Lower Bounds Estimate for the Blow-Up Time of a Slow Diffusion Equation with Nonlocal Source and Inner Absorption

We investigate a slow diffusion equation with nonlocal source and inner absorption subject to homogeneous Dirichlet boundary condition or homogeneous Neumann boundary condition. Based on an auxiliary function method and a differential inequality technique, lower bounds for the blow-up time are given if the blow-up occurs in finite time.

In the present investigation we derive a lower bound for the blow-up time  * when Ω ⊂ R 3 for the solutions that blow up.Equation (1) describes the slow diffusion of concentration of some Newtonian fluids through porous medium or the density of some biological species in many physical phenomena and biological species theories.It has been known that the nonlocal source term presents a more realistic model for population dynamics; see [1][2][3].In the nonlinear diffusion theory, there exist obvious differences among the situations of slow ( > 1), fast (0 <  < 1), and linear ( = 1) diffusions.For example, there is a finite speed propagation in the slow and linear diffusion situation, whereas an infinite speed propagation exists in the fast diffusion situation.
The bounds for the blow-up time of the blow-up solutions to nonlinear diffusion equations have been widely studied in recent years.Indeed, most of the works have dealt with the upper bounds for the blow-up time when blow-up occurs.For example, Levine [4] introduced the concavity method, Gao et al. [5] employed the way of combining an auxiliary function method and comparison method with upper-lower solutions method, and Wang et al. [6] used the regularization method and an auxiliary function method.However, the lower bounds for the blow-up time are more difficult in general.Recently, Payne and Schaefer in [7,8] used a differential inequality technique and an auxiliary function method to obtain a lower bound on blow-up time for solution of the heat equation with local source term under boundary condition (3a) or (3b).Specially, Song [9] considered the lower bounds for the blow-up time of the blow-up solution to the nonlocal problem (1)-( 2) when  = 1 and  = 0, subject to homogeneous boundary condition (3a) or (3b); for the case  = 0, we refer to [10].
Motivated by the works above, we investigate the lower bounds for the blow-up time of the blow-up solutions to the nonlocal problem (1)-( 2) with homogeneous boundary condition (3a) or (3b).Actually, it is well known that if  +  > max{, } and the initial value is large enough, then the solutions of our problem blow up in a finite time; one can see [11].Unfortunately, our results are restricted in R 3 because of the best constant of a Sobolev type inequality (see [12]).

Blow-Up Time for Dirichlet Boundary Condition
In this section, we derive a lower bound for  * if the solution (, ) ≥ 0 of ( 1)-(3a) blows up in finite time  * .
Theorem 1.Let (, ) be a classical solution of (1)-(3a) with  +  > max{, }; then a lower bound of the blow-up time for any solution which blows up in , where  is a suitable positive constant given later and (0 Proof.Define an auxiliary function of the form with Taking the derivative of () with respect to  gives where ∇ is the gradient operator.
The application of Hölder inequality to the second term on the right hand side of (6) yields where |Ω| denotes the volume of Ω.
By (7), it follows from (6) that Let then and ( 8) can be written in the from Now we seek a bound for ∫ Ω V +1  in terms of  and the first and third terms on the right in (11).First, the application of Hölder inequality yields Using the following Sobolev type inequality (see [12]): with  = 6,  = 2, and  = 4 1/3 3 −1/2  −2/3 , we obtain Then for some positive constant  1 to be determined it follows that Next, we use the fundamental inequality to obtain ] . ( Note the fact that, for some positive constant  2 , Substituting inequality (18) into (17) gives Then, by applying inequality (19), it follows from (11) that Mathematical Problems in Engineering We next choose  1 to make the coefficient of ∫ Ω V +  vanish and then choose  2 to make the coefficient of with Integrating inequality (21) from 0 to  gives from which we derive a lower bound for  * : This completes the proof of Theorem 1.

Blow-Up Time for Neumann Boundary Condition
In this final section, we discuss a lower bound for  * if the solution (, ) of ( 1), (2), and (3b) is blow-up in finite time  * .
Theorem 2. Let (, ) be a classical solution of (1), (2), and (3b) with + > max{, }; then a lower bound of the blow-up time for any solution which blows up in , where  2 and  3 are suitable positive constants given later, respectively, and Proof.We estimate ∫ Ω V (6+3 1 )/4  in inequality (14).In a similar way to the process of the derivation of (3.3) in [10], we have where  0 = min Ω   V  ,  2 = max Ω     ,  = 1, 2, 3, and V  is the th component of the unit outer normal vector V on Ω.By virtue of Hölder inequality, we get Substituting inequality (26) into (25) yields Applying the following inequality: we conclude that Mathematical Problems in Engineering 5 Applying inequality (16), we obtain where  1 and  2 are arbitrary positive constants.Recalling (12) and applying inequality (16) again, for a suitable constant  3 , we obtain By applying (30), it follows from (31) that ×  3 (∫ Ω V  ) where