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.
In this work, we consider the following porous medium equation with time-dependent coefficients under homogeneous Dirichlet boundary condition:
Global existence and nonexistence to the nonlinear parabolic equation are important topic and have been investigated extensively; please see the surveys [
In [
The local existence of classical solution to system (
Suppose that there exists a positive constant
Secondly, we give the blow-up results in the next theorem.
Suppose that there exists a positive constant
Suppose that there exists a positive constant
Furthermore, we give the following estimates to the maximal blow-up time
Suppose that
We would like to mention that the results in Theorem
The remainder of this paper is organized as follows. Global existence of the solution to problem (
In this section, we focus on the global solution of (
Obviously, if
Next, we construct a supersolution which is bounded for any
We define the function
In this section, we will discuss the blow-up solution of (
Our strategy here is to construct blow-up subsolutions in some subdomain of
Let
Set
Calculating directly, we obtain
Hence
If
Since
In this section, we will discuss upper bound for the blow-up time under some appropriate hypotheses and show Theorem
Denote
Choosing
Moreover, according to (
Choosing
Furthermore, according to
Hence, denote
In this section, we will give the lower bound to the blow-up time as long as blow-up occurs and show Theorem
Firstly, according to Theorem
Secondly, set
Clearly, making use of Hölder’s inequality, we get
Applying Sobolev’s inequality (see [
Substituting (
We suppose that
Integrating (
Thus, we complete the proof of Theorem
The authors declare that there is no conflict of interests regarding the publication of this paper.
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).