Singularity Analysis for a Class of Porous Medium Equation with Time-Dependent Coefficients

Abstract

This paper concerns the singularity and global regularity for the porous medium equation with time-dependent coefficients under homogeneous Dirichlet boundary conditions. Firstly, some global regularity results are established. Furthermore, we investigate the blow-up solution to the boundary value problem. The upper and lower estimates to the lifespan of the singular solution are also obtained here.

1. Introduction and Main Results

In this work, we consider the following porous medium equation with time-dependent coefficients under homogeneous Dirichlet boundary condition:where is a smooth bounded domain, . The coefficient is a positive bounded continuous function with for any . The nonlinearity is assumed to satisfy and . The initial value is nontrivial nonnegative continuous function and vanishes on .

Global existence and nonexistence to the nonlinear parabolic equation are important topic and have been investigated extensively; please see the surveys [1–4]. The first purpose in this paper is to investigate the sufficient conditions to the global existence of the solution to the boundary value problem (1). The second aim of this paper is to investigate the solution which blows up in finite time and estimate the lifespan of the singular solution. Singularity analysis, especially, to evaluate the lifespan of the singular solution is also an interesting research topic in this field.

In [5–7], Payne and others have considered the linear diffusion case. However, the degenerate diffusion makes the present problem more complicated and takes more essential difficulties here. We would like to refer some results on blow-up solutions to the degenerate parabolic equations and system as follows. Some global existence and nonexistence of the classical solution to degenerate parabolic equations were established in [8], and then Du and his colleagues gave the sufficient conditions to the degenerate parabolic system with nonlinear nonlocal sources in [9–11], with nonlinear localized sources in [10], and with nonlinear memory terms in [12]. Furthermore, some properties to the singular solutions, such as blow-up set, uniform blow-up profile were obtained in [13–15].

The local existence of classical solution to system (1) can be obtained by the standard method in [16]. Firstly, we give some results on the global existence of the classical solution to the boundary value problem (1) as follows.

Theorem 1. Suppose that there exists a positive constant , such that for ; then every classical solution to problem (1) is global.

Secondly, we give the blow-up results in the next theorem.

Theorem 2. Suppose that there exists a positive constant with , such that for and ; then the classical solution to problem (1) blows up in finite time, provided that the initiate data are sufficiently large.

Theorem 3. Suppose that there exists a positive constant with , such that for and ; then the classical solution to problem (1) blows up in finite time for large data , where is the first eigenvalue to the following problem:with and . Moreover, there exists a constant , which depends on , such that .

Furthermore, we give the following estimates to the maximal blow-up time .

Theorem 4. Suppose that be a convex domain in , and there exists a constant , such that , . If the solution to problem (1) blows up in finite time , then there exists a positive constant , which depends on , such that

Remark 5. We would like to mention that the results in Theorem 4 are still valid for two-dimensional case.

The remainder of this paper is organized as follows. Global existence of the solution to problem (1) is established in Section 2, by constructing some global upper solution. In Sections 3 and 4, we show that the classical solution to problem (1) will blow up in finite time and obtain upper bound of the blow-up time. In the last section, with aid of a differential inequality, we will establish lower estimate to the maximal blow-up time.

2. Global Solution for Problem (1)

In this section, we focus on the global solution of (1) and show Theorem 1.

Proof. Obviously, if , then there exists a positive constant such that ,  .
Next, we construct a supersolution which is bounded for any . Let be the solution of the following elliptic problem:Denote . Namely, .
We define the function aswhere satisfy and will be fixed later. Clearly, is bounded for any . Thus, we haveDenote As , we can choose sufficiently large that andNow, it follows from (6)–(8) that defined by (5) is a positive supersolution of (1). Hence by comparison principle, which implies exists globally.

3. Blow-Up Solution to Problem (1)

In this section, we will discuss the blow-up solution of (1) under some appropriate hypotheses and show Theorem 2.

Proof. Our strategy here is to construct blow-up subsolutions in some subdomain of in which . Some ideas are borrowed from the work [11] by Du.
Let be a nontrivial nonnegative continuous function and vanish on . Without loss of generality, we assume that and . We will construct a blow-up subsolution to complete the proof.
Setwith where , and are determined later. Clearly, and is nonincreasing since . Note thatfor sufficiently small . Obviously,   becomes unbounded as at the point .
Calculating directly, we obtainnoticing that is sufficiently small.
Case 1. If , we have ; thenHenceCase 2. If , thenIf , then there exists positive constant , such that Thus, we get Hence, for sufficiently small and , (11) holds. And (14)-(15) imply thatwhere .
Since is a nontrivial nonnegative continuous function and , there exist two positive constants and such that for all . Choose small enough to insure ; hence on . From (11), it follows that for sufficient large . By comparison principle, we have provided that , which implies that the solution of (1) blows up in finite time.

4. Upper Bound for Blow-Up Time

In this section, we will discuss upper bound for the blow-up time under some appropriate hypotheses and show Theorem 3.

Proof. Denotewhere is the solution to problem (2).
Case 1 (). Combining (1) and (19), we haveAs , making use of Hölder’s and Young’s inequality, we haveApplying (19)–(21), we obtain the inequalitywhere ,  .
Choosing sufficiently large such thatwe conclude that is increasing monotonously for any .
Moreover, according to (22) and , we can obtain that there exists , such thatwhere .
Case 2 (). According to (1) and (19), we can obtainMoreover, applying , we haveCombining (25) and (26), we obtain the inequalitywhere .
Choosing sufficiently large yieldsThus we conclude that is increasing monotonously for any .
Furthermore, according to and (27), we can obtain that there exists , such thatHence, denote , which depends on , and , and Theorem 3 is completed.

5. Lower Bound for Blow-Up Time

In this section, we will give the lower bound to the blow-up time as long as blow-up occurs and show Theorem 4.

Proof. Firstly, according to Theorem 1, we have under the conditions of Theorem 4.
Secondly, setWe computeClearly, making use of Hölder’s inequality, we getApplying Sobolev’s inequality (see [1]) in we can obtainSubstituting (23) into (22) yieldswhere .
Substituting (35) into (31), we obtainwhere We suppose thatand then there exists such that   
Integrating (36) from to , we obtainLet . Obviously, is increasing monotonously. So we can obtainwhere , is the inverse function of , and depends on , and .
Thus, we complete the proof of Theorem 4.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the reviewers for careful reading and giving many helpful comments to improve the paper. This work is supported in part by NSFC Grant (no. 61563044), the Science and Technology Major Project of Qinghai Province Natural Science Foundation (no. 2015-SF-A4-3), and SRFDP (no. 20100181110036).