. Assume that u(x) ∈ H M 0 ( ) and u(x) M = r, where r is a positive constant which will be determined later.
Summarize the above discussion, all the conditions of mountain pass theorem are satisfied. By the conclusion of mountain pass theorem, there exist a function u(x) ∈ H M 0 ( ) and u(x) = 0 such that I (u) = 0. That is, for each v(x) ∈ H M 0 ( ), we have Therefore, for each v(x) ∈ H M 0 ( ), the equality The above equality means that u(x) ∈ H M 0 ( ) is the nontrivial weak solution (nonzero weak solution) to problem (2.1) When 2M ≥ n, by a similar method, we can prove that there exists a nontrivial weak solution (nonzero weak solution) to problem (2.1).
The proof of Theorem 2.1 is completed.

Some useful lemmas
In this section, we give some properties about functionals J(u), I(u) and the potential well W with its depth d, which will be used in the following discussion.
decreases with respect to λ.
(2) An easy calculation shows that we obtain Hence, conclusion (3) (2) The depth of potential well d in (1.7), there is also a computational formula
(2) From the definition of d in (1.7), it follows that The proof of Lemma 3.2 is completed.
into inequality (4.9), we get By I(u m (x, t)) > 0 for t ∈ [0, T), it shows that |f (u m )| 2 dx = a 2 |u m | 2p dx, from the second inequality in (4.10), we can get that u m H M 0 ( ) is uniformly bounded with respect to m. By condition (H) and the Sobolev imbedding theorem, |u m | 2p dx is uniformly bounded with respect to m, it concludes that f (u m ) is uniformly bounded with respect to m.
By the theory of ordinary differential system, there exists a global solution to problem (4.2)-(4.3) on [0, T). Therefore, by the compactness principle, there exist functions u(x, t), X(x, t) and the subsequence of weakly-star, f (u m (x, t)) → X(x, t) in L ∞ (0, T; L q ( )) weakly-star (q = p+1 p ). By the first inequality of (4.10) it shows that {u mt (x, t)} +∞ m=1 is bounded in L 2 (Q T ). Since ∇ M u m is an equivalent norm in H M 0 ( ), by the second inequality of (4.10) it con- Since H 1 (Q T ) can be compactly imbedded into L 2 (Q T ), there exists a subsequence of {u m (x, t)} +∞ m=1 (still denoted by {u m (x, t)} +∞ m=1 ) such that when m → +∞, u m (x, t) is strongly convergent to u(x, t) in L 2 (Q T ) and u m (x, t) is almost everywhere convergent to u(x, t) in Q T .
Since when m → +∞, u m (x, t) is almost everywhere convergent to u(x, t) in Q T and {f (u m (x, t))} +∞ m=1 is bounded in L ∞ (0, T; L q ( )) (q = p+1 p ). {f (u m (x, t))} +∞ m=1 is also bounded in L q (Q T ). By Lemma 1.3 in [11], we can get that f (u m (x, t)) is weakly convergent to f (u(x, t)) in L q (Q T ) as m → +∞.
Making L 2 ( ) inner product by u t (x, t) in both sides of equation Integrating by parts with respect to x, it concludes that For all t ∈ [0, T), integrating from 0 to t with respect to t, we obtain That is, where [u t , u t ] = t 0 (u τ (x, τ ), u τ (x, τ )) dτ , [∇ M u t , ∇ M u t ] = t 0 (∇ M u τ (x, τ ), ∇ M u τ (x, τ )) dτ . By some computations, we get Combining