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International Journal of Differential Equations
Volume 2012 (2012), Article ID 296591, 33 pages
http://dx.doi.org/10.1155/2012/296591
Research Article

Radially Symmetric Solutions of

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

Received 31 May 2012; Accepted 10 August 2012

Academic Editor: Julio Rossi

Copyright © 2012 William C. Troy and Edward P. Krisner. 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 solutions of and focus on the regime and . Our advance is to develop a technique to efficiently classify the behavior of solutions on , their maximal positive interval of existence. Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the equation and describe their asymptotic behavior. In the subcritical case there is a well-known closed-form singular solution, , such that as and as . Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies for infinitely many values . At the critical value there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime we prove the existence of a family of “super singular” sign changing singular solutions.

1. Introduction

In this paper we investigate the behavior of solutions of where , and . Solutions of (1.1) are time-independent solutions of the nonlinear heat equation

Equation (1.1) has been widely studied as a canonical model for where is superlinear [16].

Our focus is on radially symmetric solutions of (1.1) which have the form , where , and satisfy

We distinguish two classes of solutions of (1.4). The first is nonsingular solutions which are bounded at and satisfy , where is finite. The second class consists of singular solutions that are unbounded at . Equation (1.4) has the known positive singular solution

Previous Results
(i) The positive singular solution has played a central role in analyzing (1.2). For example, when appropriately chosen, similarity solution methods show how as , where is a constant [2, 5, 7]. (ii) Chen and Derrick [8] developed comparison methods to describe the time evolution of solutions of where is superlinear [16]. Their approach is to let positive, time independent solutions act as upper and/or lower bounds for initial values of solutions of (1.6). Their comparison technique allows them to prove either global existence or finite time blowup of solutions. (iii) For the case Caffarelli et al. [9] describe the asymptotic behavior of nonnegative solutions of (1.1) that have an isolated singularity at the origin. (iv) Galaktionov [10] studied sign changing singular solutions of (1.4) on the restricted interval . He set and derived an ODE for , . He let , varied , and gave a numerical study of sign changing solutions on . (v) Other studies of nonsingular solutions of (1.4) have used Pohozaev identities, together with integral estimates which involve the independent variable [1, 2, 4].

Specific Aims
Our goal is to develop techniques to efficiently classify the behavior of solutions of (1.4) on , their maximal positive interval of existence. We study the behavior of solutions which are positive on and also sign changing solutions. In particular, our specific aims are the following.

Specific Aim I
Do positive singular solutions exist, other than , for which ? What is their asymptotic behavior as , and as ? In Section 2 we prove the existence of a second singular solution, , (see bottom right panel of Figure 1), which exists on . Also, we prove the asymptotic behavior of this solutions as and as . This result is new and different from previous analyses. In addition, in the conclusion we suggest a possible application for the role of this new solution in analyzing the time-dependent behavior of the full PDE (1.2).

fig1
Figure 1: Subcritical example: , . ((a) and (b)) The unstable manifold ( and ) and stable manifold ( and ) lead from into the phase plane. ((c)–(f)) (c) and (e) show the components of solutions on , , , ; (d) and (f) show corresponding solutions of (1.4): is generated by , is the new positive singular solution generated by , and .

Specific Aim II
Do sign changing solutions exist for which ? What is their asymptotic behavior as , and as ? In Section 4 we prove the existence of a large amplitude sign changing solution in the phaseplane (see top left panel of Figure 4). This solution forms a large amplitude outward spiral as the independent variable decreases. Such global analysis has not previously been achieved.

Our Approach
Standard methods to analyze solutions of (1.4) include Pohozaev integral estimates, or topological shooting. Obtaining global results with such methods is difficult since (1.4) is nonautonomous. Thus, to successfully address the issues in Specific Aims I-II, our advance is to develop a two-step approach which significantly simplifies the analysis. The first step is to transform the nonautonomous ODE (1.4) into a simpler, autonomous ODE. Let solve (1.4), and define [10, 11]
Then solves
Because (1.8) is autonomous, we can apply phase plane techniques to analyze the behavior of solutions. The second step of our approach is to use the “inverse” formula to analyze corresponding solutions of the equation (1.4). For example, in Section 2 we analyze (1.8) in the subcritical range and prove that there is a nonmonotonic heteroclinic orbit (labeled in Figure 1) leading from to in the phase plane. We then use (1.9) to show that, corresponding to this heteroclinic orbit, there is an entire continuum of new positive singular solutions of (1.4). Let denote a member of this continuum (Figure 1, 3rd row). Then intertwines with infinitely often as . That is, there are infinitely many positive values , with as , such that
Furthermore, there is a value such that
Thus, as , but faster than as . To our knowledge, this family of solutions has not previously been reported.
In the conclusion, Section 5, we state an open problem which gives a conjecture for the role that might play in the analysis of the full time-dependent PDE (1.2).
In Sections 3 and 4 we use similar techniques to prove the existence of new families of solutions in the critical case , and the supercritical regime , respectively. In particular, in Section 4 we prove the existence of a continuum of “super singular” sign changing solutions, each of which exists on an interval of the form . For these solutions it remains a challenging, and important, open problem (see Open Problems I and II in Section 4) to prove whether or .

2. The Subcritical Case:

In this section we consider the parameter regime and . In this range we first analyze solutions of the equation (1.8) and then show how these solutions translate into corresponding solutions of the equation (1.4). The remainder of this section consists of the following.(I) Lemmas 2.1 and 2.2 state fundamental properties of solutions of (1.8) that satisfy on an interval for some . These properties will be applied in the proof of Lemma 2.3 which asserts that there exists a solution of (1.8) such that monotonically as . In Lemma 2.4 we show that This will be used to prove Lemma 2.5 which shows that the asymptotic behavior of is for some constant .(II) Solutions along the unstable manifold, , described in part (i) of Theorem 2.9, translate into nonsingular solutions of (1.4). Of particular importance is the heteroclinic orbit solution, , described in part (ii) of Theorem 2.9. Theorem 2.10 asserts that the trajectory generates an entire continuum of strictly positive singular solutions of (1.4), each of which intertwines infinitely often with as . To our knowledge, this family of solutions is new and has not previously been reported.(III)Numerical Experiments. Figure 1 shows solutions of (1.4) and (1.8) when . However, it must be emphasized that it is illegitimate to claim that numerical results are rigorous proofs. Complete analytical proofs are needed to determine properties of solutions of (1.4) and (1.8).

The following two technical lemmas are used to help prove that monotonically as .

Lemma 2.1. Suppose that is a nonconstant solution of (1.8) such that and for some . Then there exists a constant such that . Likewise, if and , then there exists a constant such that .

Proof. The proof of this lemma relies on the property which is an immediate consequence of the case assumption and (1.8). Since is a nonconstant solution, then uniqueness of solutions [12, Chapter 1] implies that either or . This and (1.8) imply that . Hence, and on an interval provided that is sufficiently small. By (2.3), on . The increasing values of on imply the existence of such that .
In a similar manner, and implies the existence of a value such that . This completes the proof.

The following lemma is also used to show that monotonically as .

Lemma 2.2. Suppose that is a nonconstant solution of (1.8) such that on . Then on .

Proof. Suppose that for some . Since is a nonconstant solution of (1.8), then . By Lemma 2.1, there exists a constant such that if and if . This contradicts the assumption that on and concludes the proof of the lemma.

Lemma 2.3. There exists a solution of (1.8) such that(a), and(b) and on .

Proof. A linearization of (1.8) around the constant solution gives
The eigenvalues associated with (2.4) are
Since , the Stable Manifold Theorem [12, Chapter 13] ensures that there exists a one-dimensional stable manifold containing in the phase plane. Let be a solution of (1.8) such that is a point on the stable manifold. That is,
Thus, satisfies part (a).
We now show that can be chosen to satisfy (b). Since solutions of (1.8) are translation-invariant, there is no loss in generality in assuming that on . Combining this with Lemma 2.2 yields on . According to the fact that and are both solutions of (1.8) that satisfy (2.6), we may also assume that on . Lemma 2.1 implies that on . This proves (b) and concludes the proof of this lemma.

Description of the Stable Manifold
Throughout the remainder of this section we let denote the solution of (1.8) that satisfies
Furthermore, we define where denotes the maximal interval of existence of the solution . In Lemma 2.6 we will show that . In addition, we define the negative counterpart of by
The top row of Figure 1 depicts and for and .

Asymptotic Behavior of
To state Lemma 2.4 correctly we need to derive basic properties of the functional . It follows that satisfies
Also, for all implies that
It follows from (2.10) and (2.11) that
By (2.12), all that remains is to show that is bounded above. That is the purpose of the following two lemmas.

Lemma 2.4. The solution of (2.10) satisfies
Moreover,

Proof. Define . Our computations show that
Alternatively, (2.15) can be written as where and are defined by
Since as , it follows from (2.17) that
Thus, it is sufficient to show that for all . We accomplish this by process of elimination.
First, is impossible due to (2.11) and (2.12). A consequence of (2.16) is that increases to whenever and decreases to whenever . In either case, if , then
Since , then (2.19) implies that
The fact that contradicts (2.7). Therefore, is impossible. This leaves as the only possibility. Consequently,
Combining this result with yields . This completes the proof of the lemma.

We now prove that for some constant .

Lemma 2.5. The solution of (2.10) satisfies for some finite constant .

Proof. Because of (2.12) it is sufficient to show that is bounded above on . For a contradiction, assume that
By (2.13) there is a value such that for all . Hence, for all which yields where .
Integrating (2.10) over gives where . Next, we use (2.23) to show that exists and is finite. Subsequently, we divide both sides of (2.24) by and let .
Combining (2.23) with the fact that for all yields
This implies that
Dividing both sides of (2.24) by and letting we obtain as a consequence of (2.13), (2.22), and (2.26). This is an obvious contradiction. Thus, there exists a finite constant such that
This concludes the proof of the lemma.

Conclusion. It follows from (2.27) that the asymptotic behavior of described in (2.2) is now proved.

The next lemma shows that exists on and that as .

Lemma 2.6. The solution of (1.8) is defined on and satisfies . Furthermore, there is a decreasing sequence , with , such that

Proof. Let denote the maximal interval of existence of . We claim that . To prove this, we make use of the functional and the region defined by
Let denote the interior of (Figure 1, upper right). Note that the constant solution . Since , a differentiation of (2.29) gives
We conclude from (2.2), (2.6), (2.14), (2.29), and (2.31) that, when ,
Therefore, is uniformly bounded on . From this and standard theory it follows that . Thus, for the solution , we conclude that and that
From (2.31) we also conclude that (1.8) has no periodic solutions. In addition, the constant solution is the only constant solution in . Thus, it follows from (2.33) and standard phase plane arguments that
Finally, we need to determine precisely how solutions approach as . For this a linearization of (1.8) around the constant solution gives
The eigenvalues associated with (2.35) are complex, with positive real parts. Thus, solutions with initial values on the curve must spiral into the constant solution as . Property (2.28) follows as a consequence. This completes the proof of the lemma.

In Lemma 2.3, the eigenvalue defined in (2.5) led to the existence of the solution of (1.8). The decay rate of as is described in Lemmas 2.32.5. The methods used to prove Lemmas 2.32.5 can be applied to the eigenvalue defined in (2.5) to result in the following lemma.

Lemma 2.7. There exists a solution of (1.8) such that(a),(b) and on ,(c), and(d) for some constant .

Description of the Unstable Manifold
As shown in the proof of Lemma 2.3, the solution is chosen so that is a point on the unstable manifold of the constant solution . As depicted in Figure 1 we let denote the component of the unstable manifold in the and quadrant. Also, we denote the component of the unstable manifold in the and component by . Precisely, where denotes the maximal interval of existence of . Noting that is also a solution of (1.8) such that we can define
In the next lemma, we continue our analysis of the solution as .

Lemma 2.8. There is an increasing, positive sequence such that

Proof. The corresponding solution of (1.4) satisfies and it follows from parts (b) and (d) of Lemma 2.7 and (2.39) that , and
It follows from standard theory that solutions that are bounded at must satisfy . Thus, for solutions of (1.8) such that , and satisfying parts (b) and (d) of Lemma 2.7, the corresponding solution of (1.4) is nonsingular and satisfies , . Haraux and Weissler [2] showed that has at least one positive zero. Chen et al. [1] proved that has infinitely many positive zeros. These results, and the fact that , imply that solutions of (1.8) satisfying have infinitely many positive zeros. Thus, there is an increasing, positive sequence , where attains a positive relative maximum when is odd, and a negative relative minimum when is even. It follows from (1.8) that
This concludes the proof of the lemma.

The following theorem summarizes our results obtained thus far. In particular, part (i) of the following theorem summarizes the results of Lemmas 2.7 and 2.8 regarding the solution . Part (ii) summarizes the results of Lemmas 2.32.6 regarding the solution .

Theorem 2.9. Let and .(i)There is a one-dimensional unstable manifold of solutions of (1.8) leading from in the phase plane. One component, , points into the positive quadrant, and its negative counterpart, , points into the negative quadrant. If , and is sufficiently small, then there is a value such that
Furthermore, there is an increasing, positive sequence such that (ii) There is a one-dimensional stable manifold of solutions leading to in the phase plane. One component, , points into the , quadrant of the phase plane, and its negative counterpart, , points into the , quadrant. If , then
 Also, there is a decreasing sequence , with , such that
 Finally, there is a constant such that

Solutions of the Equation
Below, in Theorem 2.10, we show how to combine part (ii) of Theorem 2.9 with the formula to prove the existence and asymptotic behavior of a new family of singular solutions of the equation (1.4). Our approach is to let be an arbitrarily chosen element of the continuous curve . Since , the initial conditions for the corresponding solution of (1.4) are given at and satisfy
Since , and is a continuous curve, then (2.48) generates an entire continuum of solutions of the equation. We show how these solutions intertwine with infinitely often as . In addition, we show how to prove the limiting behavior of each solution at both ends of , its maximal positive interval of existence.

Theorem 2.10 (a continuum of new singular solutions of (1.4)). Let and . Let denote the positive singular solution of (1.4) defined in (1.5), and let be a solution of (1.8) which satisfies in part (ii) of Theorem 2.9. The corresponding solution of (1.4) has initial values
Furthermore, is the maximal interval of existence of , and there is a decreasing positive sequence, , with , such that

Numerical Example
In Figure 1 we let so that . The stable manifold (third row, left panel) is generated by solution of (1.8) with . The right panel shows the corresponding solution of (1.4). For this example the asymptotic properties (2.51) become

Proof of Theorem 2.10. Let denote a solution of (1.8) which satisfies part (ii) of Theorem 2.9. The solution of (1.4) corresponding to is
It follows from (2.45) in Theorem 2.9 that the sequence defined by is positive and decreasing in , with , and
Next, it follows from (2.53), and the definition , that
Since as , and since as , it follows from (2.56) that
This proves the first part of (2.51). It remains to prove the asymptotic behavior of the solutions as . For this we combine property (2.46) in Theorem 2.9 with (2.56) and substitute to obtain
This completes the proof of Theorem 2.10.

3. The Critical Case:

In this section we investigate the behavior of solutions of (1.4) and (1.8) when and . In this case (1.4) and (1.8) become and (1.5) reduces to

The remainder of this section consists of the following.(I)Theorem 3.1 gives a complete classification of solutions of (3.2).(II)In Theorem 3.2 we show how to combine the results of Theorem 3.1 with the formula to obtain a continuum of new positive singular solutions of (3.1), and also a continuum of new sign changing singular solutions.(III)Figures 2 and 3 illustrate our results when .

296591.fig.002
Figure 2: Critical case: , . Periodic and homoclinic orbits of (3.2) proved in Theorem 3.1.
fig3
Figure 3: Critical case: , . ((a), (c), and (e)) Homoclinic and periodic orbit solutions of (3.2) proved in Theorem 3.1. ((b), (d), and (f)) The corresponding solutions of (3.1) proved in Theorem 3.2: is the known singular solution, is a new positive singular solution, and is a new sign changing singular solution.
fig4
Figure 4: Supercritical example: , . ((a) and (b)) Trajectories of the unstable manifold ( and ) and stable manifold ( and ) leading from into the plane. ((c)–(f)) The components of solutions along , , , (left) and corresponding components along and (right): is bounded at , is the known singular solution, and is the new sign changing singular solution corresponding to .

Theorem 3.1. Let and . Each solution of (3.2) satisfies where is a constant. Define .(i)If , then there are no real solutions of (3.2) which satisfy (3.4).(ii)If , then solutions of (3.2) are constant, and either or for all .(iii)If , then solutions of (3.2) are nonconstant, periodic, they have one sign, and the interior of their trajectories in the phase plane contains one of the constant solutions .(iv)If , then there is a one parameter family of solutions
Depending on the sign of , these solutions are either strictly positive or negative. Their trajectories form homoclinic orbits in the phase plane, with one of the constant solutions in their interior, and (v) If solutions of (3.2) are nonconstant, periodic, they change sign, and the interior of their trajectories in the phase plane contains all three constant solutions and .

Proof. Let . Then , hence where is a constant, and (3.4) is proved. Properties (i)–(v) follow from (3.7).

Numerical Experiments
Figure 2 illustrates homoclinic orbits solutions, and also periodic solutions, of (3.2) in the plane when . Graphs of the components of these solutions are shown in the left column of Figure 3. The corresponding solutions of the equation (3.1) are shown in the right column of Figure 3. Proofs of their existence are given below in Theorem 3.2.

Solutions of the Equation
We now show how to combine the results of Theorem 3.1 with the formula to prove the existence and asymptotic behavior of solutions of (3.1). First, part (ii) of Theorem 3.1 shows that when , then . This and (3.8) imply that the corresponding solutions of (3.1) are . Below, in Theorem 3.2, we show how parts (iii)–(v) of Theorem 3.1 generate continuous families of strictly positive solutions of (3.1), and also a family of sign changing singular solutions.

Theorem 3.2. Let and .
(a) A Continuum of Positive Nonsingular Solutions. For each (3.1) has the nonsingular solution
(b) A Continuum of Positive “Interlacing” Singular Solutions. Let be a member of the continuum of positive periodic solutions of (3.2) which satisfy part (iii) of Theorem 3.1. The corresponding solution of (3.1) has initial values and its interval of existence is . Furthermore, the solution interlaces with ; that is, there is a positive increasing sequence, , with and such that
(c) A Continuum of Sign Changing Singular Solutions. Let be a member of the family of positive, sign changing periodic solutions of (3.2) which satisfy part (v) of Theorem 3.1. The corresponding solution of (3.1) has initial values and its interval of existence is . Furthermore, the solution changes sign infinitely often as follows: there is an positive increasing sequence, , with and such that

Remarks. (i) The solutions given in (3.9) were first derived by Joseph and Lundgren [3]. (ii) To our knowledge, the singular solutions described in parts (b) and (c) have not previously been reported.

Proof of Theorem 3.2. Part (a). For each , let denote the solution given in (3.5) in Theorem 3.1. Setting in (3.5) gives
Next, substitute (3.3) and (3.16) into (3.8) and obtain
Part (b). Let be a member of the continuum of positive periodic solutions of (3.2) which satisfy part (iii) of Theorem 3.1. The trajectory of lies in the positive quadrant of the plane and surrounds the constant solution . Thus, there are values and a positive increasing sequence , such that
The solution of the equation (3.1) corresponding to is
Define . It follows from (3.18)-(3.19)-(3.20) that and , and
This proves property (3.11). It remains to prove property (3.12). For this we combine (3.19) with (3.20), and the fact that , to conclude that
It follows from (3.3) and (3.22) that
This completes the proof of property (3.12).
Part (c). Let be a member of the continuum of positive periodic solutions of (3.2) which satisfy part (v) of Theorem 3.1. The trajectory of surrounds the constant solutions and in the plane. Thus, there exists a value , and a positive increasing sequence , such that
The solution of the equation (3.1) corresponding to is
Define . It follows from (3.24)-(3.25)-(3.27) that , , and
This proves property (3.14). It remains to prove property (3.15). For this we combine (3.24) with (3.27), and the fact that , to conclude that
It follows from (3.3), (3.28), and (3.29) that
This completes the proof of property (3.15) and of Theorem 3.2.

4. The Supercritical Case:

In this section we investigate the behavior of solutions of (1.4) and (1.8) when and . The remainder of the section consists of the following.(I)Theorem 4.1 classifies the behavior of solutions of (1.8). Again, we focus on solutions whose trajectories in the phase plane form the stable and unstable manifolds of solutions associated with the constant solution . Part (ii) of Theorem 4.1 gives a detailed proof that solutions on the stable manifold form an outgoing spiral in the phase plane as decreases from . The proof is sufficiently general as to include Galaktionov’s numerical observation of large amplitude oscillations [10]. For these spiraling solutions it remains a challenging open problem to prove their asymptotic behavior at the left endpoint of their interval of existence (see Open Problem I after the statement of Theorem 4.1).(II)Theorem 4.5 shows how to combine the results of Theorem 4.1 with the formula to obtain a continuum of positive nonsingular solutions of (3.1). In addition, we prove the existence of a continuum of new sign changing, “super singular” solutions which, to our knowledge, have not previously been reported. For these sign changing solutions it remains a challenging open problem to prove their asymptotic behavior at the left endpoint of their interval of existence (see Open Problem II after the statement of Theorem 4.5).(III)Figure 4 illustrates the behavior of solutions when .

Theorem 4.1. Let and .(i)There is a one-dimensional unstable manifold of solutions of (1.8) leading from into the phase plane. One component, , points into the positive quadrant, and its negative counterpart, , points into the negative quadrant (Figure 4, upper left). If , then , and there is a constant such that (ii)There is a one-dimensional stable manifold of solutions leading to in the phase plane. One component, , points into the , quadrant of the phase plane, and its negative counterpart, , points into the , quadrant (Figure 4, upper left). Additionally, if is sufficiently small and , then there is a such that
Let . There is a negative decreasing sequence such that
Moreover, increases as increases.

Remarks. The proof of part (i) of Theorem 4.1 uses straightforward phase plane type arguments. The proof of (ii) is admittedly more technical. Our numerical experiments (Figure 4, lower left) indicate that the amplitudes of the oscillations of the solutions described in part (ii) grow without bound as decreases. It remains a challenging open problem to determine whether these solutions exist on the entire interval , or if they exist only on a semi-infinite interval of the form . These fundamental theoretical questions are summarized in the following.
Open Problem I (Super Singular Solutions). Let denote the decreasing sequence described in part (ii) of Theorem 4.1. Prove whether , or . Second, prove whether is finite or infinite. Our numerical study suggests that .

Proof of Theorem 4.1. Part (i). First, note that properties of solutions on the component of the unstable manifold leading from into the positive quadrant of the phase plane are the same as those seen in Lemma 2.7. From these properties it again follows that if , and is sufficiently small, then , and for some , where . This proves the first part of (4.1). To complete the proof of (4.1) recall that the functional and the region defined in the proof of Lemma 2.6 are
Again, let denote the interior of , and note that contains one constant solution, . A differentiation of (4.5) gives
Thus, if , and is sufficiently small, we conclude from (4.4)–(4.7) (Figure 4, upper right) that exists for all , and
Therefore,
From (4.7) we conclude that (1.8) has no periodic solutions. Also, a linearization of (1.8) around the constant solution shows that is an asymptotically stable equilibrium point in the phase plane. Thus, if is a solution of (1.8) with initial condition , it follows from (4.7)–(4.9), and standard phase plane arguments, that
Part (ii). It follows from (2.5) that there is a one-dimensional stable manifold of solutions leading to in the phase plane. As in Theorem 2.9, one component, , points into the quadrant , , and along solutions satisfy where . Thus, if is a solution of (1.8) with sufficiently small and , there is a such that
Along this solution (Figure 4, upper right) satisfies
To complete the proof of (ii) let be the unique positive value where . We need to prove that rotates counterclockwise around and that generates an outwardly growing spiral as decreases. For this we show that there is a decreasing sequence of negative values such that