Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

Abstract

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee that exists globally or blows up at some finite time . Moreover, an upper bound for is derived. Under somewhat more restrictive conditions, a lower bound for is also obtained.

1. Introduction

We are concerned with the global existence and blow-up phenomenon for a quasilinear parabolic equation with nonlinear inner absorption term subjected to the nonlinear Neumann boundary and initial conditions where is a bounded star-shaped region of with smooth boundary , is the unit outward normal vector on , , is the blow-up time if blow-up occurs, or else , the symbol, denotes partial differentiation with respect to , , the repeated index indicates summation over the index, and is gradient operator.

Many physical phenomena and biological species theories, such as the concentration of diffusion of some non-Newton fluid through porous medium, the density of some biological species, and heat conduction phenomena, have been formulated as parabolic equation (1) (see [1–3]). The nonlinear Neumann boundary condition (2) can be physically interpreted as the nonlinear radial law (see [4, 5]).

In the past decades, there have been many works dealing with existence and nonexistence of global solutions, blow-up of solutions, bounds of blow-up time, blow-up rates, blow-up sets, and asymptotic behavior of solutions to nonlinear parabolic equations; see the books [6–8] and the survey papers [9–11]. Specially, we would like to know whether the solution blows up and at which time when blow-up occurs. A variety of methods have been used to study the problem above (see [12]), and in many cases, these methods, used to show that solutions blow up, often provide an upper bound for the blow-up time. However, lower bounds for blow-up time may be harder to be determined. For the study of the initial boundary value problem of a parabolic equation with homogeneous Dirichlet boundary condition, see [13, 14]. Payne et al. [13] considered the following quasilinear parabolic equation: where is a bounded domain in with smooth boundary . To get the lower bound for the blow-up time, the authors assumed that is a positive function which satisfies The lower bound for the blow-up time of solution to (4) with Robin boundary condition was obtained in [15], where is also assumed to satisfy the condition (5). However, under this boundary condition, the best constant of Sobolev inequality used in [13] is no longer applicable. They imposed suitable conditions on and and determined a lower bound for the blow-up time if blow-up occurs and determined when blow-up cannot occur. Marras and Vernier Piro [14] studied the nonlinear parabolic problem with time dependent coefficients where is a bounded domain in with smooth boundary . Under some conditions on the data and geometry of the spatial domain, they obtained upper and lower bounds of the blow-up time. Moreover, the sufficient conditions for global existence of the solution were derived.

For the study of the initial boundary value problem of a parabolic equation with Robin boundary condition, we refer to [15–19]. Li et al. [16] investigated the problem of the nonlinear parabolic equation where is a bounded domain in with smooth boundary . They derived the lower bound for the blow-up time when the blow-up occurs. Clearly, does not satisfy the condition (5). Enache [17] discussed the quasilinear parabolic problem where is a bounded domain in with smooth boundary . By virtue of a first-order differential inequality technique, they showed the sufficient conditions to guarantee that the solution exists globally or blows up. In addition, a lower bound for the blow-up time when blow-up occurs was also obtained. Ding [18] studied the nonlinear parabolic problem where is a bounded domain in with smooth boundary . They derived conditions on the data which guarantee the blow-up or the global existence of the solution. A lower bound on blow-up time when blow-up occurs was also obtained. For the problem of the nonlinear nonlocal porous medium equation, we read the paper of Liu [19].

Recently, for the problems with nonlinear Neumann boundary conditions, Payne et al. [20] studied the semilinear heat equation with inner absorption term They established conditions on nonlinearity to guarantee that the solution exists for all time or blows up at some finite time . Moreover, an upper bound for was derived. Under somewhat more restrictive conditions, a lower bound for was derived. Thereafter, they considered the quasilinear parabolic equation and they showed that blow-up occurs at some finite time under certain conditions on the nonlinearities and the data; upper and lower bounds for the blow-up time were derived when blow-up occurs; see [21]. Liu et al. The authors [22, 23] studied the reaction diffusion problem with nonlocal source and inner absorption terms or with local source and gradient absorption terms. Very recently, Fang et al. [24] considered lower bounds estimate for the blow-up time to nonlocal problem with homogeneous Dirichlet or Neumann boundary condition.

Motivated by the above work, we intend to study the global existence and the blow-up phenomena of problem (1)–(3), and the results of the semilinear equations are extended to the quasilinear equations. Unfortunately, the techniques used for semilinear equation to analysis of blow-up phenomena are no longer applicable to our problem. As a consequence, by using the suitable techniques of differential inequalities, we establish, respectively, the conditions on the nonlinearities and to guarantee that exists globally or blows up at some finite time. If blow-up occurs, we derive upper and lower bounds of the blow-up time.

The rest of our paper is organized as follows. In Section 2, we establish conditions on the nonlinearities to guarantee that exists globally. In Section 3, we show the conditions on data forcing the solution to blow up at some finite time and obtain an upper bound for . A lower bound of blow-up time under some assumptions is derived in Section 4.

2. The Global Existence

In this section, we establish the conditions on the nonlinearity and nonlinearity to guarantee that exists globally. We state our result as follows.

Theorem 1. Assume that the nonnegative functions and satisfy where , , , , and . Then the (nonnegative) solution of problem (1)-(3) does not blow up; that is, exists for all time .

Proof. Set Similar to Theorem 2.1 in [20], we get where , , , and where , , , .
Next, we estimate . Since it follows from Hölder inequality that Furthermore, we have which follows from (17) and membrane inequality where is the first eigenvalue in the fixed membrane problem Combining and (18), we have Making use of Hölder inequality, we obtain Combining (21), (22) with (23), we get with Applying Hölder inequality, we obtain It follows from (26) that where Combining (14), (15), (21), and (24) with (27), we obtain with We conclude from (29) that is decreasing in each time interval on which we obtain so that remains bounded for all time under the conditions in Theorem 1. This completes the proof of Theorem 1.

3. Blow-Up and Upper Bound of

In this section, needs not to be star-shaped. We establish the conditions to assure that the solution of (1)–(3) blows up at finite time and derive an upper bound for . More precisely we establish the following result.

Theorem 2. Let be the classical solution of problem (1)-(3). Assume that the nonnegative and integrable functions and satisfy with where , Moreover assume that with Then the solution of problem (1)-(3) blows up at some finite time with where is defined in (13). If , we have .

Proof. We compute Making use of the hypotheses stated in Theorem 2, we have Differentiating (35), we derive Integrating the identity over , we get Substituting (40) into (39), we have which with imply for all . Making use of the Schwarz inequality, we obtain Multiplying the above inequality by , we deduce Arguing as in Theorem 3.1 in [20], we find valid for . If , we have valid for , implying that . This completes the proof of Theorem 2.

4. Lower Bounds for

In this section, under the assumption that is a star shaped domain in , convex in two orthogonal directions, we seek a lower bound for the blow-up time . Now we state the result as follows.

Theorem 3. Let be the nonnegative solution of problem (1)-(3) and blows up at ; moreover, the nonnegative functions and satisfy with , , , , . Define where is a parameter restricted by the condition Then satisfies inequality for some computable function . It follows that is bounded from below. We have

Proof. Differentiating (47) and making use of the boundary condition (2) together with the conditions (46), we have Applying inequality in [20] to the first term on the right hand side of (51), we have Substituting (52) into (51), we obtain Making use of arithmetic-geometric mean inequality, we derive for all . Choose such that We rewrite (53) as Using Hölder inequality, we get Combining (56) with (57), we obtain where Using Sobolev type inequality derived by Payne et al. [21], we obtain We now make use of Hölder inequality to bound the second integral on the right hand side of (60) as follows: with We note that for , an inequality satisfied in view of (48). Using again Hölder's inequality, we obtain where is the volume of . Substituting (61) and (63) in (60), we obtain the following inequality: where are computable positive constants. Note that the last inequality in (64) follows from Hölder inequality under the particular form . Similarly, we can bound and get where are computable positive constants, We note that for , an inequality satisfied in view of (48). Inserting (64) and (65) in (58), we arrive at where , and are computable positive constants. Next, we want to eliminate the quantity in inequality (67). By using the following inequality: valid for , where is an arbitrary positive constant, then we have with arbitrary positive constants , and computable positive constants , . Substitute (69) in (67) and choose the arbitrary (positive) constants , such that . We obtain We eliminate the last term in (70), by using the following inequality: valid for and arbitrary , and choose such that Then (70) can be rewritten as Integrating (73) over , we conclude This completes the proof of Theorem 3.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally to the paper and read and approved the final paper.

Acknowledgments

This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018) and the Fundamental Research Funds for the Central Universities (no. 201362032). The authors would like to deeply thank all the reviewers for their insightful and constructive comments.