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

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.

Global existence and nonexistence to the nonlinear parabolic equation are important topic and have been investigated extensively; please see the surveys [1][2][3][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][6][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][10][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][14][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  > 0; then every classical solution to problem (1) is global.Theorem 2. Suppose that there exists a positive constant  with  ≥ , such that () ≥   for  > 0 and flinf() > 0; 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  > 0 and flinf() >  1 ; then the classical solution to problem (1) blows up in finite time  for large data  0 (), where  1 is the first eigenvalue to the following problem: with  > 0 ( ∈ Ω) and ∫ Ω   = 1.Moreover, there exists a constant  0 , which depends on , , ,  0 (), such that  ≤  0 .
Furthermore, we give the following estimates to the maximal blow-up time .Theorem 4. Suppose that Ω be a convex domain in R 3 , and there exists a constant  > 0, such that () ≤   ,  > 0. If the solution to problem (1) blows up in finite time , then there exists a positive constant  0 , which depends on , , ,  0 (), such that  ≥  0 . ( 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.

Global Solution for Problem (1)
In this section, we focus on the global solution of (1) and show Theorem 1.
We define the function (, ) as where  > 0 satisfy  < 1 and  > 0 will be fixed later.Clearly,  is bounded for any  > 0. Thus, we have Denote As  −  > 0, we can choose  sufficiently large that  >  1 and Now, it follows from ( 6)-( 8) that  defined by ( 5) is a positive supersolution of (1).Hence  ≤  by comparison principle, which implies  exists globally.

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 Ω in which  > 0. Some ideas are borrowed from the work [11] by Du.
Calculating directly, we obtain noticing that  > 0 is sufficiently small.

Upper Bound for Blow-Up Time
In this section, we will discuss upper bound for the blowup time under some appropriate hypotheses and show Theorem 3.

Proof. Denote
where  is the solution to problem (2).
Choosing  0 () sufficiently large such that we conclude that () is increasing monotonously for any  > 0.

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.
Thus, we complete the proof of Theorem 4.