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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2011, Article IDΒ 608576, 22 pages
http://dx.doi.org/10.1155/2011/608576
Research Article

Radially Symmetric Solutions of a Nonlinear Elliptic Equation

1Department of Mathematics, University of Pittsburgh at Greensburg, Greensburg, PA 15601, USA
2Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Received 30 December 2010; Accepted 18 April 2011

Academic Editor: FrankΒ Werner

Copyright Β© 2011 Edward P. Krisner and William C. Troy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the existence and asymptotic behavior of positive, radially symmetric singular solutions of π‘€ξ…žξ…ž+((π‘βˆ’1)/π‘Ÿ)π‘€ξ…žβˆ’|𝑀|π‘βˆ’1𝑀=0, π‘Ÿ>0. We focus on the parameter regime 𝑁>2 and 1<𝑝<𝑁/(π‘βˆ’2) where the equation has the closed form, positive singular solution 𝑀1=(4βˆ’2(π‘βˆ’2)(π‘βˆ’1)/(π‘βˆ’1)2)1/(π‘βˆ’1)π‘Ÿβˆ’2/(π‘βˆ’1), π‘Ÿ>0. Our advance is to develop a technique to efficiently classify the behavior of solutions which are positive on a maximal positive interval (π‘Ÿmin,π‘Ÿmax). Our approach is to transform the nonautonomous 𝑀 equation into an autonomous ODE. This reduces the problem to analyzing the behavior of solutions in the phase plane of the autonomous equation. We then show how specific solutions of the autonomous equation give rise to the existence of several new families of singular solutions of the 𝑀 equation. Specifically, we prove the existence of a family of singular solutions which exist on the entire interval (0,∞), and which satisfy 0<𝑀(π‘Ÿ)<𝑀1(π‘Ÿ) for all π‘Ÿ>0. An important open problem for the nonautonomous equation is presented. Its solution would lead to the existence of a new family of β€œsuper singular” solutions which lie entirely above 𝑀1(π‘Ÿ).

1. Introduction

We investigate the behavior of solutions ofΞ”π‘€βˆ’|𝑀|π‘βˆ’1𝑀=0,(1.1) where 𝑀=𝑀(π‘₯1,…,π‘₯𝑁), 𝑁>1 and 𝑝>1. Solutions of (1.1) are time-independent solutions of the nonlinear heat equationπœ•π‘€πœ•π‘‘=Ξ”π‘€βˆ’|𝑀|π‘βˆ’1𝑀.(1.2) In the mid 1980’s, Brezis et al. [1], and Kamin and Peletier [2], investigated the existence and asymptotic behavior of positive, time-dependent singular solutions of (1.2). This led to the classical 1989 study by Kamin et al. [3], whose goal was to completely classify all positive, time-dependent solutions of (1.2). A natural extension of their study is to classify positive, time-independent solutions. Such solutions play an important role in analyzing the large time behavior of solutions of the time-dependent equation (1.2) (e.g., see the discussion following (1.5) below). Thus, in this paper, our goal is to extend the results in [1–3], and develop a method to efficiently classify the behavior of positive, time-independent solutions of (1.1). Our focus is on radially symmetric solutions, which have the form 𝑀=𝑀(π‘Ÿ), where π‘Ÿ=(π‘₯21+β‹―+π‘₯2𝑁)1/2, and satisfyπ‘€ξ…žξ…ž+π‘βˆ’1π‘Ÿπ‘€ξ…žβˆ’|𝑀|π‘βˆ’1𝑀=0,π‘Ÿ>0.(1.3) Equation (1.3) has the closed form, positive singular solution (see Figure 2)𝑀1ξ‚΅(π‘Ÿ)=4βˆ’2(π‘βˆ’2)(π‘βˆ’1)(π‘βˆ’1)2ξ‚Ά1/(π‘βˆ’1)π‘Ÿ2/(1βˆ’π‘)𝑁,𝑁>2,1<𝑝<π‘βˆ’2.(1.4)

A Related Equation
A second, widely studied nonlinear heat equation is πœ•π‘£πœ•π‘‘=Δ𝑣+|𝑣|π‘βˆ’1𝑣.(1.5) Equation (1.5) has the closed form, stationary, positive singular solution 𝑣1ξ‚΅(π‘Ÿ)=2(π‘βˆ’2)(π‘βˆ’1)βˆ’4(π‘βˆ’1)2ξ‚Ά1/(π‘βˆ’1)π‘Ÿ2/(1βˆ’π‘)𝑁,𝑁>2,π‘βˆ’2<𝑝<𝑁+2π‘βˆ’2.(1.6) This well-known singular solution plays an important role in the analysis of blowup of solutions of (1.5). For example, when 𝑣(π‘₯1,…,π‘₯𝑁,0) is appropriately chosen, similarity solution methods developed by Haraux and Weissler [4], and Souplet and Weissler [5], show how 𝑣(π‘₯1,…,π‘₯𝑁,𝑑)→𝑐𝑣1(π‘Ÿ) as π‘‘β†’βˆž, where 𝑐>0 is a constant [4, 5]. In 1999, Chen and Derrick [6] developed comparison methods to determine the large time behavior of solutions of the general equation πœ•π‘€πœ•π‘‘=Δ𝑀+𝑓(𝑀),(1.7) where 𝑓(𝑀) is super linear, as in (1.7) and (1.5). Their approach is to let positive, time independent solutions act as upper and/or lower bounds for initial values of solutions of (1.7). Their comparison technique allows them to prove either global existence or finite time blowup of solutions. It is hoped that the methods described above, combined with the new singular solutions found in this paper, will lead to future analytical insights into the behavior of solutions of the time-dependent equation (1.2).

Specific Aims
We have three specific aims. The first two are listed below. The third is given later in this section. We assume throughout that 𝑁>2 and 1<𝑝<𝑁/(π‘βˆ’2), the parameter regime where 𝑀1(π‘Ÿ) exists. In order to study properties of positive solutions of (1.3), our approach is to let π‘Ÿ0>0 be arbitrarily chosen and analyze solutions with initial values π‘€ξ€·π‘Ÿ0ξ€Έ=𝛼>0,π‘€ξ…žξ€·π‘Ÿ0ξ€Έ=π›½βˆˆπ‘….(1.8) Let (π‘Ÿmin,π‘Ÿmax) denote the largest interval containing π‘Ÿ0 over which the solution of (1.3)–(1.8) is positive.

Specific Aim 1. For each solution of (1.3)–(1.8), prove whether π‘Ÿmin=0 or π‘Ÿmin>0, and determine limπ‘Ÿβ†’π‘Ÿ+min(𝑀(π‘Ÿ),𝑀′(π‘Ÿ)).

Specific Aim 2. For each solution of (1.3)–(1.8), prove whether π‘Ÿmax<∞ or π‘Ÿmax=∞, and determine limπ‘Ÿβ†’π‘Ÿβˆ’max(𝑀(π‘Ÿ),𝑀′(π‘Ÿ)).

Analytical Methods
To address the issues raised in Specific Aims 1 and 2, we need to determine the behavior of each solution of (1.3)–(1.8) over the entire interval (π‘Ÿmin,π‘Ÿmax), where π‘Ÿminξ€½ξ€·=infΜ‚π‘Ÿβˆˆ0,π‘Ÿ0ξ€Έξ€·βˆ£π‘€(π‘Ÿ)>0βˆ€π‘ŸβˆˆΜ‚π‘Ÿ,π‘Ÿ0,π‘Ÿξ€»ξ€Ύmaxξ€½=supΜ‚π‘Ÿ>π‘Ÿ0ξ€Ίπ‘Ÿβˆ£π‘€(π‘Ÿ)>0βˆ€π‘Ÿβˆˆ0.,Μ‚π‘Ÿξ€Έξ€Ύ(1.9)

Numerical Experiments
In Figure 1, we set (𝑁,𝑝,π‘Ÿ0,𝛼)=(3,2,2,2) and illustrate solutions of (1.3)–(1.8) for various 𝛽 values. For example, when 𝛽≀0, panels (a)–(d) show that both π‘Ÿmin=0 and π‘Ÿmin>0 are possible, and that π‘€ξ…žξ€·π‘Ÿ(π‘Ÿ)<0βˆ€π‘Ÿβˆˆmin,π‘Ÿ0ξ€Έ,limπ‘Ÿβ†’π‘Ÿ+min𝑀(π‘Ÿ),π‘€ξ…žξ€Έ=(π‘Ÿ)(∞,βˆ’βˆž).(1.10) Panels (a)–(f) and also Figure 2 show that solutions can satisfy either π‘Ÿmax<∞ or π‘Ÿmax=∞.

fig1
Figure 1: Solutions of (1.3)–(1.8) for various 𝛽 values when (𝑁,𝑝,π‘Ÿ0,𝛼)=(3,2,2,2).
608576.fig.002
Figure 2: Solutions of (1.3)–(1.8) such that π‘Ÿmin=0 when (𝑁,𝑝)=(3,2); 𝑀1(π‘Ÿ) is defined in (1.4); 𝑀2(π‘Ÿ) and 𝑀3(π‘Ÿ) satisfy parts (ii) and (iii) of Specific Aim 3. Also, 𝑀2(π‘Ÿ) satisfies part (i) of Theorem 2.2, with π‘Ÿmin=0 and π‘Ÿmax=∞. 𝑀3(π‘Ÿ) is a solution satisfying part (iv) of Theorem 2.2, with π‘Ÿmin=0 and π‘Ÿmax=2.2.

Remark 1.1. It must be emphasized that it is illegitimate to claim that numerical results are rigorous proofs. Complete analytical proofs are needed to determine properties such as (1.10).

We now give a brief discussion of (1.10) which demonstrates the difficulties that arise in studying only the 𝑀 equation to resolve Specific Aims 1 and 2. The proof of the first property in (1.10) follows from (1.3), which implies that π‘€ξ…žξ…ž(π‘Ÿ)>0 at any π‘Ÿβˆˆ(π‘Ÿmin,π‘Ÿ0], where 𝑀′(π‘Ÿ)=0 and 𝑀(π‘Ÿ)>0. However, the fact that 𝑀′(π‘Ÿ)<0 for all (π‘Ÿmin,π‘Ÿ0) is not sufficient by itself to prove whether π‘Ÿmin=0 or π‘Ÿmin>0. Nor does it prove the second part of (1.10), that limπ‘Ÿβ†’π‘Ÿ+min(𝑀(π‘Ÿ),𝑀′(π‘Ÿ))=(∞,βˆ’βˆž). In fact, our study suggests that there is a unique 𝛽crit<0 where π‘Ÿmin=0 (see Figure 1(d)), and that π‘Ÿmin>0 at all other negative 𝛽 values. The proof of these claims requires the development of further estimates. Such estimates might be obtained using Pohozaev-type identities [7] or topological shooting techniques [8]. Once the location of π‘Ÿmin and limπ‘Ÿβ†’π‘Ÿ+min(𝑀(π‘Ÿ),𝑀′(π‘Ÿ)) have been determined, we need to turn our attention to the interval π‘Ÿ>π‘Ÿ0. As Figure 1 shows, there are several different types of behavior when π‘Ÿ>π‘Ÿ0. For example, consider the solutions in panels (a), (b), and (c) in Figure 1. In each case,π‘Ÿmin>0,limπ‘Ÿβ†’π‘Ÿ+min𝑀(π‘Ÿ)=∞.(1.11) When π‘Ÿ>π‘Ÿ0, panels (a), (b), and (c) show three different behaviors of solutions, namely,π‘Ÿmax<∞,limπ‘Ÿβ†’π‘Ÿβˆ’max𝑀(π‘Ÿ),π‘€ξ…žξ€Έ=π‘Ÿ(π‘Ÿ)(0,βˆ’.25),max=∞,limπ‘Ÿβ†’π‘Ÿβˆ’max𝑀(π‘Ÿ),π‘€ξ…žξ€Έπ‘Ÿ(π‘Ÿ)=(0,0),max<∞,limπ‘Ÿβ†’π‘Ÿβˆ’max𝑀(π‘Ÿ),π‘€ξ…žξ€Έ(π‘Ÿ)=(∞,∞).(1.12) These results lead to the following analytical challenge: given only the fact that a solution satisfies property (1.11) when π‘Ÿ<π‘Ÿ0, how can we prove which of the possibilities (1.12) occurs when π‘Ÿ>π‘Ÿ0? It is not at all clear how to answer this question using standard methods such as Pohozaev identities or topological shooting.

Solutions with π‘Ÿmin=0
It is particularly important to understand the global behavior of solutions for which π‘Ÿmin=0 since such solutions may play an important role in analyzing the asymptotic behavior of blowup of solutions of the time-dependent equation (1.2). Figure 1(d) shows one such solution for which π‘Ÿmin=0. This solution lies entirely above 𝑀1(π‘Ÿ), that is, 𝑀(π‘Ÿ)>𝑀1(π‘Ÿ) for all π‘Ÿβˆˆ(0,π‘Ÿmax). Figure 2 shows two other solutions, labeled 𝑀2(π‘Ÿ) and 𝑀3(π‘Ÿ), for which π‘Ÿmin=0. These solutions lie entirely below 𝑀1(π‘Ÿ) on (0,π‘Ÿmax). Our computations indicate that 𝑀2(π‘Ÿ) satisfies π‘Ÿmax=∞, and that π‘Ÿmax<∞ for 𝑀3(π‘Ÿ). These numerical experiments lead to

Specific Aim 3. Let 𝑁>2 and 1<𝑝<𝑁/(π‘βˆ’2). Prove that there are at least three families of solutions, other than 𝑀1(π‘Ÿ), with π‘Ÿmin=0. The solutions in these families have the following properties:(i)(see Figure 1(d)). For each 𝛼0>𝑀1(π‘Ÿ0) there exists 𝛽0<0 such that if 𝑀0(π‘Ÿ) is the solution of (1.3) with (𝑀0(π‘Ÿ0),𝑀0β€²(π‘Ÿ0))=(𝛼0,𝛽0), then rmin=0, π‘Ÿmax<∞,limπ‘Ÿβ†’0+𝑀(π‘Ÿ),π‘€ξ…žξ€Έ=(π‘Ÿ)(∞,βˆ’βˆž),limπ‘Ÿβ†’π‘Ÿ+βˆ’max𝑀(π‘Ÿ),π‘€ξ…žξ€Έ=(π‘Ÿ)(∞,∞).(1.13)(ii)(See Figure 2). For each 𝛼2∈(0,𝑀1(π‘Ÿ0)) there exists 𝛽2<0 such that if 𝑀2(π‘Ÿ) is the solution of (1.3) with (𝑀2(π‘Ÿ0),𝑀2β€²(π‘Ÿ0))=(𝛼2,𝛽2), then π‘Ÿmin=0, π‘Ÿmax=∞,0<𝑀2(π‘Ÿ)<𝑀1(π‘Ÿ)βˆ€π‘Ÿ>0,limπ‘Ÿβ†’0+𝑀2(π‘Ÿ),π‘€ξ…ž2𝑀(π‘Ÿ)=(∞,βˆ’βˆž),2(π‘Ÿ),π‘€ξ…ž2ξ€ΈβˆΌξ€·π‘€(π‘Ÿ)1(π‘Ÿ),π‘€ξ…ž1ξ€Έ(π‘Ÿ)asπ‘ŸβŸΆβˆž.(1.14)(iii)(See Figure 2). For each 𝛼3∈(0,𝑀1(π‘Ÿ0)) there exists 𝛽3<0 such that if 𝑀3(π‘Ÿ) is the solution of (1.3) with (𝑀3(π‘Ÿ0),𝑀3β€²(π‘Ÿ0))=(𝛼3,𝛽3), then π‘Ÿmin=0,π‘Ÿmax<∞,limπ‘Ÿβ†’0+𝑀3(π‘Ÿ),π‘€ξ…ž3ξ€ΈβˆΌξ€·π‘€(π‘Ÿ)1(π‘Ÿ),π‘€ξ…ž1ξ€Έ(π‘Ÿ)asπ‘ŸβŸΆ0+,π‘€ξ€·π‘Ÿmaxξ€Έξ€·π‘Ÿ=0,π‘€ξ…žmaxξ€Έ<0.(1.15)

Our Analytical Approach
Our goal is to develop techniques to efficiently prove the existence of solutions of the 𝑀 equation (1.3) satisfying the properties described in Specific Aims 1, 2, and 3. Our experience shows that the analysis of (1.3) is especially complicated since useful estimates must include the independent variable π‘Ÿ. Our advance is to significantly simplify the analysis by transforming (1.3) into an equation which is autonomous, that is, independent or π‘Ÿ. For this, let 𝑀(π‘Ÿ) denote any solution of (1.3), and define β„Ž(𝜏)=𝑀(exp(𝜏))𝑀1(exp(𝜏)),βˆ’βˆž<𝜏<∞.(1.16) Then β„Ž(𝜏) solves β„Žξ…žξ…ž+π‘βˆ’2ξ‚€π‘βˆ’1π‘βˆ’π‘+2ξ‚β„Žπ‘βˆ’2ξ…ž+2(π‘βˆ’2)(π‘βˆ’1)2ξ‚€π‘π‘βˆ’||β„Ž||π‘βˆ’2ξ‚ξ‚€π‘βˆ’1ξ‚βˆ’1β„Ž=0.(1.17)

Remark 1.2. The effect of transformation (1.16) is to change (1.3) into (1.17). Transformation (1.16) is similar to the classical Emden-Fowler transformation 𝑦=𝑀/𝑑, π‘₯=1/𝑑, which changes the Emden-Fowler equation π‘¦ξ…žξ…ž=𝐴π‘₯π‘›π‘¦π‘š(1.18) to the new equation π‘€ξ…žξ…ž=π΄π‘‘βˆ’π‘›βˆ’π‘šβˆ’3π‘€π‘š.(1.19)

Because (1.17) is autonomous, we can apply phase plane techniques to prove the behavior of its solutions. We then use the β€œinverse” formula𝑀(π‘Ÿ)=β„Ž(ln(π‘Ÿ))𝑀1(π‘Ÿ),0<π‘Ÿ<∞(1.20) to determine the global behavior of corresponding solutions of the 𝑀 equation (1.3). In Section 2, we demonstrate the utility of this two step procedure. First, in Theorem 2.1, we analyze the β„Ž equation (1.17), and prove the existence and global behavior of four new classes of solutions. Secondly, in Theorem 2.2, we demonstrate how these families generate four new families of singular solutions of the 𝑀 equation (1.3). In parts (i), (iii), and (iv) of Theorem 2.2 we show how the formula 𝑀(π‘Ÿ)=β„Ž(ln(π‘Ÿ))𝑀1(π‘Ÿ) can be efficiently used to prove the precise asymptotic behavior of each solution as π‘Ÿβ†’π‘Ÿ+min, and as π‘Ÿβ†’π‘Ÿβˆ’max. These three solutions satisfy parts (i), (ii), and (iii) of Specific Aim 3. The final family of solutions in Theorem 2.2 (see part (ii)), is a family of β€œsuper singular solutions,” which satisfy π‘Ÿmax=∞,𝑀(π‘Ÿ)>𝑀1ξ€·π‘Ÿ(π‘Ÿ)βˆ€π‘Ÿβˆˆminξ€Έ,∞,limπ‘Ÿβ†’π‘Ÿ+min𝑀(π‘Ÿ)𝑀1(π‘Ÿ)=∞.(1.21) However, it remains a challenging open problem (see Open Problems 1 and 2 in Section 2) to prove whether π‘Ÿmin=0 or π‘Ÿmin>0. If the first possibility holds, then we have a fourth family of singular solutions, other than 𝑀1(π‘Ÿ), which satisfy π‘Ÿmin=0.

2. The Main Result

In this section, we show how to make use of the autonomous β„Ž equation (1.17) to address the issues raised in Specific Aims 1, 2, and 3 for solutions of the nonautonomous 𝑀 equation (1.3). In particular, our technique shows how the analysis of a solution of (1.17) can be used to completely determine the behavior of the corresponding solution of the 𝑀 equation (1.3) on the maximal interval (π‘Ÿmin,π‘Ÿmax), where 𝑀 is positive. To demonstrate the utility of our method, we restrict our focus to four specific branches of solutions of the β„Ž equation (1.17). Our approach consists of two steps.

First, in Theorem 2.1, we classify the behavior of solutions of (1.17) whose trajectories lie on the stable and unstable manifolds leading to and from the constant solution (β„Ž,β„Žξ…ž)=(1,0) in the (β„Ž,β„Žξ…ž) plane. The stable manifold has two components, 𝐡1 nd 𝐢1, and the unstable manifold has two components, 𝐷1 and 𝐸1. Solutions on 𝐡1, 𝐢1, 𝐷1, and𝐸1 are illustrated in Figure 4(a).

Secondly, in Theorem 2.2, we make use of the link𝑀(π‘Ÿ)=β„Ž(ln(π‘Ÿ))𝑀1(π‘Ÿ),(2.1) to show how solutions with initial values on 𝐡1, 𝐢1, 𝐷1, and 𝐸1 translate into four new continuous families of singular solutions of the 𝑀 equation (1.3). For three of the four cases, we completely prove the behavior of solutions of the 𝑀 equation on the maximal interval (π‘Ÿmin,π‘Ÿmax), where they are positive. For the fourth case, it remains a challenging open problem (see Open Problems 1 and 2 below) to prove the asymptotic behavior of the solution at the left end point π‘Ÿ=π‘Ÿmin. The important consequences of resolving these open problems is described in Section 3.

Theorem 2.1. Let 𝑁>2 and 1<𝑝<𝑁/(π‘βˆ’2). Then (i)There is a one-dimensional stable manifold Ξ“ of solutions of (1.17) leading to (1,0) in the (β„Ž,β„Žξ…ž) phase plane. One component, 𝐡1, of Ξ“ points into the region β„Ž<1, β„Žβ€²>0. If (β„Ž(0),β„Žξ…ž(0))∈𝐡1, then0<β„Ž(𝜏)<1,0<β„Žξ…ž(𝜏)<π‘βˆ’2ξ‚€π‘π‘βˆ’1ξ‚π‘βˆ’2βˆ’π‘β„Ž(𝜏)βˆ€πœβˆˆβ„,(2.2)limπœβ†’βˆ’βˆžξ€·β„Ž(𝜏),β„Žξ…žξ€Έ(𝜏)=(0,0),limπœβ†’βˆ’βˆžβ„Žξ…ž(𝜏)β„Ž=(𝜏)π‘βˆ’2ξ‚€π‘π‘βˆ’1ξ‚π‘βˆ’2βˆ’π‘,(2.3)limπœβ†’βˆžξ€·β„Ž(𝜏),β„Žξ…žξ€Έ=(𝜏)(1,0).(2.4)(ii)The second component, 𝐢1, of Ξ“ points into the region β„Ž>1, β„Žξ…ž<0 of the (β„Ž,β„Žξ…ž) plane. If a solution satisfies (β„Ž(0),β„Žξ…ž(0))∈𝐢1, and (𝜏min,∞) is its interval of existence, thenβ„Ž(𝜏)>1,β„Žξ…ž(𝜏)<0,β„Žξ…žξ…žξ€·πœ(𝜏)>0βˆ€πœβˆˆminξ€Έ,∞,(2.5)limπœβ†’βˆžξ€·β„Ž(𝜏),β„Žξ…ž(ξ€Έπœ)=(1,0)limπœβ†’πœ+minβ„Ž(𝜏)=∞.(2.6)(iii)There is a one-dimensional unstable manifold Ξ© of solutions of (1.17) leading from (1,0) into the (β„Ž,β„Žξ…ž) plane. One component, 𝐷1, of Ξ© points into the region β„Ž>1, β„Žξ…ž>0. If a solution satisfies (β„Ž(0),β„Žξ…ž(0))∈𝐷1, and (βˆ’βˆž,𝜏max) is its interval of existence, then 𝜏max<∞,β„Ž(𝜏)>1,β„Žξ…ž(𝜏)>0,β„Žξ…žξ…žξ€·(𝜏)>0βˆ€πœβˆˆβˆ’βˆž,𝜏maxξ€Έ,(2.7)limπœβ†’βˆ’βˆžξ€·β„Ž(𝜏),β„Žξ…ž(ξ€Έπœ)=(1,0),limπœβ†’πœβˆ’maxξ€·β„Ž(𝜏),β„Žξ…ž(ξ€Έπœ)=(∞,∞).(2.8)(iv)The second component, 𝐸1, of Ξ© points into the region β„Ž<1,β„Žξ…ž<0 of the (β„Ž,β„Žξ…ž) plane. If (β„Ž(0),β„Žξ…ž(0))∈𝐸1, with 0<β„Ž(0)<1 and β„Žξ…ž(0)<0, then there exists a value πœβˆ—>0 such that0<β„Ž(𝜏)<1,β„Žξ…žξ€·(𝜏)<0βˆ€πœβˆˆβˆ’βˆž,πœβˆ—ξ€Έ,limπœβ†’βˆ’βˆžξ€·β„Ž(𝜏),β„Žξ…ž(ξ€Έξ€·πœπœ)=(1,0),β„Žβˆ—ξ€Έ=0,β„Žξ…žξ€·πœβˆ—ξ€Έ<0.(2.9)

Proof of (i). We need to prove properties (2.2)–(2.4). The first step is to linearize (1.17) around the constant solution (β„Ž,β„Žξ…ž)=(0,0). This gives β„Žξ…žξ…ž+π‘βˆ’2ξ‚€π‘βˆ’1π‘βˆ’π‘+2ξ‚β„Žπ‘βˆ’2ξ…žβˆ’2(π‘βˆ’2)(π‘βˆ’1)2ξ‚€π‘π‘βˆ’ξ‚π‘βˆ’2β„Ž=0.(2.10) The eigenvalues associated with (2.10) satisfy πœ‡1=π‘βˆ’2ξ‚€π‘π‘βˆ’1ξ‚π‘βˆ’2βˆ’π‘>0,πœ‡2=2π‘βˆ’1>0.(2.11) We will make use of the observation that (1.17) can be written as β„Žξ…žξ…žβˆ’ξ€·πœ‡1+πœ‡2ξ€Έβ„Žξ…ž+πœ‡1πœ‡2β„Ž=πœ‡1πœ‡2||β„Ž||π‘βˆ’1β„Ž.(2.12) Next, a linearization of (1.17) about the constant solution (β„Ž,β„Žξ…ž)=(1,0) gives β„Žξ…žξ…ž+π‘βˆ’2ξ‚€π‘βˆ’1π‘βˆ’π‘+2ξ‚β„Žπ‘βˆ’2ξ…ž+2(π‘βˆ’2)ξ‚€π‘π‘βˆ’1π‘βˆ’ξ‚π‘βˆ’2(β„Žβˆ’1)=0.(2.13) Define π‘˜=βˆ’2/(π‘βˆ’1). Then (2.13) becomes β„Žξ…žξ…ž+π›Ύβ„Žξ…ž+2(π›Ύβˆ’π‘˜)(β„Žβˆ’1)=0,(2.14) where 𝛾=π‘βˆ’2ξ‚€π‘βˆ’1π‘βˆ’π‘+2ξ‚π‘βˆ’2<0,π›Ύβˆ’π‘˜=π‘βˆ’2ξ‚€π‘π‘βˆ’1π‘βˆ’ξ‚π‘βˆ’2<0.(2.15) Thus, the eigenvalues associated with (2.13) and (2.14) satisfy πœ†1=βˆšβˆ’π›Ύβˆ’π›Ύ2βˆ’8(π›Ύβˆ’π‘˜)2<0,πœ†2=βˆšβˆ’π›Ύ+𝛾2βˆ’8(π›Ύβˆ’π‘˜)2>0.(2.16) It follows from (2.16) and the Stable Manifold Theorem that there is a one-dimensional stable manifold Ξ“ of solutions leading to (1,0) in the (β„Ž,β„Žξ…ž) phase plane. Additionally, limπœβ†’βˆžβ„Žξ…ž(𝜏)β„Ž(𝜏)βˆ’1=πœ†1(2.17) for (β„Ž(𝜏),β„Žξ…ž(𝜏))βˆˆΞ“. Thus, for sufficiently large 𝜏, solutions on Ξ“ satisfy β„Ž(𝜏)>1 if β„Žξ…ž(𝜏)<0 and β„Ž(𝜏)<1 if β„Žξ…ž(𝜏)>0. Let 𝐡1 denote the component of Ξ“ pointing into the region β„Ž<1, β„Žξ…ž>0 of the (β„Ž,β„Žξ…ž) plane. Assume that (β„Ž(0),β„Žξ…ž(0))∈𝐡1. Then (2.4) holds. It remains to prove (2.2)-(2.3). Because of (2.11) and (2.17), and the translation invariance of (2.12), we can choose 1βˆ’β„Ž(0)>0 and β„Žξ…ž(0)>0 small enough so that 0<β„Ž(𝜏)<1,0<β„Žξ…ž(𝜏)<πœ‡1[β„Ž(𝜏)βˆ€πœβˆˆ0,∞).(2.18) The definition of 𝐡1, together with (2.18), imply that the maximal interval of existence is of the form (𝜏min,∞), where 𝜏min<0.
Next, we show that 𝐡1βŠ‚π‘ˆπ‘œ, where π‘ˆπ‘œ is the bounded open triangular region π‘ˆπ‘œ=β„Žξ€½ξ€·1,β„Ž2ξ€Έβˆ£0<β„Ž1<1,0<β„Ž2<πœ‡1β„Ž1ξ€Ύ.(2.19) Figure 4(b) shows π‘ˆπ‘œ when (𝑁,𝑝)=(3,2). Because of (2.18), it suffices to show that (β„Ž(𝜏),β„Žξ…ž(𝜏))βˆˆπ‘ˆπ‘œ for all 𝜏∈(𝜏min,0]. For contradiction, assume that (β„Ž(𝜏),β„Žξ…ž(𝜏)) leaves π‘ˆπ‘œ at some point in (𝜏min,0). Define 𝐻=π‘‘β„Žπ‘‘πœβˆ’πœ‡1β„Ž.(2.20) It follows from (2.12) that 𝐻 satisfies π»ξ…žβˆ’πœ‡2𝐻=πœ‡1πœ‡2||β„Ž||π‘βˆ’1β„Ž.(2.21) Suppose that (β„Ž(𝜏),β„Žξ…ž(𝜏)) leaves π‘ˆπ‘œ across the line 𝐻=0. That is, (see Figure 3(a)) suppose that there exists 𝜏0∈(𝜏min,0) such that π»ξ€·πœ(𝜏)<0,0<β„Ž(𝜏)<1on0ξ€Έξ€·πœ,0,𝐻0ξ€Έ=0.(2.22) If β„Ž(𝜏0)=0, then (2.20) implies that β„Žξ…ž(𝜏0)=0, contradicting uniqueness of the constant solution (β„Ž,β„Žξ…ž)=(0,0). Thus, β„Ž(𝜏0)>0. Also, (2.22) implies that π»ξ…žξ€·πœ0≀0.(2.23) The fact that β„Ž(𝜏0)>0, combined with (2.21), results in π»ξ…žξ€·πœ0ξ€Έ=πœ‡1πœ‡2ξ€·β„Žξ€·πœ0𝑝>0,(2.24) contradicting (2.23). Thus, (β„Ž(𝜏),β„Žξ…ž(𝜏)) can only leave π‘ˆπ‘œ across the line segment 0<β„Ž<1,β„Žξ…ž=0. If so, there is a 𝜏1∈(𝜏min,0) such that 𝐻(𝜏)<0,0<β„Ž(𝜏)<1,β„Žξ…žξ€·πœ(𝜏)>0βˆ€πœβˆˆ1ξ€Έξ€·πœ,0,(2.25)0<β„Ž1ξ€Έ<1,β„Žξ…žξ€·πœ1ξ€Έ=0,(2.26) as depicted in the right panel of Figure 3. Hence, β„Žξ…žξ…žξ€·πœ1ξ€Έβ‰₯0.(2.27) It follows from (2.12) and (2.26) that β„Žξ…žξ…žξ€·πœ1ξ€Έ=πœ‡1πœ‡2ξ‚€ξ€·β„Žξ€·πœ1ξ€Έξ€Έπ‘βˆ’1ξ‚β„Žξ€·πœβˆ’11ξ€Έ<0,(2.28) contradicting (2.27). We conclude that (β„Ž(𝜏),β„Žξ…ž(𝜏)) cannot leave π‘ˆπ‘œ on (𝜏min,∞), hence 𝐡1βŠ‚π‘ˆπ‘œ as claimed. Moreover, since (β„Ž(𝜏),β„Žξ…ž(𝜏)) is bounded, then 𝜏min=βˆ’βˆž follows from standard ODE theory. Thus, (β„Ž(𝜏),β„Žξ…ž(𝜏))βˆˆπ‘ˆπ‘œ for all πœβˆˆβ„, and, therefore, β„Žξ…ž(𝜏)>0 for all πœβˆˆβ„.Proof of the first part (2.3). First, we prove that β„Žβ†’0+ as πœβ†’βˆ’βˆž. Since β„Žξ…ž(𝜏)>0 and 0<β„Ž(𝜏)<1 on ℝ, then 0β‰€β„Ž<1 where, β„Ž=limπœβ†’βˆ’βˆžβ„Ž. To obtain a contradiction suppose that β„Ž>0. Then 0<β„Ž<1 and (2.12) yield 𝑑2β„Žπ‘‘πœ2βˆ’ξ€·πœ‡1+πœ‡2ξ€Έπ‘‘β„Žπ‘‘πœβŸΆπœ‡1πœ‡2ξ‚€β„Žπ‘βˆ’1ξ‚βˆ’1β„Ž<0asπœβŸΆβˆ’βˆž.(2.29) It follows from (2.29) that β„Žξ…ž(𝜏)βˆ’(πœ‡1+πœ‡2)β„Ž(𝜏)β†’βˆž as πœβ†’βˆ’βˆž which contradicts the fact that π‘ˆπ‘œ is bounded and (β„Ž(𝜏),β„Žξ…ž(𝜏))βˆˆπ‘ˆπ‘œ for all πœβˆˆβ„. Thus, β„Ž(𝜏)β†’0+ as πœβ†’βˆ’βˆž. Next, we show that β„Žξ…ž(𝜏)β†’0+ as πœβ†’βˆ’βˆž. Note that 0<β„Žξ…ž(𝜏)<πœ‡1β„Ž(𝜏) on (βˆ’βˆž,0] is an immediate consequence of 𝐻(𝜏)<0 and β„Žξ…ž(𝜏)>0 on (βˆ’βˆž,0]. Therefore, β„Žξ…ž(𝜏)β†’0+ as πœβ†’βˆ’βˆž follows from the fact that β„Ž(𝜏)β†’0+ as πœβ†’βˆ’βˆž.Proof of second part of (2.3). Finally, we need to prove that 𝜌=(β„Žξ…ž/β„Ž)β†’πœ‡1 as πœβ†’βˆ’βˆž. The definition of 𝜌 together with (2.12) gives πœŒξ…ž+𝜌2βˆ’ξ€·πœ‡1+πœ‡2ξ€ΈπœŒ=πœ‡1πœ‡2ξ€·β„Žπ‘βˆ’1ξ€Έβˆ’1.(2.30) We now show that πœŒβ†’πœ‡1 monotonically as πœβ†’βˆ’βˆž. Differentiating (2.30) yields πœŒξ…žξ…ž+ξ€·2πœŒβˆ’πœ‡1βˆ’πœ‡2ξ€ΈπœŒξ…ž=πœ‡1πœ‡2(π‘βˆ’1)β„Žπ‘βˆ’2β„Žξ…ž.(2.31) Hence, if πœŒξ…ž(πœβˆ—)=0 for some πœβˆ—βˆˆβ„, then πœŒξ…žξ…žξ€·πœβˆ—ξ€Έ=πœ‡1πœ‡2(π‘βˆ’1)β„Žπ‘βˆ’2ξ€·πœβˆ—ξ€Έβ„Žξ…žξ€·πœβˆ—ξ€Έ>0.(2.32) This implies that πœŒξ…ž has at most one zero on ℝ. Furthermore, β„Ž0<𝜌(𝜏)=ξ…ž(𝜏)β„Ž(𝜏)<πœ‡1βˆ€πœβˆˆβ„(2.33) since ξ€·β„Ž(𝜏),β„Žξ…žξ€Έ(𝜏)βˆˆπ‘ˆπ‘œβˆ€πœβˆˆβ„.(2.34) Thus, 𝜌=limπœβ†’βˆ’βˆžπœŒ exists and 0β‰€πœŒβ‰€πœ‡1. Moreover, the fact that 𝜌 is finite ensures the existence of an unbounded decreasing sequence {πœπ‘›} such that limπœπ‘›β†’βˆ’βˆžπœŒξ…ž(πœπ‘›)=0. Substituting limπœπ‘›β†’βˆ’βˆžπœŒξ…žξ€·πœπ‘›ξ€Έ=0=limπœπ‘›β†’βˆ’βˆžβ„Žξ€·πœπ‘›ξ€Έ,𝜌=limπœπ‘›β†’βˆ’βˆžπœŒ(2.35) into (2.30) results in 𝜌2βˆ’ξ€·πœ‡1+πœ‡2ξ€ΈπœŒ+πœ‡1πœ‡2=0.(2.36) The bound 0β‰€πœŒβ‰€πœ‡1 and (2.36) imply that 𝜌=πœ‡1. Thus, πœŒβ†’πœ‡1 as πœβ†’βˆ’βˆž as claimed.

fig3
Figure 3: (a) solid curve illustrates properties (2.22). (b) solid curve illustrates properties (2.25) and (2.26).
fig4
Figure 4: 𝑁=3, 𝑝=2. Row 1: Solutions on the stable and unstable manifolds associated with (β„Ž,β„Žξ…ž)≑(Β±1,0) and (β„Ž,β„Žξ…ž)≑(0,0). Rows 2 and 3: β„Ž components on 𝐴1, 𝐴2, 𝐡1, 𝐡2 and 𝑀 components along 𝐴1 and 𝐡1; 𝑀0(π‘Ÿ) is bounded at π‘Ÿ=0, 𝑀1(π‘Ÿ)=2π‘Ÿβˆ’2 is the known singular solution; 𝑀2(π‘Ÿ) is the new, positive singular solution corresponding to heteroclinic orbit 𝐡1.

Proof of (ii). It follows from the Stable Manifold Theorem and (2.17) that there is a second component, 𝐢1, of Ξ“ which points into the region β„Ž>1,β„Žξ…ž<0 of the (β„Ž,β„Žξ…ž) plane (Figure 4(a)). Thus, if (β„Ž(0),β„Žξ…ž(0))∈𝐢1, and β„Ž(0)βˆ’1>0 is sufficiently small, then β„Ž(𝜏)>1β„Žξ…ž[(𝜏)<0βˆ€πœβˆˆ0,∞),(2.37)limπœβ†’βˆžξ€·β„Ž(𝜏),β„Žξ…žξ€Έ(𝜏)=(1,0).(2.38) Let (𝜏min,∞) denote the interval of existence of this solution. It remains to prove (2.5) and the second part of (2.6), that is, that β„Ž(𝜏)>1,β„Žξ…ž(𝜏)<0,β„Žξ…žξ…žξ€·πœ(𝜏)>0βˆ€πœβˆˆminξ€Έ,∞,(2.39)limπœβ†’πœ+minβ„Ž(𝜏)=∞.(2.40) Let (πœβˆ—,∞) denote the maximal subinterval of (𝜏min,∞) such that β„Žξ…ž(𝜏)<0 for all 𝜏∈(πœβˆ—,∞). From the definition of πœβˆ— and (2.37), it follows that β„Ž(𝜏)>1 for all 𝜏>πœβˆ—. Next, we prove that πœβˆ—=𝜏min. Suppose, for contradiction, that πœβˆ—>𝜏min. Then β„Žξ€·πœβˆ—ξ€Έ>1β„Žξ…žξ€·πœβˆ—ξ€Έ=0,β„Žξ…žξ…žξ€·πœβˆ—ξ€Έβ‰€0.(2.41) From (1.17), and the fact that β„Ž(πœβˆ—)>1 and β„Žξ…ž(πœβˆ—)=0, it follows that β„Žξ…žξ…žξ€·πœβˆ—ξ€Έ=2(π‘βˆ’2)(π‘βˆ’1)2ξ‚€π‘π‘βˆ’||β„Žξ€·πœπ‘βˆ’2ξ‚ξ‚€βˆ—ξ€Έ||π‘βˆ’1ξ‚β„Žξ€·πœβˆ’1βˆ—ξ€Έ>0,(2.42) which contradicts (2.41). We conclude that πœβˆ—=𝜏min, hence β„Ž(𝜏)>1 and β„Žξ…ž(𝜏)<0 for all 𝜏∈(𝜏min,∞). Finally, suppose that β„Žξ…žξ…ž(Μ‚πœ)=0 at some Μ‚πœβˆˆ(𝜏min,∞). A differentiation (1.17) gives β„Žξ…žξ…žξ…ž(Μ‚πœ)=βˆ’2(π‘βˆ’2)(π‘βˆ’1)2ξ‚€π‘π‘βˆ’π‘||||π‘βˆ’2ξ‚ξ‚€β„Ž(Μ‚πœ)π‘βˆ’1ξ‚β„Žβˆ’1ξ…ž(Μ‚πœ)<0.(2.43) Thus, since β„Žξ…žξ…žξ…ž<0 whenever β„Žξ…žξ…ž=0, we conclude that β„Žξ…žξ…ž(𝜏)<0 for all 𝜏>Μ‚πœ. This implies that β„Žξ…ž(∞)<0, contradicting (2.38). Therefore, it must be the case that β„Žξ…žξ…ž(𝜏)>0 for all 𝜏∈(𝜏min,∞). This completes the proof of (2.39). It then follows from (2.39) and standard theory that limπœβ†’πœ+minβ„Ž(𝜏)=∞, and (2.40) is proved.

Open Problem 1. The issue of whether 𝜏min=βˆ’βˆž or 𝜏min>βˆ’βˆž remains unresolved. Its resolution may lead to new classes of solutions of the 𝑀 equation (1.3). Precise details of the implications for solutions of (1.3) are given below, both in the proof of Theorem 2.2, and in the discussion which follows its proof.

Proof of (iii). It follows from (2.16) and the Stable Manifold Theorem that there is a one-dimensional unstable manifold Ξ© of solutions leading from (1,0) into the (β„Ž,β„Žξ…ž) plane. Additionally, solutions on Ξ© satisfy limπœβ†’βˆ’βˆžβ„Žξ…ž(𝜏)β„Ž(𝜏)βˆ’1=πœ†2>0.(2.44) Thus, for sufficiently large 𝜏, solutions on Ξ© satisfy β„Ž(𝜏)>1 if β„Žξ…ž(𝜏)>0, and β„Ž(𝜏)<1 if β„Žξ…ž(𝜏)<0. Let 𝐷1 denote the component of Ξ© pointing into the region β„Ž>1, β„Žξ…ž>0 of the (β„Ž,β„Žξ…ž) plane (Figure 4(a)). Let (β„Ž(0),β„Žξ…ž(0))∈𝐷1. Then limπœβ†’βˆ’βˆž(β„Ž(𝜏),β„Žξ…ž(𝜏))=(1,0), hence, the first part of (2.8) is proved. Next, because of (2.44) and the translation invariance of (2.12), we can choose β„Ž(0)βˆ’1>0 and β„Žξ…ž(0)>0 small enough so that β„Ž(𝜏)>1,β„Žξ…ž](𝜏)>0,βˆ€πœβˆˆ(βˆ’βˆž,0.(2.45) The interval of existence of this solution is of the form (βˆ’βˆž,𝜏max), where 𝜏max>0. It remains to prove that finite time blowup occurs, that is, that 𝜏max<∞, and β„Ž(𝜏)>1,β„Žξ…ž(𝜏)>0,β„Žξ…žξ…žξ€·(𝜏)>0βˆ€πœβˆˆβˆ’βˆž,𝜏maxξ€Έ,(2.46)limπœβ†’πœβˆ’maxξ€·β„Ž(𝜏),β„Žξ…ž(ξ€Έπœ)=(∞,∞).(2.47) Let (βˆ’βˆž,πœβˆ—) denote the maximal subinterval of (βˆ’βˆž,𝜏max) such that β„Žξ…ž(𝜏)>0 for all 𝜏∈(βˆ’βˆž,πœβˆ—). It follows from (2.45) and the definition of πœβˆ— that β„Ž(𝜏)>1 for all 𝜏∈(βˆ’βˆž,πœβˆ—). We claim that πœβˆ—=𝜏max. Suppose, for contradiction, that πœβˆ—<𝜏max. Then β„Žξ€·πœβˆ—ξ€Έξ€·πœ>1,β„Žβˆ—ξ€Έ=0,β„Žξ…žξ…žξ€·πœβˆ—ξ€Έβ‰€0.(2.48) However, (1.17) and the fact that β„Ž(πœβˆ—)>1 and β„Žξ…ž(πœβˆ—)=0, imply that β„Žξ…žξ…žξ€·πœβˆ—ξ€Έ=2(π‘βˆ’2)(π‘βˆ’1)2ξ‚€π‘π‘βˆ’||β„Žξ€·πœπ‘βˆ’2ξ‚ξ‚€βˆ—ξ€Έ||π‘βˆ’1ξ‚β„Žξ€·πœβˆ’1βˆ—ξ€Έ>0,(2.49) contradicting (2.48). Thus, πœβˆ—=𝜏max, hence β„Ž(𝜏)>1 and β„Žξ…ž(𝜏)>0 for all 𝜏∈(βˆ’βˆž,𝜏max). Also, it follows exactly as in the Proof of (ii) that β„Žξ…žξ…ž(𝜏) does not change sign on (βˆ’βˆž,𝜏max), and that β„Žξ…žξ…ž(𝜏)>0 for all 𝜏∈(βˆ’βˆž,𝜏max). This completes the proof of (2.46). Next, we prove that 𝜏max<∞. Suppose, however, that 𝜏max=∞. Then β„Žξ…žξ…ž(𝜏)>0 for all 𝜏∈(βˆ’βˆž,∞). This implies that β„Žξ…ž(𝜏)β‰₯β„Žξ…ž(0)>0βˆ€πœβ‰₯0,limπœβ†’βˆžβ„Ž(𝜏)=∞.(2.50) To use (2.50) to contradict the assumption that 𝜏max<∞, we analyze ξ€·β„Žπ‘†=ξ…žξ€Έ22+2(π‘βˆ’2)(π‘βˆ’1)2ξ‚€π‘π‘βˆ’ξ‚ξ‚΅β„Žπ‘βˆ’2𝑝+1βˆ’β„Žπ‘+122ξ‚Ά,(2.51) which satisfies π‘†ξ…ž=π‘βˆ’2ξ‚€π‘βˆ’1𝑁+2ξ‚ξ€·β„Žπ‘βˆ’2βˆ’π‘ξ…žξ€Έ2.(2.52) Since β„Žξ…ž(𝜏)β‰₯β„Žξ…ž(0)>0 for all 𝜏β‰₯0, it follows from an integration of (2.52) that 𝑆(𝜏)β†’βˆž as πœβ†’βˆž. These, (2.51) and the fact that β„Ž(𝜏)β†’βˆž as πœβ†’βˆž, imply that there is a 𝜏1β‰₯0 such that 𝑆(𝜏)β‰₯0 for all 𝜏β‰₯𝜏1, that is, that ξ€·β„Žξ…žξ€Έ2β‰₯2(π‘βˆ’2)(π‘βˆ’1)2ξ‚€π‘ξ‚β„Žπ‘βˆ’2βˆ’π‘π‘+1𝑝+1βˆ€πœβ‰₯𝜏1.(2.53) An integration of (2.53) gives (β„Ž(𝜏))(1βˆ’π‘)/2β‰€ξ€·β„Žξ€·πœ1ξ€Έξ€Έ(1βˆ’π‘)/2+π‘Ž(1βˆ’π‘)2ξ€·πœβˆ’πœ1ξ€Έ,𝜏β‰₯𝜏1,(2.54) where π‘Ž=(2(π‘βˆ’2)/(π‘βˆ’1)(π‘βˆ’1)2)(𝑁/(π‘βˆ’2)βˆ’π‘))1/2>0 since 1<𝑝<𝑁/(π‘βˆ’2). The right side of (2.54) is negative when 𝜏>𝜏2=𝜏1+(2(π‘βˆ’1)/π‘Ž)(β„Ž(𝜏1))(1βˆ’π‘)/2. Thus, (2.54) reduces to (β„Ž(𝜏))(1βˆ’π‘)/2<0 when 𝜏>𝜏2, a contradiction. We conclude that 𝜏max<∞, as claimed. Since 𝜏max<∞, it follows from (1.17), (2.7), and standard theory that (β„Ž(𝜏),β„Žξ…ž(𝜏))β†’(∞,∞) as πœβ†’πœβˆ’max. This proves property (2.47).

Proof of (iv). It follows from the Stable Manifold Theorem and (2.44) that there is a second component, 𝐸1, of Ξ© which points into the region 0<β„Ž<1, β„Žξ…ž<0 of the (β„Ž,β„Žξ…ž) plane. Thus, if (β„Ž(0),β„Žξ…ž(0))∈𝐸1, and 1βˆ’β„Ž(0)>0 is sufficiently small, then 0<β„Ž(𝜏)<1,β„Žξ…ž],(𝜏)<0βˆ€πœβˆˆ(βˆ’βˆž,0limπœβ†’βˆ’βˆžξ€·β„Ž(𝜏),β„Žξ…žξ€Έ(𝜏)=(1,0).(2.55) Define πœβˆ—ξ€½=supΜ‚πœ>0∣0<β„Ž(𝜏)<1,β„Žξ…ž[)ξ€Ύ(𝜏)<0βˆ€πœβˆˆ0,Μ‚πœ.(2.56) We need to prove that πœβˆ—<∞, that β„Ž(πœβˆ—) and β„Žξ…ž(πœβˆ—) are finite, β„Žξ€·πœβˆ—ξ€Έ=0,β„Žξ…žξ€·πœβˆ—ξ€Έ<0.(2.57) For this, integrate (1.17) and get β„Žξ…ž(𝜏)π‘’π΄πœ=β„Žξ…žξ€œ(0)+𝐡𝜏0π‘’π΄πœ‚ξ‚€||||β„Ž(πœ‚)π‘βˆ’1ξ‚βˆ’1β„Ž(πœ‚)π‘‘πœ‚,0β‰€πœ<πœβˆ—,(2.58) where 𝐴=((π‘βˆ’2)/(π‘βˆ’1))(π‘βˆ’(𝑁+2)/(π‘βˆ’2))<0 and 𝐡=(2(π‘βˆ’2)/(π‘βˆ’1)2)(𝑁/(π‘βˆ’2)βˆ’π‘)>0. Because (|β„Ž|π‘βˆ’1βˆ’1)β„Ž>βˆ’1 for all β„Žβˆˆ[0,1], it follows that ξ€œπœ0π‘’π΄πœ‚ξ‚€||||β„Ž(πœ‚)π‘βˆ’1ξ‚ξ€œβˆ’1β„Ž(πœ‚)π‘‘πœ‚β‰₯βˆ’πœ0π‘’π΄πœ‚π‘‘πœ‚=βˆ’1π΄ξ€·π‘’π΄ξ€Έξ€Ίπœβˆ’1βˆ€πœβˆˆ0,πœβˆ—ξ€Έ.(2.59) Combining (2.58) and (2.59) gives 0>β„Žξ…ž(𝜏)π‘’π΄πœβ‰₯β„Žξ…žπ΅(0)βˆ’π΄ξ€·π‘’π΄πœξ€Έξ€Ίβˆ’1βˆ€πœβˆˆ0,πœβˆ—ξ€Έ.(2.60) We conclude from (2.60) that if πœβˆ— is finite, then β„Ž(𝜏) and β„Žξ…ž(𝜏) are bounded on the closed interval [0,πœβˆ—]. This, (2.58), the definition of πœβˆ—, and (2.60) imply that β„Žξ…ž(πœβˆ—)<0 and β„Ž(πœβˆ—)=0 if πœβˆ— is finite. Thus, (2.57) is proved if πœβˆ— is shown to be finite. We assume, for contradiction, that πœβˆ—=∞. Then 0<β„Ž(𝜏)<1,β„Žξ…ž(𝜏)<0βˆ€πœβ‰₯0.(2.61) Since the integral term in (2.58) is negative for all 𝜏β‰₯0, then (2.58) reduces to β„Žξ…ž(𝜏)π‘’π΄πœβ‰€β„Žξ…ž(0) for all 𝜏β‰₯0. An integration gives β„Žβ„Ž(𝜏)β‰€β„Ž(0)βˆ’ξ…ž(0)π΄ξ€·π‘’βˆ’π΄πœξ€Έ[βˆ’1βˆ€πœβˆˆ0,∞).(2.62) The right side of (2.62) is negative when 𝜏>βˆ’(1/𝐴)ln(β„Ž(0)𝐴/β„Žξ…ž(0)+1), contradicting (2.61). We conclude that πœβˆ—<∞, as claimed. This completes the proof of Theorem 2.1.

Solutions of the 𝑀 equation
Below, in Theorem 2.2, we show how to combine parts (i)–(iv) of Theorem 2.1 together with the formula 𝑀(π‘Ÿ)=β„Ž(ln(π‘Ÿ))𝑀1(π‘Ÿ),(2.63) to generate new families of solutions of the 𝑀 equation (1.3). In each of the four cases (i)–(iv), we show how to use (2.63) to prove the existence of an entire continuum of new singular solutions of (1.3). In each case, our approach is to let (β„Ž(0),β„Žξ…ž(0)) be an arbitrarily chosen element of one of the four continuous curves 𝐡1,𝐢1,𝐷1 or 𝐸1. Since π‘Ÿ=π‘’πœ, the initial conditions for the corresponding solution of (1.3) are given at π‘Ÿ=𝑒0=1, and satisfy 𝑀(1)=β„Ž(0)𝑀1(1),π‘€ξ…ž(1)=β„Žξ…ž(0)𝑀(1)+β„Ž(0)π‘€ξ…ž(1).(2.64) Because the curves 𝐡1, 𝐢1, 𝐷1, and 𝐸1 are continuous, this technique generates four new continua of solutions of the 𝑀 equation. In addition, for cases (i), (iii), and (iv), our analytical technique allows us to completely resolve the issues raised in Specific Aims 1, 2, and 3 in Section 1. That is, for each of the solutions described in (i), (iii), and (iv) we show how to efficiently prove the limiting behavior of the solution at both ends of the maximal interval (π‘Ÿmin,π‘Ÿmax), where it is positive. For part (ii), our analysis of the behavior of solutions at π‘Ÿmin is incomplete, and this leads to Open Problem 2 which is stated at the end of the proof of (ii). This problem is directly related to Open Problem 1 described above at the end of the proof of part (ii) of Theorem 2.1.
Open Problem 3 Prove the existence of other families of solutions of (1.3). For example, the existence and limiting behavior of the solutions labeled (c) and (f) in Figure 1 have not yet been proved. It is our hope that our analytical techniques can be extended to prove the existence and limiting behavior of these and many other new families of solutions.

Theorem 2.2. Let 𝑁>2 and 1<𝑝<𝑁/(π‘βˆ’2), and let 𝑀1(π‘Ÿ) denote the positive singular solution of (1.3) defined in (1.4).

(1) A Continuum of Singular Solutions Generated by 𝐡1
Let β„Ž2(𝜏) denote a solution of (1.17) which satisfies (β„Ž2(0),β„Žξ…ž2(0))∈𝐡1 in part (i) of Theorem 2.1. The corresponding solution 𝑀2(π‘Ÿ)=β„Ž2(ln(π‘Ÿ))𝑀1(π‘Ÿ) of (1.3) has initial values 𝑀2(1)=β„Ž2(0)𝑀1(1),π‘€ξ…ž2(1)=β„Žξ…ž2(0)𝑀(1)+β„Ž2(0)π‘€ξ…ž(1),(2.65) and satisfies 0<𝑀2(π‘Ÿ)<𝑀1𝑀(π‘Ÿ)βˆ€π‘Ÿ>0,2(π‘Ÿ)𝑀1(π‘€π‘Ÿ)⟢1asπ‘ŸβŸΆβˆž,2ξ‚΅(π‘Ÿ)∼4βˆ’2(π‘βˆ’2)(π‘βˆ’1)(π‘βˆ’1)2ξ‚Ά1/(π‘βˆ’1)π‘Ÿβˆ’(π‘βˆ’2)asπ‘ŸβŸΆ0+.(2.66) Figures 2 and 4(d) show solutions of (1.3) with these properties.

(2) A Continuum of Singular Solutions Generated by 𝐢1
Let β„Ž3(𝜏) denote a solution of (1.17) which satisfies (β„Ž3(0),β„Žξ…ž3(0))∈𝐢1 in part (ii) of Theorem 2.1. The corresponding solution 𝑀3(π‘Ÿ)=β„Ž3(ln(π‘Ÿ))𝑀1(π‘Ÿ) of (1.3) has initial values 𝑀3(1)=β„Ž3(0)𝑀1(1),π‘€ξ…ž3(1)=β„Žξ…ž3(0)𝑀(1)+β„Ž3(0)π‘€ξ…ž(1).(2.67) Let (π‘Ÿmin,π‘Ÿmax) be the maximal interval where 𝑀3(π‘Ÿ)>0. Then π‘Ÿmax=∞, 𝑀3(π‘Ÿ)>𝑀1(π‘Ÿ)βˆ€π‘Ÿ>π‘Ÿmin,(2.68)limπ‘Ÿβ†’π‘Ÿ+min𝑀3(π‘Ÿ)𝑀1(π‘Ÿ)=∞,limπ‘Ÿβ†’βˆžπ‘€3(π‘Ÿ)𝑀1(π‘Ÿ)=1.(2.69) Figure 1(b) shows a solution of (1.3) with these properties.

(3) A Continuum of Singular Solutions Generated by 𝐷1
Let β„Ž4(𝜏) denote a solution of (1.17) which satisfies (β„Ž4(0),β„Žξ…ž4(0))∈𝐷1 in part (iii) of Theorem 2.1. The corresponding solution 𝑀4(π‘Ÿ)=β„Ž4(ln(π‘Ÿ))𝑀1(π‘Ÿ) of (1.3) has initial values 𝑀4(1)=β„Ž4(0)𝑀1(1),π‘€ξ…ž4(1)=β„Žξ…ž4(0)𝑀(1)+β„Ž4(0)π‘€ξ…ž(1).(2.70) Let (π‘Ÿmin,π‘Ÿmax) be the maximal interval where 𝑀4(π‘Ÿ)>0. Then π‘Ÿmin=0 and π‘Ÿmax<∞, 𝑀4(π‘Ÿ)>𝑀1ξ€·(π‘Ÿ)βˆ€π‘Ÿβˆˆ0,π‘Ÿmaxξ€Έ,(2.71)limπ‘Ÿβ†’0+𝑀4(π‘Ÿ)𝑀1(π‘Ÿ)=1,limπ‘Ÿβ†’π‘Ÿmax𝑀4(π‘Ÿ)=∞.(2.72) Figure 1(d) shows a solution of (1.3) with these properties.

(4) A Continuum of Singular Solutions Generated by 𝐸1
Let β„Ž5(𝜏) denote a solution of (1.17) which satisfies (β„Ž5(0),β„Žξ…ž5(0))∈𝐸1 in part (iv) of Theorem 2.1. The corresponding solution 𝑀5(π‘Ÿ)=β„Ž5(ln(π‘Ÿ))𝑀1(π‘Ÿ) of (1.3) has initial values 𝑀5(1)=β„Ž5(0)𝑀1(1),π‘€ξ…ž5(1)=β„Žξ…ž5(0)𝑀(1)+β„Ž5(0)π‘€ξ…ž(1).(2.73) Let (π‘Ÿmin,π‘Ÿmax) be the maximal interval where 𝑀5(π‘Ÿ)>0. Then π‘Ÿmin=0 and π‘Ÿmax<∞, 0<𝑀5(π‘Ÿ)<𝑀1ξ€·(π‘Ÿ)βˆ€π‘Ÿβˆˆ0,π‘Ÿmaxξ€Έ,(2.74)limπ‘Ÿβ†’0+𝑀5(π‘Ÿ)𝑀1(π‘Ÿ)=1,limπ‘Ÿβ†’π‘Ÿmπ‘Žπ‘₯𝑀4(π‘Ÿ)=0.(2.75) Figure 2 shows a solution of (1.3) with these properties.

Proof of (1). Let β„Ž2 denote a solution of (1.3) which satisfies part (i) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to β„Ž2 is 𝑀2(π‘Ÿ)=β„Ž2(ln(π‘Ÿ))𝑀1(π‘Ÿ).(2.76) It follows from (2.2) in Theorem 2.1 that 0<β„Ž2(ln(π‘Ÿ))<1 for all π‘Ÿ>0. This, as well as (2.76), implies that 0<𝑀2(π‘Ÿ)<𝑀1(π‘Ÿ)βˆ€π‘Ÿ>0(2.77) (see Figure 4(d)). We claim that 𝑀2 is singular at π‘Ÿ=0. The first step in proving this claim is to observe that (2.3) and (2.11) imply that (β„Žξ…ž2(𝜏)/β„Ž2(𝜏))β†’πœ‡1 as πœβ†’βˆ’βˆž. Thus, ln(β„Ž2(𝜏))βˆΌπœ‡1𝜏 as πœβ†’βˆ’βˆž. This and the fact that 𝜏=ln(π‘Ÿ) lead to β„Ž2(𝜏)=β„Ž2(ln(π‘Ÿ))βˆΌπ‘Ÿπœ‡1asπ‘Ÿβ†’0+.(2.78) Substituting (1.4) and (2.78) into (2.76) gives 𝑀2ξ‚΅(π‘Ÿ)∼4βˆ’2(π‘βˆ’2)(π‘βˆ’1)(π‘βˆ’1)2ξ‚Ά1/(π‘βˆ’1)π‘Ÿπœ‡1βˆ’πœ‡2asπ‘ŸβŸΆ0+.(2.79) Our claim that 𝑀2 is singular at π‘Ÿ=0 follows from (2.79) and the fact that πœ‡1βˆ’πœ‡2=2βˆ’π‘<0. It remains to determine the asymptotic behavior of 𝑀2(π‘Ÿ) as π‘Ÿβ†’βˆž. Since β„Ž2(ln(π‘Ÿ))β†’1βˆ’ as π‘Ÿβ†’βˆž, then (𝑀2(π‘Ÿ)/𝑀1(π‘Ÿ))β†’1 as π‘Ÿβ†’βˆž. This completes the proof of properties (2.66).

Proof of (2). Let β„Ž3 denote a solution of (1.3) which satisfies part (ii) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to β„Ž3 is 𝑀3(π‘Ÿ)=β„Ž3(ln(π‘Ÿ))𝑀1(π‘Ÿ).(2.80) Initial conditions (2.67) follow exactly as in the proof of part (i). Let (π‘Ÿmin,π‘Ÿmax) denote the maximal interval over which 𝑀3(π‘Ÿ)>0. It follows from (2.5) in Theorem 2.1 that π‘Ÿmax=∞ and β„Ž4(ln(π‘Ÿ))>1 for all π‘Ÿ>π‘Ÿmin. This, together with (2.80), implies that 𝑀3(π‘Ÿ)>𝑀1ξ€·π‘Ÿ(π‘Ÿ)βˆ€π‘Ÿβˆˆminξ€Έ,∞.(2.81) This proves (2.68). Property (2.6) in Theorem 2.1, as well as (2.80), imply that limπ‘Ÿβ†’βˆžπ‘€3(π‘Ÿ)𝑀1(π‘Ÿ)=limπ‘Ÿβ†’βˆžβ„Ž3(ln(π‘Ÿ))=1,(2.82)limπ‘Ÿβ†’π‘Ÿ+min𝑀3(π‘Ÿ)𝑀1(π‘Ÿ)=limπ‘Ÿβ†’π‘Ÿ+minβ„Ž3(ln(π‘Ÿ))=∞.(2.83)

Open Problem 2. Prove whether π‘Ÿmin=0 or π‘Ÿmin>0. This problem arises as a direct consequence of Open Problem 1 described above at the end of the proof of part (ii) of Theorem 2.1. If it can be proved that π‘Ÿmin=0, then 𝑀3(π‘Ÿ)>𝑀1(π‘Ÿ)βˆ€π‘Ÿβˆˆ(0,∞),limπ‘Ÿβ†’0+𝑀3(π‘Ÿ)𝑀1(π‘Ÿ)=∞.(2.84) Because 𝑀3(π‘Ÿ)β†’βˆž much faster than 𝑀1(π‘Ÿ), we refer to any solution satisfying either (2.83) of (2.84) as a Super Singular Solution. This class of solutions has not previously been reported.

Proof of (3). Let β„Ž4 denote a solution of (1.3) which satisfies part (iii) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to β„Ž4 is 𝑀4(π‘Ÿ)=β„Ž4(ln(π‘Ÿ))𝑀1(π‘Ÿ).(2.85) Initial conditions (2.70) follow exactly as in the proof of part (i). Let (π‘Ÿmin,π‘Ÿmax) denote the maximal interval over which 𝑀4(π‘Ÿ)>0. It follows from (2.7) in Theorem 2.1 that π‘Ÿmin=0 and π‘Ÿmax<∞, and β„Ž4(ln(π‘Ÿ))>1 for all π‘Ÿβˆˆ(0,π‘Ÿmax). This, together with (2.85), implies that 𝑀4(π‘Ÿ)>𝑀1ξ€·(π‘Ÿ)βˆ€π‘Ÿβˆˆ0,π‘Ÿmaxξ€Έ.(2.86) This proves (2.71). Property (2.8), in Theorem 2.1, as well as (2.85), implies that limπ‘Ÿβ†’π‘Ÿβˆ’max𝑀4(π‘Ÿ)=limπ‘Ÿβ†’π‘Ÿβˆ’maxβ„Ž4(ln(π‘Ÿ))𝑀1(π‘Ÿ)=∞,limπ‘Ÿβ†’0+𝑀4(π‘Ÿ)𝑀1(π‘Ÿ)=limπ‘Ÿβ†’0+β„Ž4(ln(π‘Ÿ))=1.(2.87) This completes the proof of (2.72).

Proof of (4). Let β„Ž5 denote a solution of (1.3) which satisfies part (iv) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to β„Ž5 is 𝑀5(π‘Ÿ)=β„Ž5(ln(π‘Ÿ))𝑀1(π‘Ÿ).(2.88) Initial conditions (2.73) follow exactly as in the proof of part (i). Let (π‘Ÿmin,π‘Ÿmax) be the maximal interval over which 𝑀5(π‘Ÿ)>0. It follows from (2.9) in Theorem 2.1 that π‘Ÿmin=0 and π‘Ÿmax<∞, and β„Ž5(ln(π‘Ÿ))<1 for all π‘Ÿβˆˆ(0,π‘Ÿmax). This, together with (2.88), implies that 0<𝑀5(π‘Ÿ)<𝑀1ξ€·(π‘Ÿ)βˆ€π‘Ÿβˆˆ0,π‘Ÿmaxξ€Έ.(2.89) This proves (2.74). Properties (2.9) in Theorem 2.1, and (2.88), imply that limπ‘Ÿβ†’0+𝑀3(π‘Ÿ)𝑀1(π‘Ÿ)=limπ‘Ÿβ†’0+β„Ž3(ln(π‘Ÿ))=1,limπ‘Ÿβ†’π‘Ÿβˆ’max𝑀5(π‘Ÿ)=limπ‘Ÿβ†’π‘Ÿβˆ’maxβ„Ž5(ln(π‘Ÿ))𝑀1(π‘Ÿ)=0.(2.90) This completes the proof of (2.75). Therefore, Theorem 2.2 is proved.

3. Conclusions

In this paper, our analytic advance is the development of methods to efficiently prove the existence and asymptotic behavior of families of positive singular solutions of (1.3). Our approach consists of the following three steps.

Step 1. Transform the nonautonomous 𝑀 equation (1.3) into the autonomous β„Ž equation (1.17) by setting β„Ž(𝜏)=𝑀(exp(𝜏))𝑀1(exp(𝜏)),βˆ’βˆž<𝜏<∞.(3.1)

Step 2. Analyze the existence and asymptotic behavior of solutions of (1.17) which are positive on a maximal interval (𝜏min,𝜏max).

Step 3. For each such solution of the β„Ž equation, make use of the inverse transformation 𝑀(π‘Ÿ)=β„Ž(ln(π‘Ÿ))𝑀1(π‘Ÿ),0<π‘Ÿ<∞(3.2) to prove the existence and asymptotic behavior of the associated solution (3.2) of the 𝑀 equation on the maximal interval (π‘Ÿmin,π‘Ÿmax), where 𝑀(π‘Ÿ)>0.

In Section 2, we used this three-step procedure (see Theorems 2.1 and 2.2) to prove the existence and asymptotic behavior of three new families of solutions of (1.3). Open Problems 1 and 2 describe a fourth family of solutions whose existence is also proved in these theorems, and which satisfy𝑀(π‘Ÿ)>𝑀1(π‘Ÿ),π‘€ξ…žξ€·π‘Ÿ(π‘Ÿ)<0βˆ€π‘Ÿβˆˆminξ€Έ,∞,limπ‘Ÿβ†’βˆžπ‘€(π‘Ÿ)=0.(3.3) The unresolved issue is to prove whether π‘Ÿmin=0 or π‘Ÿmin>0. If π‘Ÿmin=0, then solutions in this family satisfy the limiting propertylimπ‘Ÿβ†’0+𝑀(π‘Ÿ)𝑀1(π‘Ÿ)=∞.(3.4) Thus, as π‘Ÿβ†’0+, these β€œsuper singular” solutions approach ∞ much faster than the closed form solution 𝑀1(π‘Ÿ).

Open Problem 3. Prove the existence and asymptotic behavior of families of positive solutions of (1.3) other than those found in Theorem 2.2. For example, the existence and limiting behavior of the solutions labelled (c) and (f) in Figure 1 have not yet been proved. It is our hope that our techniques can be extended to prove the existence and limiting behavior of these and many other new families of solutions.

Open Problem 4. Determine the role that the singular solutions proved in Theorem 2.2 play in the analysis of the full time-dependent PDE (1.2). Can the analytic techniques developed by Souplet and Weissler [5], and those of Chen and Derrick [6], be extended to apply to these new solutions?

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