#### Abstract

We investigate the existence and asymptotic behavior of positive, radially symmetric singular solutions of , . We focus on the parameter regime and where the equation has the closed form, positive singular solution , . Our advance is to develop a technique to efficiently classify the behavior of solutions which are positive on a maximal positive interval . Our approach is to transform the nonautonomous 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 equation. Specifically, we prove the existence of a family of singular solutions which exist on the entire interval , and which satisfy for all . An important open problem for the nonautonomous equation is presented. Its solution would lead to the existence of a new family of “super singular” solutions which lie entirely above .

#### 1. Introduction

We investigate the behavior of solutions of
where , and . Solutions of (1.1) are time-independent solutions of the nonlinear heat equation
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 , where , and satisfy
Equation (1.3) has the closed form, positive singular solution (see Figure 2)

*A Related Equation*

A second, widely studied nonlinear heat equation is
Equation (1.5) has the closed form, stationary, positive singular solution
This well-known singular solution plays an important role in the analysis of blowup of solutions of (1.5). For example, when is appropriately chosen, similarity solution methods developed by Haraux and Weissler [4], and Souplet and Weissler [5], show how as , where 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
where 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 and , the parameter regime where exists. In order to study properties of positive solutions of (1.3), our approach is to let be arbitrarily chosen and analyze solutions with initial values
Let denote the largest interval containing over which the solution of (1.3)–(1.8) is positive.

*Specific Aim 1. *For each solution of (1.3)–(1.8), prove whether or , and determine .

*Specific Aim 2. *For each solution of (1.3)–(1.8), prove whether or , and determine .

*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 , where

*Numerical Experiments*

In Figure 1, we set and illustrate solutions of (1.3)–(1.8) for various values. For example, when , panels (a)–(d) show that both and are possible, and that
Panels (a)–(f) and also Figure 2 show that solutions can satisfy either or .

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

*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 equation to resolve Specific Aims 1 and 2. The proof of the first property in (1.10) follows from (1.3), which implies that at any , where and . However, the fact that for all is *not* sufficient by itself to prove whether or . Nor does it prove the second part of (1.10), that . In fact, our study suggests that there is a unique where (see Figure 1(d)), and that at all other negative 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 and have been determined, we need to turn our attention to the interval . As Figure 1 shows, there are several different types of behavior when . For example, consider the solutions in panels (a), (b), and (c) in Figure 1. In each case,
When , panels (a), (b), and (c) show three different behaviors of solutions, namely,
These results lead to the following analytical challenge: given *only* the fact that a solution satisfies property (1.11) when , how can we prove which of the possibilities (1.12) occurs when ? It is not at all clear how to answer this question using standard methods such as Pohozaev identities or topological shooting.

*Solutions with *

It is particularly important to understand the global behavior of solutions for which 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 . This solution lies entirely *above *, that is, for all . Figure 2 shows two other solutions, labeled and , for which . These solutions lie entirely *below * on . Our computations indicate that satisfies , and that for . These numerical experiments lead to

*Specific Aim 3. *Let and . Prove that there are at least three families of solutions, other than , with . The solutions in these families have the following properties:(i)(see Figure 1(d)). For each there exists such that if is the solution of (1.3) with , then , ,(ii)(See Figure 2). For each there exists such that if is the solution of (1.3) with , then , ,(iii)(See Figure 2). For each there exists such that if is the solution of (1.3) with , then ,

*Our Analytical Approach*

Our goal is to develop techniques to efficiently prove the existence of solutions of the 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 . Our advance is to significantly simplify the analysis by transforming (1.3) into an equation which is *autonomous*, that is, independent or . For this, let denote any solution of (1.3), and define
Then solves

*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 , , which changes the Emden-Fowler equation
to the new equation

Because (1.17) is autonomous, we can apply phase plane techniques to prove the behavior of its solutions. We then use the “inverse” formula to determine the global behavior of corresponding solutions of the equation (1.3). In Section 2, we demonstrate the utility of this two step procedure. First, in Theorem 2.1, we analyze the 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 equation (1.3). In parts (i), (iii), and (iv) of Theorem 2.2 we show how the formula can be efficiently used to prove the precise asymptotic behavior of each solution as , and as . 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 “super singular solutions,” which satisfy , However, it remains a challenging open problem (see Open Problems 1 and 2 in Section 2) to prove whether or . If the first possibility holds, then we have a fourth family of singular solutions, other than , which satisfy .

#### 2. The Main Result

In this section, we show how to make use of the autonomous equation (1.17) to address the issues raised in Specific Aims 1, 2, and 3 for solutions of the nonautonomous 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 equation (1.3) on the maximal interval , where is positive. To demonstrate the utility of our method, we restrict our focus to four specific branches of solutions of the 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 in the plane. The stable manifold has two components, nd , and the unstable manifold has two components, and . Solutions on , , , and are illustrated in Figure 4(a).

Secondly, in Theorem 2.2, we make use of the link to show how solutions with initial values on , , , and translate into four new continuous families of singular solutions of the equation (1.3). For three of the four cases, we completely prove the behavior of solutions of the equation on the maximal interval , 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 . The important consequences of resolving these open problems is described in Section 3.

Theorem 2.1. *Let and . Then *(i)*There is a one-dimensional stable manifold of solutions of (1.17) leading to in the phase plane. One component, , of points into the region , . If , then**(ii)**The second component, , of points into the region , of the plane. If a solution satisfies , and is its interval of existence, then**(iii)**There is a one-dimensional unstable manifold of solutions of (1.17) leading from into the plane. One component, , of points into the region , . If a solution satisfies , and is its interval of existence, then ,**(iv)**The second component, , of points into the region of the plane. If , with and , then there exists a value such that*

*Proof of (i). *We need to prove properties (2.2)–(2.4). The first step is to linearize (1.17) around the constant solution . This gives
The eigenvalues associated with (2.10) satisfy
We will make use of the observation that (1.17) can be written as
Next, a linearization of (1.17) about the constant solution gives
Define . Then (2.13) becomes
where
Thus, the eigenvalues associated with (2.13) and (2.14) satisfy
It follows from (2.16) and the Stable Manifold Theorem that there is a one-dimensional stable manifold of solutions leading to in the phase plane. Additionally,
for . Thus, for sufficiently large , solutions on satisfy if and if . Let denote the component of pointing into the region , of the plane. Assume that . 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 and small enough so that
The definition of , together with (2.18), imply that the maximal interval of existence is of the form , where .

Next, we show that , where is the bounded open triangular region
Figure 4(b) shows when . Because of (2.18), it suffices to show that for all . For contradiction, assume that leaves at some point in . Define
It follows from (2.12) that satisfies
Suppose that leaves across the line . That is, (see Figure 3(a)) suppose that there exists such that
If , then (2.20) implies that , contradicting uniqueness of the constant solution . Thus, . Also, (2.22) implies that
The fact that , combined with (2.21), results in
contradicting (2.23). Thus, can only leave across the line segment . If so, there is a such that
as depicted in the right panel of Figure 3. Hence,
It follows from (2.12) and (2.26) that
contradicting (2.27). We conclude that cannot leave on , hence as claimed. Moreover, since is bounded, then follows from standard ODE theory. Thus, for all , and, therefore, for all .*Proof of the first part (2.3). *First, we prove that as . Since and on , then where, . To obtain a contradiction suppose that . Then and (2.12) yield
It follows from (2.29) that as which contradicts the fact that is bounded and for all . Thus, as . Next, we show that as . Note that on is an immediate consequence of and on . Therefore, as follows from the fact that as .*Proof of second part of (2.3). *Finally, we need to prove that as . The definition of together with (2.12) gives
We now show that monotonically as . Differentiating (2.30) yields
Hence, if for some , then
This implies that has at most one zero on . Furthermore,
since
Thus, exists and . Moreover, the fact that is finite ensures the existence of an unbounded decreasing sequence such that . Substituting
into (2.30) results in
The bound and (2.36) imply that . Thus, as as claimed.

**(a)**

**(b)**

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

*Proof of (ii). *It follows from the Stable Manifold Theorem and (2.17) that there is a second component, , of which points into the region of the plane (Figure 4(a)). Thus, if , and is sufficiently small, then
Let denote the interval of existence of this solution. It remains to prove (2.5) and the second part of (2.6), that is, that
Let denote the maximal subinterval of such that for all . From the definition of and (2.37), it follows that for all . Next, we prove that . Suppose, for contradiction, that . Then
From (1.17), and the fact that and , it follows that
which contradicts (2.41). We conclude that , hence and for all . Finally, suppose that at some . A differentiation (1.17) gives
Thus, since whenever , we conclude that for all . This implies that , contradicting (2.38). Therefore, it must be the case that for all . This completes the proof of (2.39). It then follows from (2.39) and standard theory that , and (2.40) is proved.

*Open Problem 1. *The issue of whether or remains unresolved. Its resolution may lead to new classes of solutions of the 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 of solutions leading from into the plane. Additionally, solutions on satisfy
Thus, for sufficiently large , solutions on satisfy if , and if . Let denote the component of pointing into the region , of the plane (Figure 4(a)). Let . Then , hence, the first part of (2.8) is proved. Next, because of (2.44) and the translation invariance of (2.12), we can choose and small enough so that
The interval of existence of this solution is of the form , where . It remains to prove that finite time blowup occurs, that is, that , and
Let denote the maximal subinterval of such that for all . It follows from (2.45) and the definition of that for all . We claim that . Suppose, for contradiction, that . Then
However, (1.17) and the fact that and , imply that
contradicting (2.48). Thus, , hence and for all . Also, it follows exactly as in the Proof of (ii) that does not change sign on , and that for all . This completes the proof of (2.46). Next, we prove that . Suppose, however, that . Then for all . This implies that
To use (2.50) to contradict the assumption that , we analyze
which satisfies
Since for all , it follows from an integration of (2.52) that as . These, (2.51) and the fact that as , imply that there is a such that for all , that is, that
An integration of (2.53) gives
where since . The right side of (2.54) is negative when . Thus, (2.54) reduces to when , a contradiction. We conclude that , as claimed. Since , it follows from (1.17), (2.7), and standard theory that as . This proves property (2.47).

*Proof of (iv). *It follows from the Stable Manifold Theorem and (2.44) that there is a second component, , of which points into the region , of the plane. Thus, if , and is sufficiently small, then
Define
We need to prove that , that and are finite,
For this, integrate (1.17) and get
where and . Because for all , it follows that
Combining (2.58) and (2.59) gives
We conclude from (2.60) that if is finite, then and are bounded on the closed interval . This, (2.58), the definition of , and (2.60) imply that and if is finite. Thus, (2.57) is proved if is shown to be finite. We assume, for contradiction, that . Then
Since the integral term in (2.58) is negative for all , then (2.58) reduces to for all . An integration gives
The right side of (2.62) is negative when , contradicting (2.61). We conclude that , as claimed. This completes the proof of Theorem 2.1.

*Solutions of the equation*

Below, in Theorem 2.2, we show how to combine parts (i)–(iv) of Theorem 2.1 together with the formula
to generate new families of solutions of the 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 be an arbitrarily chosen element of one of the four continuous curves or . Since , the initial conditions for the corresponding solution of (1.3) are given at , and satisfy
Because the curves , , , and are continuous, this technique generates four new continua of solutions of the 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 , where it is positive. For part (ii), our analysis of the behavior of solutions at 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 and , and let denote the positive singular solution of (1.3) defined in (1.4).*

*
(1) A Continuum of Singular Solutions Generated by *

Let denote a solution of (1.17) which satisfies in part (i) of Theorem 2.1. The corresponding solution of (1.3) has initial values
and satisfies
Figures 2 and 4(d) show solutions of (1.3) with these properties.

*
(2) A Continuum of Singular Solutions Generated by *

Let denote a solution of (1.17) which satisfies in part (ii) of Theorem 2.1. The corresponding solution of (1.3) has initial values
Let be the maximal interval where . Then ,
Figure 1(b) shows a solution of (1.3) with these properties.

*
(3) A Continuum of Singular Solutions Generated by *

Let denote a solution of (1.17) which satisfies in part (iii) of Theorem 2.1. The corresponding solution of (1.3) has initial values
Let be the maximal interval where . Then and ,
Figure 1(d) shows a solution of (1.3) with these properties.

*
(4) A Continuum of Singular Solutions Generated by *

Let denote a solution of (1.17) which satisfies in part (iv) of Theorem 2.1. The corresponding solution of (1.3) has initial values
Let be the maximal interval where . Then and ,
Figure 2 shows a solution of (1.3) with these properties.

*Proof of (1). *Let denote a solution of (1.3) which satisfies part (i) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to is
It follows from (2.2) in Theorem 2.1 that for all . This, as well as (2.76), implies that
(see Figure 4(d)). We claim that is singular at . The first step in proving this claim is to observe that (2.3) and (2.11) imply that as . Thus, as . This and the fact that lead to
Substituting (1.4) and (2.78) into (2.76) gives
Our claim that is singular at follows from (2.79) and the fact that . It remains to determine the asymptotic behavior of as . Since as , then as . This completes the proof of properties (2.66).

*Proof of (2). *Let denote a solution of (1.3) which satisfies part (ii) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to is
Initial conditions (2.67) follow exactly as in the proof of part (i). Let denote the maximal interval over which . It follows from (2.5) in Theorem 2.1 that and for all . This, together with (2.80), implies that
This proves (2.68). Property (2.6) in Theorem 2.1, as well as (2.80), imply that

*Open Problem 2. *Prove whether or . 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 , then
Because much faster than , 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 denote a solution of (1.3) which satisfies part (iii) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to is
Initial conditions (2.70) follow exactly as in the proof of part (i). Let denote the maximal interval over which . It follows from (2.7) in Theorem 2.1 that and , and for all . This, together with (2.85), implies that
This proves (2.71). Property (2.8), in Theorem 2.1, as well as (2.85), implies that
This completes the proof of (2.72).

*Proof of (4). *Let denote a solution of (1.3) which satisfies part (iv) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to is
Initial conditions (2.73) follow exactly as in the proof of part (i). Let be the maximal interval over which . It follows from (2.9) in Theorem 2.1 that and , and for all . This, together with (2.88), implies that
This proves (2.74). Properties (2.9) in Theorem 2.1, and (2.88), imply that