In this paper, we investigate the structure and properties of the set of positive blow-up solutions of the differential equation , , where . The differential equation is studied together with the boundary conditions , . We specify conditions for the data function h which guarantee that the set of all positive solutions to the above boundary value problem is nonempty. Further properties of the solutions are discussed and results of numerical simulations are presented.
MSC: 34B18, 34B16, 34A12.
Keywords:singular ordinary differential equation of the second order; time singularities; blow-up, positive solutions; existence of solutions; polynomial collocation
In this paper, we investigate the structure and properties of the set of positive blow-up solutions of the differential equation
Models in the form of (1) arise in many applications. Among others, they occur in the study of phase transitions of Van der Waals fluids [1-3], in population genetics, where they characterize the spatial distribution of the genetic composition of a population [4,5], in the homogeneous nucleation theory , in relativistic cosmology for particles which can be treated as domains in the universe , and in the nonlinear field theory, in particular, in context of bubbles generated by scalar fields of the Higgs type in Minkowski spaces .
Here, we assume that h is positive and satisfies the Carathéodory conditions on . We define a positive solution of (1) as a function v which satisfies (1) for a.e. , is positive on , and has absolutely continuous first derivative on each compact subinterval in .
According to Lemma 4, if v is a positive solution of (1), then either
In the literature, bounded solutions of (1) have been widely investigated; see e.g.[9-13]. Such solutions are characterized by the initial condition . In contrast to this, some real problems lead to the investigation of unbounded solutions which are characterized by the condition for some and which are called blow-up solutions. We refer to [14-16]. Here, we are interested in blow-up solutions of (1), where . In particular, (1) will be considered together with the boundary conditions
Our main goal is to find conditions for the data function h in (1), which guarantee that the set is nonempty for each and then to investigate the properties of this set. For example, we prove that the difference of any two functions in , , retains its sign on , and that there exist minimal and maximal solutions for each , cf. Theorem 5. If the interior of the set is nonempty, we show that this interior is fully covered by ordered graphs of other functions belonging to for each ; see Theorem 6. Finally, in Theorem 7, the existence of a positive constant such that is shown and all functions v from ℛ are uniquely characterized by the condition
Similarly, means the Banach space of functions having a continuous first derivative on with the corresponding maximum norm . By we denote the set of functions which are Lebesgue integrable on . Moreover, is the set of functions whose first derivative is absolutely continuous on , while is the set of functions having absolutely continuous first derivative on each compact subinterval of . We say that satisfies the Carathéodory conditions on , if the following three conditions hold:
Structure of the paper
The paper is organized as follows. In Section 2 we discuss properties of the solutions of the auxiliary Dirichlet problem (3), (4). We recapitulate previous results from  and also present new results in Theorems 1, 2, and 3. The main results of the paper can be found in Section 3, where we describe a relation between solutions of problem (3), (4) and blow-up solutions of problem (1), (2); see Theorem 4. Using this relation and the results of Section 2, we obtain various interesting properties of blow-up solutions; see Theorems 5 to 9. Section 4 contains three examples illustrating the theoretical findings. Final remarks and open problems are formulated in Section 5.
2 Auxiliary results
In this section we consider the auxiliary singular differential equation,
We now study (3) subject to the boundary conditions
and require that
holds. We call a function a positive solution of the Dirichlet problem (3), (4) if , on , u satisfies the boundary conditions (4), and (3) holds for a.e. . Clearly, for each positive solution u of (3), (4), there exists such that (5) is satisfied.
In the following lemma we cite those results from  which will be used in the analysis of problem (1), (2).
Lemma 1Let (H1)-(H3) hold. Then the following statements hold:
We now formulate new results which complete those from . We first establish a relation between and the set if its interior is nonempty. This question was a short time ago an open problem [, Remark 4.4]. We note that the relation between and the set with having nonempty interior is described in Lemma 1(d).
ProofStep 1. Auxiliary Dirichlet problem.
We claim that there exists a solution v to problem (3), (7) such that
We now consider the auxiliary differential equation
for a.e. and all , . Since and for , , , and , solve (3) on , we conclude that and are lower and upper functions of problem (3), (7) (see e.g. or ). This fact together with (11) implies the existence of a solution v to problem (3), (7) satisfying (8), cf. [, Lemma 3.7]. Moreover,
From (8) and (9) we conclude that
By the Gronwall lemma,
Step 2. Continuation of the solutionv.
Let u be a solution of problem (3), (7) on an interval J which is a left-continuation of v. Let us assume that u is not continuable. Let . Then and (12) with v replaced by u holds for a.e. . The integration now yields
and we claim that
holds. This is a contradiction.
In the next corollary we extend the statement (d) from Lemma 1 to a large set of A values.
Proof The result follows immediately from Lemma 1(d) and Theorem 1. □
By Lemma 1(b), (c) we know that functions from are uniquely determined by the values −c of their derivatives at the right end point only in the case that is a singleton set for each . Since we cannot uniquely determine all functions from via their derivatives at , see Lemma 1(e), we discuss their derivatives at the singular point .
and (15) follows. □
Using integration by parts we obtain
On the other hand, it follows from Corollary 2 that
Combining the above two equalities yields the result. □
Hence, by Lemma 2,
It remains to prove that . Assume that the equality does not hold. Then, from the structure of bounded and closed subsets of ℝ the existence of an open interval , , follows. Let and , where . Due to Lemma 1(c), there exists such that on , on . Choose . By Corollary 1, there exists such that . Then on and on . Consequently, on , that is, , which is not possible. □
Proof We first show
It follows from Lemma 3 that
Proof Let . Then because for by Corollary 4. Since and is compact in by Lemma 1(f), the sequence is relatively compact in . Let be a subsequence of which is convergent in , and let . Then for a and . Therefore, and hence any subsequence of converging in has the same limit . Consequently, is convergent in and is its limit. □
3 Blow-up solutions and their properties
Proof By Corollary 3.5 in , if , then v satisfies (19). Now, assume that
Then v has to satisfy
because otherwise, if (22) was not true, then , contradicting (21). Since h is positive, (1) indicates that the function is increasing on and hence there exists . The next part of the proof is divided into three cases.
(i) Let us assume that
which is a contradiction to (22).
(ii) Assume that
By integration, we obtain from (24)
which again is a contradiction.
(iii) Assume that
Now, we investigate the existence and properties of blow-up solutions of (1), for the case that the function h has the form
In particular, we study the equation
Together with (27) we consider the conditions
Define a set by
Moreover, let us define sets
we can rewrite (27) and obtain the form
We now introduce a function f,
Under conditions ()-(), the function f satisfies (H1)-(H3); see the proof of [, Theorem 5.1]. Therefore, the results of Section 2 hold for problem (34), (4). As in Section 2, we define the sets and for solutions of (34).
The following result is the key-stone to the analysis of the structure of the set ℛ and describes the relation between the sets and ℛ.
We now argue as in the proof of [, Lemma 3.3] and arrive at
for . Since u is bounded on , we have . Then, by [, Lemma 2.1],
for , and hence, . Now, [, Corollary 1] guarantees that u can be extended on with such that the equality (35) holds for . Consequently, .
Now, we are in the position to provide results on the solvability of problem (27), (28) and formulate the properties of its solutions.
Proof The result follows by combining results from Lemma 1(a), (b), (c), and (e) with those from Theorem 4. □
The next theorem shows that the set
is covered by graphs of the functions from ℛ.
Proof Choose and . Define . Then, using (33) and (36), we deduce . Therefore, by Corollary 1, there exists such that . Consequently, (33) and Theorem 4 give and . The last statement follows from Theorem 5(c). □
and therefore the second condition in (39) holds.
(b) Choose . Theorem 2 guarantees that there exists a unique satisfying . Using (33) and Theorem 4, we conclude . Then v satisfies (40) and (41) which results in (39). It remains to prove that v is unique. Assume that there exists a function , , such that w satisfies (39). Let . Then and . Consequently, by Theorem 2, we arrive at and , which is a contradiction. □
Proof According to (42), there exist , , and such that , for . Also, and, since , it follows that in a right neighborhood of . Therefore, there exists such that and hence, . (Note that if , then Theorem 5(b) yields for , which is not possible.) The result now follows from Theorem 5(b) for and from Theorem 5(c) for . □
Let and . Let be countable. Then there exists a decreasing subsequence of . By Theorem 8, the sequence is not increasing on and on this interval. Using (33) and (42), we obtain , . Since in it follows from Theorem 3, note that , that and that for . This fact together with the monotonicity of and gives in , which contradicts (43). Hence in . □
According to (33), (42), and Theorem 4 we have
Let us denote
Consider a sequence . Then there exists a sequence with . By Lemma 1(f), the set is compact in . Therefore, there exists a subsequence which converges in to . In particular, . Let us denote , then we have . This yields, by Corollary 6, in and . Thus we showed that for any sequence in , there exists a uniformly converging subsequence on with a limit in . □
Using the open domain MATLAB Code bvpsuite, we numerically simulate three model problems in order to illustrate theoretical statements made above.
4.1 MATLAB Code bvpsuite
The MATLABTM software package bvpsuite is designed to solve BVPs in ODEs and differential algebraic equations. The solver routine is based on a class of collocation methods whose orders may vary from 2 to 8. Collocation has been investigated in context of singular differential equations of first and second order in [24,25], respectively. This method could be shown to be robust with respect to singularities in time and retains its high convergence order in case that the analytical solution is appropriately smooth. The code also provides an asymptotically correct estimate for the global error of the numerical approximation. To enhance the efficiency of the method, a mesh adaptation strategy is implemented, which attempts to choose grids related to the solution behavior, in such a way that the tolerance is satisfied with the least possible effort. The error estimate procedure and the mesh adaptation work dependably provided that the solution of the problem and its global error are appropriately smooth.b The code and the manual can be downloaded from http://www.math.tuwien.ac.at/~ewa webcite. For further information see . This software proved useful for the approximation of numerous singular BVPs important for applications; see e.g.[9,27-29].
Example 1 The first example is used to comment on Theorem 5 and to show that if a function in (27) is increasing in x, the graphs of positive solutions of problem (27), (28) cannot intersect. Here, we choose
and thus problem (27), (28) becomes
Using substitution (33), which has the form
we transform (48) onto the form, cf. (34),
and we solve the terminal value problem given by
where , and 4. Then, using (49), we recalculate these solution to their blow-up form and obtain different positive solutions v of problem (48). Note that their graphs are ordered after c, cf. Figure 1.
Figure 1. Example 1. Solutions of (50) (first) and of (48) (second) for different values of c, orange, green, red, dark blue, magenta. We used a collocation method of order 8 based on Gaussian points and grid adaptation strategy to satisfy the tolerances .
Then problem (27), (28) has the form
Using substitution (49), we rewrite (51) and have
The resulting terminal value problem reads
where , and 4. Then, using (49), we obtain different positive blow-up solutions of problem (51). However, their graphs are again ordered; see Figure 2. Hence, condition () seems not to be necessary for statement (b) of Theorem 5 to hold. On the other hand, the question, if another model with a decreasing nonlinearity g exists, where solutions cross each other and are not ordered after c, remains open.
Figure 2. Example 2. Solutions of (52) (first) and of (51) (second) for different values of c, orange, green, red, dark blue, magenta. We used a collocation method of order 8 based on Gaussian points and grid adaptation strategy to satisfy the tolerances .
then problem (27), (28) has the form
Using (49), we again transform (53) and solve the terminal value problem
where , and 4. Then, using (49), we calculate the related positive blow-up solutions of problem (53). As can be seen in Figure 3, their graphs are again ordered after c and the question of solution crossings remains open even if we consider oscillating nonlinearities.
Figure 3. Example 3. Solutions of (54) (first) and of (53) (second) for different values of c, orange, green, red, dark blue, magenta. We used a collocation method of order 8 based on Gaussian points and grid adaptation strategy to satisfy the tolerances .
In Figure 4, solutions of (54) and of (55), together with the final grids are shown on the interval (left) and (right). Due to the different smoothness of u and v, their accuracy strongly varies. While the absolute and relative errors of u (or rather their estimates) on the grid with 52 grid points is 10−13, the absolute and relative errors of v on the grid with exactly the same number of points is only 10−9. Note that the code has automatically adapted the location of the grid points to correctly reflect the solution behavior.
Figure 4. Example 3,. Solutions of (54) and of (55), shown on the interval (first) and (second). We used a collocation method of order 8 based on Gaussian points and grid adaptation strategy.
We have described the set ℛ of all positive solutions of problem (1), (2), where h has the form and assumptions ()-() hold. By Theorem 5 we know that ℛ is nonempty and that for each there exists at least one function fulfilling . In addition, graphs of functions from ℛ do not intersect and ℛ has a minimal element .
If we choose an arbitrary and denote , then there exists a maximal element in . Clearly and . According to Theorem 6, the interior of the set is fully covered by graphs of functions from ℛ. Finally, we deduce from Theorems 7-9 that the set is compact in .
Example 1 illustrates the results of Section 3 and hence, according to (), we have chosen the increasing function in (48). Figure 1 shows ordered graphs of solutions.
In contrast to this, in Example 2 and Example 3, we have chosen the decreasing function in (51) and the non-monotonous function in (53), respectively. We can see on Figure 2 and Figure 3 that the graphs of solutions are ordered in both cases. But to prove such order of solutions for non-increasing g is an open problem. On the other hand, the question of the construction of problem (27), (28) whose positive solutions cross each other remains open, as well.
The authors declare that they have no competing interests.
The authors read and approved the final draft. IR and SS contributed to the analytical part of the paper and JG and EW contributed to the numerical part of the paper.
This research was supported by the grants IGA PrF_2013_013 and IGA-PrF_2014028. The authors are grateful to the referees for useful comments and suggestions which improved the paper.
Gouin, H, Rotoli, H: An analytical approximation of density profile and surface tension of microscopic bubbles for van der Waals fluids. Mech. Res. Commun.. 24, 255–260 (1997). Publisher Full Text
Fischer, RA: The wave of advance of advantageous genes. Ann. Eugenics. 7, 355–369 (1937). Publisher Full Text
Kitzhofer, G, Koch, O, Lima, P, Weinmüller, EB: Efficient numerical solution of the density profile equation in hydrodynamics. J. Sci. Comput.. 32, 411–424 (2007). Publisher Full Text
Lima, PM, Chemetov, NV, Konyukhova, NB, Sukov, AI: Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems. J. Comput. Appl. Math.. 189, 260–273 (2006). Publisher Full Text
Rachůnková, I, Koch, O, Pulverer, G, Weinmüller, EB: On a singular boundary value problem arising in the theory of shallow membrane caps. J. Math. Anal. Appl.. 332, 523–541 (2007). Publisher Full Text
Rachůnková, I, Staněk, S, Weinmüller, EB, Zenz, M: Neumann problems with time singularities. Comput. Math. Appl.. 60, 722–733 (2010). Publisher Full Text
Rachůnková, I, Tomeček, J: Bubble-type solutions of nonlinear singular problems. Math. Comput. Model.. 51, 658–669 (2010). Publisher Full Text
Tanigawa, T: On the structure of positive solutions of a class of fourth order nonlinear differential equations. Ann. Mat. Pura Appl.. 185, 521–536 (2006). Publisher Full Text
Rachůnková, I, Staněk, S: Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities. Cent. Eur. J. Math.. 11, 112–132 (2013). Publisher Full Text
Lomtatidze, A, Malaguti, L: On a two-point boundary value problem for the second order differential equations with singularities. Nonlinear Anal.. 52, 1553–1567 (2003). Publisher Full Text
de Hoog, F, Weiss, R: Collocation methods for singular boundary value problems. SIAM J. Numer. Anal.. 15, 198–217 (1978). Publisher Full Text
Weinmüller, EB: Collocation for singular boundary value problems of second order. SIAM J. Numer. Anal.. 23, 1062–1095 (1986). Publisher Full Text
Budd, C, Koch, O, Weinmüller, EB: From nonlinear PDEs to singular ODEs. Appl. Numer. Math.. 56, 413–422 (2006). Publisher Full Text
Rachůnková, I, Pulverer, G, Weinmüller, EB: A unified approach to singular problems arising in the membrane theory. Appl. Math.. 55, 47–75 (2010). Publisher Full Text