Singularity and Decay Estimates for a Degenerate Parabolic Equation

Abstract

In this paper, a degenerate parabolic equation with and , is considered. Based on rescaling arguments combined with a doubling property, the space-time singularity and decay estimates are established. Moreover, a universal and a priori bound of global nonnegative solutions for the corresponding initial boundary value problem is derived.

1. Introduction

In this paper, we consider the degenerate parabolic equation of the formwhere , is a domain of with , and is an interval of . Throughout this paper, solutions are considered in the class (unless otherwise specified) We denote

In order to prove the existence and multiplicity of positive solutions, it is important to obtain a priori bounds for their positive solutions. It is well known that Liouville theorem would imply optimal universal estimates for solutions of related initial or initial boundary value problems, including estimates of their singularities and decay via the rescaling method (see [1, 2]). The following Liouville result [3] will play an important role in our proof of singularity estimates.

Theorem 1. Let and . Equation (1) has no nontrivial solution in .

In the special case , equation (1) has been widely studied by many authors. Here, the space-time singularity and decay estimates of positive solutions of equation (1) have been established by Peter et al. in [2] for the case and by Phan [4] for the case . Motivated by the aforementioned work, we establish the space-time singularity and decay estimates of solutions of equation (1).

Theorem 2. (i)Assume that , and . Let be a nonnegative solution of (1) on , where . Then, for all and , there holds(ii)Assume that , and . Let be a nonnegative solution of (1) on , where . Then, for all and , there holds

The proof of Theorem 1 is based on rescaling arguments combined with a key doubling property. The classical rescaling technique argument was first introduced by Gidas and Spruck [5] for the elliptic problem; it was then significantly improved in [2, 6–11] for elliptic and parabolic that enables one to obtain a variety of important results. Moreover, it also needs the nonexistence of nontrivial solutions on the whole space for equation (1) with .

Remark 1.   In Theorem 2, if we replace the interval by , then we have the spatial decay estimateAs the next topic, we consider the associated initial boundary value problem:Here, we assume that is a smoothly bounded domain containing the origin. We have the following result.

Theorem 3. Let , and . Assume that is any global solution of (6) with initial data and satisfies the estimate

Then, there exists a constant independent of , and such that

Remark 2.  Phan [2] proved a priori bound of nonnegative solutions for the parabolic problem under the condition . Theorem 3 says that a priori bound for a degenerate parabolic equation in the form of (6) also holds under condition (7). It should be mentioned that the universal and a priori bound of global solutions was shown by Quittner [12], Xing [13], and Földers [14] for sign changing to (6) in the special case .
Our proof of Theorem 3 appeals to rescaling techniques and argues by contradiction following the ideas introduced by Phan [2] for the parabolic equation. We also use this powerful idea and introduce some rescalings to deal with some new difficulties arising due to the degeneracy and singularity.
The rest of the paper is organized as follows. Section 2 is devoted to the proof of singularity and decay estimates (Theorem 2). Section 3 contains the proof of Theorem 3 which is an a priori bound of global solutions and blow-up estimates.

2. Singularity and Decay Estimates

In this section, we give the proof of Theorem 2.

Proof of Theorem 2. Assume either and or and . Denote and observe that, for all , so that . Let us define Then, is a solution ofNotice that for all . We first claim that, for all , there exists a constant (independent of and ) such thatEstimate (10) can be rewritten in a more concise equivalent form by using the parabolic distance , namely, let , , and . Then, estimate (10) can be restated asAssume that estimate (10) fails. Then, there exist sequences , points , and that solveswhere such that the functions satisfyBy the doubling lemma in [11], it follows that there exists such thatIt follows from (14) that . Therefore, any satisfying the condition is automatically contained in . Since , we haveNow, we define the scaled functionwhere . Clearly,due to (15). We see that satisfiesNow, for each and , by (19), (20), and interior parabolic estimates, the sequence is uniformly bounded in . Note that for all . By the interior parabolic estimates and standard embeddings, we can find subsequences (still denoted by ), and we may assume that and in . Therefore, is a classical solution of and . Since , after a space shift, we know that this contradicts Theorem 1 and concludes our claim (11). This also implies (10) holds. Therefore,that is,which completes the proof of Theorem 2.

3. A Priori Bound

Proof of Theorem 3. Our proof is by contradiction. Suppose that Theorem 3 is false. Then, there exist satisfying (6) and sequences of points such thatSince is compact, we may assume that as . Denote . We divide the situation into three cases: Case 1: Let . Since , evidently, . Define the scaled function where . Clearly, It follows that is a solution to where . Using parabolic estimates and standard embedding theorems, we can find a subsequence (still denoted by ) which converges uniformly to some function in , and Moreover, By our assumption (7), we have that Since , we get . This yields because in . After a space shift, this gives a contradiction with Theorem 1. Hence, . This contradicts the fact . Hence, case 1 cannot occur. Case 2: . Without loss of generality, we may assume that, near , the boundary is contained in the hyperplane . We rescale the solution according to Note that, for large , is well defined in for some and satisfies Clearly, By (30) and the fact on , we have . If , then after a choice of the subsequence, we have the same contradiction as in the first case. If , using parabolic estimates and standard embedding theorems, we can find a subsequence (still denoted by ) which converges uniformly to some function satisfying , and As in case 1, with the necessary changes of the domain of integration, we yield . This contradicts Theorem 2.1 in [2]. Case 3: . We discuss the situation in two possibilities:(a)If , let ; then, we have is bounded. We may assume that as by choosing subsequences. Denote by ; then, is a solution to where . A similar limiting procedure as in case 1 then produces a solution of Moreover, Using (7) and , we have Since , we get . This yields because in . Hence, After a space shift, this gives a contradiction with Theorem 2 in [15]. Hence, .(b)If there exists a subsequence of , still denoted by , such that , then we can choose such thatLetwhere ; then, solves the problemSince and , we may assume that with and . A similar limiting procedure as in case 1 then produces a solution ofWe will show that ; indeed,Sincethus, . This implies that is a solution ofThis contradicts with Theorem 2 in [15].

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11801431).