Blow-up phenomena for a pseudo-parabolic system with variable exponents

In this paper, we consider a pseudo-parabolic system with nonlinearities of variable exponent type { ut − ∆ut − div(|∇u|m(x)−2∇u) = |uv|p(x)−2uv2 in Ω× (0, T), vt − ∆vt − div(|∇v|n(x)−2∇v) = |uv|p(x)−2u2v in Ω× (0, T) associated with initial and Dirichlet boundary conditions, where the variable exponents p(·), m(·), n(·) are continuous functions on Ω. We obtain an upper bound and a lower bound for blow-up time if variable exponents p(·), m(·), n(·) and the initial data satisfy some conditions.


Introduction
In this paper, we consider the initial-boundary value problem where Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω, the nonlinear term div(|∇u| m(x)−2 ∇u) is called m(x)-Laplace operator, and the variable exponents p(•), m(•), n(•) are continuous functions on Ω, later specified.It is well known that nonlinear pseudo-parabolic equations appear in the study of various problems of the hydrodynamics, filtration theory, electrorheological fluids and others Corresponding author.Email: nttccyj@ntu.edu.cn(see [1,4,6]).Recently, Di et al. [2] has been studied the following initial-boundary value problem u t − ν∆u t − div(|∇u| m(x)−2 ∇u) = |u| p(x)−2 u in Ω × (0, T) (1.2) with Dirichlet boundary condition.By means of a differential inequality technique, they obtained an upper bound and a lower bound for blow-up time if variable exponents p(•), m(•) and the initial data satisfy some conditions.Obviously, if ν = 1, m(x) = 2, p(x) = p, (1.2) reduces to the following pseudo-parabolic equation As for (1.3), there are many results concerning asymptotic behavior [7,14], the existence and uniqueness [1,13] of solutions, blow-up [8,14] property and so on.Especially, Xu [14] prove that the solutions blow up in finite time in H 1 0 (Ω)-norm.Luo [8] obtain an upper bound and a lower bound of the blow-up rate.More generally, Peng et al. [10] considered the following initial-boundary value problem A lower bound for blow-up time is determined if blow-up does occur.Furthermore, they establish an upper bound for blow-up time to a special class.
As we know, on the bounds, has been less studied the case of blow-up time to the system (1.1).Our objective in this paper is to study the blow-up phenomenon of solutions of the system (1.1) in the framework of the Lebesgue and Sobolev spaces with variable exponents.In details, this paper is organized as follows: in Section 2, we introduce the function spaces of Orlicz-Sobolev type and present a brief description of their main properties.In Section 3, a criterion for blow-up to the system (1.1) that leads to the upper bound for blow-up time is obtained.In Section 4, we give the lower bound of blow-up time to the system (1.1).

Function spaces
As in [2], we first recall some known results about the Lebesgue and Sobolev spaces with variable exponents (see [3,5,11,12]) which will be needed in this paper.

Upper bound for blow-up time
Since p(•), m(•), n(•) are continuous functions on Ω, we denote by where stands for p(•), m(•) and n(•) respectively.Assume that and Firstly, we start with the following local existence theorem for the solutions of system (1.1) which can be obtained by Faedo-Galerkin method.
Next, we seek the upper bound for the blow-up time of the system (1.1).
Theorem 3.2.Assume that (2.1), (3.1) and Then, the solution (u, v) of the system (1.1) blows up in finite time T * in H 1 0 (Ω)-norm.Moreover, an upper bound for blow-up time is given by where β and b are suitable positive constants given later and F(0) = u 0 2 Proof.Replacing ϕ by u t , ψ by v t in (3.3) respectively, and adding, we have Let us define the energy as follows Hence, by (3.6) and (3.7), we have We define an auxiliary function Multiplying u and v on two sides of two equations of the system (1.1) respectively, and integrating by part, we have Thanks to E (t) ≤ 0, we have By (3.13) and (3.14), we see where This implies that , where λ 1 is the first eigenvalue of the problem Thus, the follow relations hold, where u p = ( Ω u p dx) Combining (3.15) and (3.16), we conclude ). Consequently, which implies By (3.17) and the fact that Therefore, we have that F (t) ≥ α, where α = min{C 8 (F(0)) 2), it is easy to see a − b ≤ 0. So, combining with (3.18), we get where the constant β = (3.20) Integrating the inequality (3.20) from 0 to t, we see which implies that Thus, (3.22) shows that F(t) blows up at some finite time T * such that Finally, we get the solution (u, v) blows up in H 1 0 (Ω)-norm in finite time.
Remark 3.3.From (3.23), we see that the larger F(0) is, the smaller the blow-up time T * is.

Lower bound for blow-up time
In this section, our aim is to determine a lower bound for blow-up time of the system (1.1).
The technique is the same as [2].
Theorem 4.1.Suppose that (2.1) and (3.1) hold.Furthermore assume that 2 (Ω) and the solution (u, v) of the system (1.1) becomes unbounded at finite time T * in H 1 0 (Ω)-norm, then a lower bounded T * for blow-up time is given by where M and N are suitable positive constants given later and F(0) = u 0 2 Proof.We define the function F(t) the same as (3.9).By (3.13), it is easy to get where B + , B − are the Sobolev embedding constants for H 1 0 (Ω) → L p + (Ω) and H 1 0 (Ω) → L p − (Ω), respectively.From the Cauchy-Schwarz inequality, we have