Blow-Up of Solutions for a Class of Nonlinear Pseudoparabolic Equations with a Memory Term

and Applied Analysis 3 We now construct a space of functions as follows. Let H m (Ω) denote the Sobolev space with the norm ‖u‖ H m = (∑ |α|≤m ‖D α u‖ 2 2 ) 1/2 .C 0 (Ω) denotes the class ofC functions with the compact support inΩ.H 0 (Ω) denotes the closure in H m (Ω) of C∞ 0 (Ω). The Hilbert spaceH 0 (Ω) is a subspace of the Sobolev spaceH(Ω). The following are the basic hypotheses to establish the main results of this paper: (a) 2 < p ≤ 2 + α < ∞; (b) λ is a C1 function satisfying λ (τ) ≥ 0, λ 󸀠 (τ) ≤ 0, (10)

Equation (1) includes many important mathematical physics models.
In the absence of the memory term and -Laplace operator term ( = 0, () = 0),  =  = 1, the model reduces to semilinear pseudoparabolic equation: Kaikina et al. [10] discussed the periodic boundary value problem of (3) under some assumption forms of nonlinear function .Cao et al. [11] investigated a class of periodic problems of pseudoparabolic type equations with nonlinear periodic sources.A rather complete classification of the exponent  was given, in terms of the existence and nonexistence of nontrivial and nonnegative periodic solutions.Cao et al. [12] dealt with the Cauchy problem for semilinear pseudoparabolic equations.Existenceand uniqueness of local solutions were proved, and the large-time behavior was investigated.Kaikina [13] and Xu and Su [14] discussed the initial boundary value problems of pseudoparabolic equation (3) under some classes of nonlinear function ().
They obtained some sufficient conditions of existence and uniqueness of local solutions and the large-time behavior of global solutions.
(5) He investigated the initial boundary value problem of (5) and established the global existence of a strong solution of the problem.
In the absence of the viscous term and -Laplace operator ( = 0,  = 0), as  = 1, the model reduces to the equation Yin [20] obtained the global existence of a classical solution of (7) under the assumption of a one-sided growth condition.Messaoudi [21] investigated a semilinear parabolic equation with the viscoelastic memory term.He established the finite time blow-up result for the solution with negative or vanishing initial energy for nonlinear function () = || −2 .
To the best of our knowledge, there are few works on the study of nonlinear pseudoparabolic equation with memory term of Volterra integral type.Shang and Guo [22][23][24] investigated the initial boundary value problem and initial value problem of the nonlinear pseudoparabolic equations with Volterra integral term: They proved the existence, uniqueness, and regularities of the global strong solution and gave some conditions of the nonexistence of global solution.In 2007, Ptashnyk [25] investigated the initial boundary value of degenerate quasilinear pseudoparabolic equations with memory term.He obtained some existence results of global solutions.Up to now, there are not any research works on the multidimensional nonlinear pseudoparabolic equations with memory term.
In the present work, we deal with the initial boundary problem of the nonlinear pseudoparabolic equation with the memory term of Volterra integral type, the damping term, and -Laplace operator: where Ω ⊂ R  is a bounded domain, () : R + → R is a given continuous function, , , , > 0, and div(|∇| −2 ∇) is the so-called -Laplace operator.By using the concavity method first introduced by Levine [5], under negative initial energy and suitable conditions on , , and the relaxation function (), we prove that there exists finite-time blow-up solution.
Without loss of generality, we choose  =  =  = 1 in the following discussion.

Preliminaries and Main Results
In this section, we introduce some notations, basic definitions, and important lemmas which will be needed in this paper.
For functions (, ), V(, ) defined on Ω, we introduce Abstract and Applied Analysis 3 We now construct a space of functions as follows.Let   (Ω) denote the Sobolev space with the norm .  ∞ 0 (Ω) denotes the class of  ∞ functions with the compact support in Ω.   0 (Ω) denotes the closure in   (Ω) of  ∞ 0 (Ω).The Hilbert space   0 (Ω) is a subspace of the Sobolev space   (Ω).
The following are the basic hypotheses to establish the main results of this paper: To obtain the results of this paper, we will introduce the "modified" energy function: where The following lemma is similar to the lemma of [21] with slight modification.
Lemma 1. Assume that (10) hold.Let p satisfy () and let  be a solution of (8).Then () is nonincreasing function; that is Moreover, the following energy inequality holds: Proof.By multiplying the equation in (8) For the last term on the left side of ( 16), Inserting ( 17) into ( 16), we have for regular solution.The proof of Lemma 1 is completed.This result is valid for weak solutions by a simple density argument.Now we consider the finite time blow-up of solutions with (0) < 0 for the problem (8).
A direct computation yields By multiplying (8) with  and integrating over Ω, This implies that and we have where Using Schwartz's inequality, we have this implies that where  is a positive constant.From the discussion above, we see that (39) Hence, this proves that ()