International Journal of Differential Equations

Volume 2012 (2012), Article ID 296591, 33 pages

doi:10.1155/2012/296591

Research Article

Radially Symmetric Solutions of

William C. Troy¹ and Edward P. Krisner²

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 [1–6].

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 [1–6]. 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).

(a)

(b)

(c)

(d)

(e)

(f)

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 2.3–2.5. The methods used to prove Lemmas 2.3–2.5 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 2.3–2.6 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 .

Figure 2: Critical case: , . Periodic and homoclinic orbits of (3.2) proved in Theorem 3.1.

(a)

(b)

(c)

(d)

(e)

(f)

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.

(a)

(b)

(c)

(d)

(e)

(f)

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

and that

The proof of (4.15)-(4.16) is in two steps. First, we prove three technical results, Lemmas 4.2–4.4. Secondly, we use these lemmas to follow as decreases. The proofs of Lemmas 4.2 and 4.4 are straightforward. The proof of Lemma 4.3 is admittedly technical.

Lemma 4.2. Let . (a) If a solution of (1.8) satisfies

then there is a such that

(b) If a solution of (1.8) satisfies

then there is a such that

Proof. (a) Let . Since (4.7) implies that , then increases monotonically as decreases. It follows from the definition of in (4.5) that for as long as . An integration of shows that at some first . This proves (4.18). The proof of part (b) is the same and is omitted.

Lemma 4.3. Let . (a) If a solution of (1.8) satisfies

then there is a such that

(b) If a solution of (1.8) satisfies

then there is a such that

Proof. (a) The first step is to assume, for contradiction, that does not exist and that

where denotes the maximal left interval of existence of the solution. We claim that . Suppose that . Then standard theory implies that must be unbounded on . To show that this cannot happen we use the function . Then satisfies

since , and . Integrating (4.26) gives

From (4.27) it follows that is bounded on , contradicting the previous conclusion that is unbounded on . We conclude that . Therefore, (4.25) becomes

Next, to obtain a contradiction of (4.28) we need to prove two technical properties. The first is that

To prove the first part of (4.29), differentiate (1.8) and get

for as long as . If there is an , where , then an integration of (4.30) gives

An integration of (4.31) gives for , contradicting (4.28). We conclude that , and the first part of (4.29) is proved. In turn, this implies that

and the proof of (4.29) is complete. The second property we need is

where satisfies the ODE

Property (4.33) follows immediately from (4.34), since (4.34) implies that whenever .

We now show how to use properties (4.29) and (4.33) to obtain a contradiction of (4.28). First, an integration of (4.34) gives

Integrating both sides of (4.35) from to , we obtain

where

Our goal is to prove that

Once we prove these properties, we combine (4.39) with (4.36) and conclude that when . Since , this implies that when , which contradicts (4.28).

To prove the first part of (4.39), evaluate the right side of (4.37) and get

since and . To prove the second part of (4.39), recall from (4.32) that . Thus, since and , we conclude that

Next, it follows from (4.33) that . This, (4.41), and the facts that and as allow us to apply L’Hospital’s Rule to . This gives

This proves part (a). The proof of part (b) is the same and is omitted.

Lemma 4.4. Let . (a) If a solution of (1.8) satisfies

then there is a such that

(b) If a solution of (1.8) satisfies

then there is a such that

Proof. (a) Since , it follows from (1.8) that . Thus, there is an such that

Since , a calculation shows that

Recall from (4.7) that . Thus, , and (4.48) implies that

for as long as . Integrating shows that there is a such that

This proves part (a). The proof of part (b) is the same and is omitted.

We now return to the proof of Theorem 4.1. It remains to prove that rotates counterclockwise around in the plane as decreases from . To accomplish this we use Lemmas 4.2–4.4 to show that passes infinitely often through the sets

Recall that and that (4.12) is satisfied, consequently . Lemma 4.2 implies that there is a first such that , that is, and . This and Lemma 4.3 imply that there is a such that , , and . Thus, . It follows from Lemma 4.4 that there is a such that and , hence . This and part (b) of Lemma 4.2 imply that there is a such that and , hence . This and part (b) of Lemma 4.3 imply that there is a such that , , and , hence . This and part (b) of Lemma 4.4 imply that there is a such that and , hence . We have shown how passes sequentially through as decreases. Since is contained in it follows from a repetition of the steps given above, and mathematical induction, that passes sequentially through infinitely often as decreases from . This produces a decreasing sequence where ,

Finally, since increases as decreases, it follows that increases as increases. This completes the proof of Theorem 4.1.

Solutions of the Equation

Below, in Theorem 4.5, we show how to combine part (ii) of Theorem 4.1 with the formula

to prove the existence and asymptotic behavior of families of nonsingular and singular solutions of the equation (1.4). In particular, in part (b) of Theorem 4.5 we show how a new family of “super singular” solutions is generated. Open Problem II stated after Theorem 4.5 describes important, and as yet unproven, properties of this continuum of singular solutions.

Theorem 4.5. Let and .

(a) A Continuum of Positive Nonsingular Solutions. Let denote a solution of (1.8) which satisfies in part (i) of Theorem 4.1. The corresponding solution of (1.4) is strictly positive and satisfies

Furthermore, its interval of existence is ,

(b) A Continuum of Sign Changing Singular Solutions. Let be a member of the family of “spiraling” solutions of (3.2) which satisfy in part (ii) of Theorem 4.1, and let denote the decreasing sequence of values which satisfy property (4.3). The corresponding solution of (3.1) satisfies

Its interval of existence is of the form . As , faster than . That is, there exists such that.

As decreases from , changes sign infinitely often. That is, along the decreasing sequence , one has ,

Open Problem II (Super Singular Solutions). Let denote the decreasing sequence in part (b), which satisfies . Prove whether or . Secondly, prove whether is finite or infinite. Our numerical experiments suggest that . As in Open Problem I, our analytical methods have not allowed us to resolve these fundamental theoretical issues.

Proof of Theorem 4.5. Part (a). Let be a solution of (1.8) satisfying in part (i) of Theorem 4.1. The corresponding solution of (1.4) is

where we recall that

Because , the initial point for translates to for . This, (4.59) and (4.60) give (4.54). Next, it follows from property (4.1) and the fact that that there exists such that

We conclude from (4.59), (4.60), and the first part of (4.61) that

It then follows from standard theory that . This agrees with a result of Haraux and Weissler (see Theorem  4 in [2]). Finally, we conclude from (4.59), (4.60), and the second part of (4.61) that

This completes the proof of part (a).

Part (b). Let be a solution of (1.8) satisfying in part (ii) of Theorem 4.1. The corresponding solution of (1.4) is

It follows from (4.2) that there exists such that

Combining (4.60), (4.64), and (4.65) gives

This proves the first part of property (4.57). Next, let denote a sequence of values satisfying property (4.3) in Theorem 4.1. Then decreases as increases, with , and

Define . Setting in (4.67) gives

Finally, we combine with (4.64) and (4.68) and obtain

Since , this completes the proof of (4.58) and of Theorem 4.5.

5. Conclusions

In this paper we have developed an analytical method to classify the behavior of radially symmetric, time-independent solutions of the nonlinear heat equation (1.2). These solutions satisfy the ODE (1.4). We have studied solutions which remain strictly positive on their entire intervals of existence, and also solutions which change sign. There have been few analyses in the literature of sign changing solutions. Our analytical method follows a three-step approach:

Step 1. Transform the nonautonomous equation (1.4) into the autonomous equation (1.8).

Step 2. Analyze (1.8) using phase plane methods.

Step 3. Use the inverse transformation

to translate results for (1.8) into new results for the equation (1.4).

Our Advance

This approach has allowed us to prove the existence and asymptotic behavior of several new families of solutions of (1.4). In particular, we mention two important classes of solutions which, to our knowledge, have not previously been reported.

(I)When and , we proved (see part (ii) of Theorem 2.10) the existence and asymptotic behavior of a continuum of positive, singular solutions which “interlace” with the known singular solutions .(II)When and , we proved (see part (ii) of Theorem 4.1) the existence of sign changing solutions of the equation which form outward spirals in the phase plane. These solutions transform, by means of (5.1), into “super singular” sign changing solutions of the equation (1.4). Open Problems I and II (see Section 4) summarize important issues for these solutions which have not yet been resolved.

Below, we describe challenging problems for further research.

Open Problem III. When , do the new positive singular solutions, which interlace with , play an important role similar to that of (e.g., see [5]) in proving the large time behavior of solutions of the time-dependent problem (1.2)?

Open Problem IV. Equation (1.1) is a canonical model for the general equation

where is positive and superlinear [1–5]. A natural extension of our investigation is to use our new singular solutions of (1.4) as a guide in analyzing (5.2) for the existence of new classes of solutions.

Open Problem V. Gazzola and Grunau [13] investigate the behavior of solutions of the biharmonic equation

This equation has the singular solution , . It is hoped that our approach can be used to look for new classes of solutions of (5.3).

Open Problem VI. Develop a comparison technique which allows one to use the new singular solution to establish blowup of solutions of the full time-dependent PDE.

References

S. Chen, W. R. Derrick, and J. A. Cima, “Positive and oscillatory radial solutions of semilinear elliptic equations,” Journal of Applied Mathematics and Stochastic Analysis, vol. 10, no. 1, pp. 95–108, 1997.

A. Haraux and F. B. Weissler, “Nonuniqueness for a semilinear initial value problem,” Indiana University Mathematics Journal, vol. 31, no. 2, pp. 167–189, 1982.

D. D. Joseph and T. S. Lundgren, “Quasilinear Dirichlet problems driven by positive sources,” Archive for Rational Mechanics and Analysis, vol. 49, pp. 241–269, 1972/73.

W. M. Ni and J. Serrin, “Existence and nonexistence theorems for ground states of quasilinear partial differential equations: the anomalous case,” Accademia Nazionale dei Lincei, vol. 77, pp. 231–257, 1986.

P. Souplet and F. B. Weissler, “Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state,” Annales de l'Institut Henri Poincaré, vol. 20, no. 2, pp. 213–235, 2003.

F. Merle and H. Zag, “Blowup estimates for nonlinear heat equations,” Methods and Applications of Analysis, vol. 8, pp. 551–556, 2001.

V. A. Galaktionov and J. L. Vazquez, “Continuation of blowup solutions of nonlinear heat equations in several space dimensions,” Communications on Pure and Applied Mathematics, vol. 50, no. 1, pp. 1–67, 1997.

S. Chen and W. R. Derrick, “Global existence and blow-up of solutions for a semilinear parabolic system,” The Rocky Mountain Journal of Mathematics, vol. 29, no. 2, pp. 449–457, 1999.

L. A. Caffarelli, B. Gidas, and J. Spruck, “Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,” Communications on Pure and Applied Mathematics, vol. 42, no. 3, pp. 271–297, 1989.

V. A. Galaktionov, “On blow-up “twistors” for the Navier Stokes equations in R₃: a view from reaction-diffusion theory,” http://arxiv.org/abs/0901.4286

X. Wang, “On the Cauchy problem for reaction-diffusion equations,” Transactions of the American Mathematical Society, vol. 337, no. 2, pp. 549–590, 1993.

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, NY, USA, 1955.

F. Gazzola and H.-C. Grunau, “Radial entire solutions for supercritical biharmonic equations,” Mathematische Annalen, vol. 334, no. 4, pp. 905–936, 2006.