Blow-Up Results for a Class of Quasilinear Parabolic Equation with Power Nonlinearity and Nonlocal Source

where Ω ⊂ RNðN ≥ 1Þ is a bounded domain with smooth boundary ∂Ω and Δpu = div ðj∇ujp−2∇uÞ is the standard p-Laplace operator with p > 2, μ, k > 0, u0ðxÞ ∈W1,p 0 ðΩÞ \ f0g. In the past decades, many physical phenomena have been expressed as nonlocal mathematical models (see [1, 2]). It is also suggested that the nonlocal growth term provides a more realistic model for the physical model of compressible reaction gas. Problem (1) appears in the study of fluid flow through porous media with integral source (see [3, 4]) and population dynamics (see [5, 6]). Actually, equations of the above form are mathematical models occurring in studies of the p-Laplace equation ([7–15] and references therein), generalized reaction-diffusion theory [16], nonNewtonian fluid theory [17, 18], non-Newtonian filtration theory [19, 20], and the turbulent flow of a gas in porous medium [7]. Media with p > 2 are called dilatant fluids and those with p < 2 are called pseudoplastics. If p = 2, they are Newtonian fluids. When p ≠ 2, the problem becomes more complicated since certain nice properties inherent to the case p = 2 seem to be lose or at least difficult to verify. Blow-up results of parabolic equations with nonlocal sources have been studied as well. For example, the problem of the form


Introduction
In this paper, we consider the following quasilinear parabolic equation with power nonlinearity and nonlocal source term: where Ω ⊂ R N ðN ≥ 1Þ is a bounded domain with smooth boundary ∂Ω and Δ p u = div ðj∇uj p−2 ∇uÞ is the standard p-Laplace operator with p > 2, μ, k > 0, u 0 ðxÞ ∈ W 1,p 0 ðΩÞ \ f0g. In the past decades, many physical phenomena have been expressed as nonlocal mathematical models (see [1,2]). It is also suggested that the nonlocal growth term provides a more realistic model for the physical model of compressible reaction gas. Problem (1) appears in the study of fluid flow through porous media with integral source (see [3,4]) and population dynamics (see [5,6]). Actually, equations of the above form are mathematical models occurring in studies of the p-Laplace equation ( [7][8][9][10][11][12][13][14][15] and references therein), generalized reaction-diffusion theory [16], non-Newtonian fluid theory [17,18], non-Newtonian filtration theory [19,20], and the turbulent flow of a gas in porous medium [7]. Media with p > 2 are called dilatant fluids and those with p < 2 are called pseudoplastics. If p = 2, they are Newtonian fluids. When p ≠ 2, the problem becomes more complicated since certain nice properties inherent to the case p = 2 seem to be lose or at least difficult to verify.
Blow-up results of parabolic equations with nonlocal sources have been studied as well. For example, the problem of the form was studied by Li and Xie [7]. They established global existence of solutions and discussed the blow-up properties of solutions.
The authors in [8] studied the following p-Laplacian equation with power nonlinearity By using an efficient technique and according to some sufficient conditions, the global existence and decay estimates of solutions under some sufficient conditions are discussed.
The authors in [9] considered the Neumann problem to the following initial parabolic equation with logarithmic source: in a bounded domain with smooth boundary, p > 2. By using the logarithmic Sobolev inequality and potential wells method, they obtain the decay, blow-up, and nonextinction of solutions under some conditions. In [10], the following model of a quasilinear diffusion equation with interior logarithmic source has been studied: in which p > 2, u 0 ðxÞ ∈ W 1,p 0 ðΩÞ \ f0g: By using the potential well method and a logarithmic Sobolev inequality, the authors obtained results of existence or nonexistence of global weak solution. They also provided sufficient conditions for the large time decay of global weak solutions and for the finite time blow-up of weak solutions. Among some other interesting results, they showed that the weak solution uðx, tÞ of problem (5) blows up at finite time under the condition Jðu 0 Þ ≤ M and Iðu 0 Þ < 0, where M > 0 is a constant; the energy functional JðuÞ and Nehari functional IðuÞ are defined as follows: in which k·k p = ð Ð Ω j·j p dxÞ 1/p .
Motivated by the above studies, in this paper, we investigate blow-up results of problem (1). We will give the conditions for the blow-up results and establish the lower bounds for the blow-up rate. Our main results are as follows.
Theorem 2. Assume that u 0 ∈ H 1 0 ðΩÞ and J 1 ðu 0 Þ < 0, let u = uðx, tÞ be the nonnegative solution of problem (1), then u blows up in finite time Theorem 3. Assume that u 0 ∈ H 1 0 ðΩÞ, J 1 ðu 0 Þ ≤ 0, Ð t 0 0 ku s ð·, sÞk 2 2 ds > 0 for any t 0 > 0, then, the weak solution u = uðx, tÞ of problem (1) blows up at infinity. Moreover, if ku 0 k 2 ≤ ð−Jðu 0 ÞÞ 2/p , the lower bound for blow-up rate can be estimated by 2. Criterions of Blow-Up 2.1. Preliminaries. In this section, we start with the definition of weak solution and blow-up at infinity of (1). Definition 5 (blow-up at infinity). Let uðx, tÞ be a weak solution of (1), we call uðx, tÞ blow-up at +∞ if the maximal existence time T = +∞ and To obtain the blow-up results, define the potential energy functional and the Nehari's functional as follows: ð Þu p+1 y, t ð Þdxdy, To prove the main result, we need the following lemmas.

Journal of Function Spaces
Lemma 6. Assume that uðx, tÞ is a weak solution of (1). Then, J 1 ðuÞ is nonincreasing with respect to t and satisfies the energy inequality Proof. Similar to the proof in [8,10,11], we can get the result.

Lemma 7.
[8] J 1 ðuÞ is a nonincreasing function, for t ≥ 0, Lemma 8 [12]. Let Φ be a positive, twice differentiable function satisfying the following conditions: for some t ∈ ½0, TÞ, and the inequality where α > 1. Then, we have in whichΦ is a positive constant, and This implies Lemma 9 [9]. Suppose that θ > 0, α > 0, β > 0, and hðtÞ is a nonnegative and absolutely continuous function satisfying h ′ðtÞ + αh θ ðtÞ ≥ β, then for 0 < t < +∞, it holds 2.2. Proof of the Main Results. We will consider the finite time blow-up results of problem (1) under the condition of nonpositive initial energy. The theorems are proved as follows.
Proof of Theorem 1. Assume that uðx, tÞ is the weak solution of problem (1), for any T > 0, we define the functional It is obvious that ΓðtÞ > 0 for all t ∈ ½0, T. Since Γ is continuous, there exists ρ > 0 (independent of T) such that ΓðtÞ > ρ for all t ∈ ½0, T.
Then, we have By using (12) in Lemma 7, we have From J 1 ðu 0 Þ < 0, (22) and (23), we get Now, multiplying (24) by ΓðtÞ, we have Noticing that With the help of Cauchy-Schwarz inequality, we have 3 Journal of Function Spaces Using (25) and (28), we further get for all t ∈ ½0, T: By Lemma 8, there exists T * > 0 such that which implies As a consequence, we get this means uðx, tÞ blows up at finite time T * .