Blow-Up in a Slow Diffusive p-Laplace Equation with the Neumann Boundary Conditions

and Applied Analysis 3 Proof. An easy computation using (1) and the fact∫ Ω u dx = 0 and by parts shows that M 󸀠 (t) = ∫ Ω uu t dx = ∫ Ω u (div (|∇u|p−2∇u) + |u|q−1u − − ∫ Ω |u| q−1 u dx) = −∫ Ω |∇u| p dx + ∫ Ω |u| q+1 dx = − (q + 1) E (t) + q + 1 − p p ∫ Ω |∇u| p dx. (15) The last equality implies M 󸀠 (t) ≥ − (q + 1) E (t) = − (q + 1) E (0) + (q + 1) ∫ t 0 ∫ Ω (u t ) 2 dx ds

The problem (1) with  = 2 can be used to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved [1,2].If  > 2, the problem (1) is the degenerate parabolic equation and appears to be relevant in the theory of non-Newtonian fluids (see [3]).Here, we are mainly interested in the case  > 2, namely, the degenerate one.
As an important feature of many evolutionary equations, the properties of blow-up solution have been the subject of intensive study during the last decades.Among those investigations in this area, it was Fujita [7] who first established the so-called theory of critical blow-up exponents for the heat equation with reaction sources in 1966, which can be, of course, regarded as the elegant description for either blowup or global existence of solutions.From then on, there has been increasing interest in the study of critical Fujita exponents for different kinds of evolutionary equations; see [8,9] for a survey of the literature.In recent years, special attention has been paid to the blow-up property to nonlinear degenerate or singular diffusion equations with different nonlinear sources, including the inner sources, boundary flux, or multiple sources; see, for example, [3,10,11].
In some situations, we have to deal with changing sign solutions.For instance, the changing sign solutions were considered in [1] for the nonlocal and quadratic equation with the Neumann boundary condition.The study in [5] for a natural generalization of (2), proposed with 1 <  ≤ 2 a global existence result (for small initial data) and a new blow-up criterion (based on the partial maximum principle and a Gamma-convergence argument).The authors also conjectured that the solutions blow up when  > 2, which was then proved with a positive answer [4].The changing sign solutions to the reaction-diffusion equation were discussed in [2], with such as (, ()) = || −1  − ().
The blow-up of solutions was obtained even under the source with − ∫ Ω   = 0.The semilinear parabolic equation [12] with a homogeneous Neumann's boundary condition is studied.A blow-up result for the changing sign solution with positive initial energy is established.In [6], a fast diffusive -Laplace equation with the nonlocal source (, ) ∈ Ω × (0, ) ,   = 0, (, ) ∈ Ω × (0, ) ,  (, 0) =  0 () ,  ∈ Ω, was considered.The authors showed that a critical blowup criterion was determined for the changing sign weak solutions, depending on the size of  and the sign of the natural energy associated.The relationship between the finite time blow-up and the nonpositivity of initial energy was discussed, based on an energy technique.Notice that (1) is degenerate if  > 2 at points where ∇ = 0; therefore, there is no classical solution in general.For this, a weak solution for problem (1) is defined as follows.
The local existence of the weak solutions can be obtained via the standard procedure of regularized approximations [10].Throughout the paper, we always assume that the weak solution is appropriately smooth for convenience of arguments, instead of considering the corresponding regularized problems.
This paper is organized as follows.In Section 2, we show that the solutions to (1) blow up with nonpositive initial energy.In Section 3, under some assumptions of initial data, we prove that the solutions with bounded initial energy also blow up in finite time.

Lemma 3. The energy 𝐸(𝑡) is a nonincreasing function and
Proof.A direct computation using (1) and by parts yields Integrate from 0 to  to get (12).
Then, there exists  1 with 0 <  1 < ∞, such that First, we prove the following two Lemmas, similar to the idea in [13].
Lemma 7. Assume that  is a solution of the system (1).If (0) <  1 and ‖∇ 0 ‖  >  1 .Then, there exists a positive constant  2 >  1 , such that Proof.Let ‖∇‖  =  and by (32), we have For convenience, we define It is easy to find that  increases if 0 <  < which implies that Therefore, (36) is concluded.
Then, we have the following.
At the end, let us finish the proof of Theorem 6.