### Abstract

In this paper, we investigate the structure and properties of the set of positive
blow-up solutions of the differential equation
*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### 1 Introduction

In this paper, we investigate the structure and properties of the set of positive blow-up solutions of the differential equation

where

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 [6], in relativistic cosmology for particles which can be treated as domains in the universe [7], and in the nonlinear field theory, in particular, in context of bubbles generated by scalar fields of the Higgs type in Minkowski spaces [8].

Here, we assume that *h* is positive and satisfies the Carathéodory conditions on
*positive solution of* (1) as a function *v* which satisfies (1) for a.e.

According to Lemma 4, if *v* is a positive solution of (1), then either

or

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 this case, we speak about a *positive solution of problem (**1**), (**2**).* Let us denote by the set of all positive solutions to (1), (2). Moreover, let

Our main goal is to find conditions for the data function *h* in (1), which guarantee that the set
*cf.* Theorem 5. If the interior of the set
*v* from ℛ are uniquely characterized by the condition

If we denote such *v* by
*cf.* Theorem 8, and that for each

The study of a structure of positive solutions to other types of ordinary differential equations can be found for example in [17-19].

#### Notation

Let us denote by

Similarly,

(i) The function

(ii) The function

(iii) For each compact set

For functions satisfying above conditions, we use the notation

#### 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 [20] 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,

where
*f* satisfies the following conditions:

(H_{1})

(H_{2})

(H_{3})
*x* for a.e.

We now study (3) subject to the boundary conditions

and require that

holds. We call a function
*positive solution of the Dirichlet problem* (3), (4) if
*u* satisfies the boundary conditions (4), and (3) holds for a.e.
*u* of (3), (4), there exists

We now denote by the set of all positive solutions of problem (3), (4), and

In the following lemma we cite those results from [20] which will be used in the analysis of problem (1), (2).

**Lemma 1***Let* (H_{1})-(H_{3}) *hold*. *Then the following statements hold*:

(a) *For each*
*the set*
*is nonempty and there exist functions*
*such that*
*for*
*and*

(b) *If*
*then*
*for*

(c) *If*
*and*
*for some*
*then either*
*for*
*or there exists*
*such that*
*for*
*and*
*for*

(d) *For each*
*and each*
*there exists*
*satisfying*

(e)
*is a one*-*point set for each*
*where*
*is at most countable*.

(f) *For each*
*the set*
*is compact in*

(g) *If*
*then*
*if and only if it is a solution of the equation*

*in the set*

We now formulate new results which complete those from [20]. We first establish a relation between

**Theorem 1***Let* (H_{1})-(H_{3}) *hold*. *Let us assume that there exists*
*such that*
*Then for any*
*there exists*
*satisfying*

*Proof**Step* 1. *Auxiliary Dirichlet problem*.

Choose

We claim that there exists a solution *v* to problem (3), (7) such that

We show this result utilizing the method of lower and upper functions. It follows
from (H_{1}) that there exists

Let

and let

We now consider the auxiliary differential equation

It is not difficult to verify that
*cf.* (9),

for a.e.
*e.g.*[21] or [22]). This fact together with (11) implies the existence of a solution *v* to problem (3), (7) satisfying (8), *cf.* [[21], Lemma 3.7]. Moreover,

and taking the limit
^{a}

We now prove that

From (8) and (9) we conclude that

Since

we have

In particular,

By the Gronwall lemma,

Therefore

which contradicts

Consequently,
*v* is a solution of problem (3), (7).

*Step* 2. *Continuation of the solution**v*.

It follows from the arguments given in Step 1 that *v* is a solution of problem (3), (7) on

is satisfied for a.e.

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
*v* replaced by *u* holds for a.e.

and we claim that

We show inequality (14) indirectly. Let us assume that (14) does not hold. Then there
exists

and either

Since, by (H_{3}),

which is not possible. The case

Suppose that
*u* is bounded on

over

Since

Using the Gronwall lemma, we deduce that for

holds. This is a contradiction.

Therefore

In the next corollary we extend the statement (d) from Lemma 1 to a large set of *A* values.

**Corollary 1***For each*
*and each*
*there exists*
*satisfying*

*Proof* The result follows immediately from Lemma 1(d) and Theorem 1. □

**Remark 1** Corollary 1 says that the set

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

**Lemma 2***Let* (H_{1})-(H_{3}) *hold*. *Let*
*Then*

*Proof* It follows from Lemma 1(g) that (6) holds for

and (15) follows. □

**Corollary 2***Let*
*Then*

*Proof* The result follows from Lemma 2, since

**Corollary 3***Let*
*Then*

*Proof* Since

which together with Corollary 2 gives

**Corollary 4***Let*
*Then*

*In particular*,

*Proof* Inequality (16) follows from Lemma 2 and the fact that
_{2}). □

**Corollary 5***Let*
*Then*

*Proof* The integration of (3) over

Using integration by parts we obtain

and, therefore,

Taking the limit

On the other hand, it follows from Corollary 2 that

Combining the above two equalities yields the result. □

**Lemma 3***Let* (H_{1})-(H_{3}) *hold*. *Let*
*and*
*be not a singleton set*. *Then for each*
*there exists a unique*
*such that*
*Consequently*,

*Proof* Let
_{3}) that

Hence, by Lemma 2,

Since

It remains to prove that

Since functions from belong to

Lemma 3 implies that functions from can be uniquely determined by the values of their derivatives at the singular point

**Theorem 2***Let* (H_{1})-(H_{3}) *hold*. *Then there exists a unique*
*satisfying*
*if and only if*

*Proof* We first show

It follows from Lemma 3 that

for each

Let us now choose
*u* follows from Corollary 3. □

For

**Theorem 3***Let* (H_{1})-(H_{3}) *hold*. *Assume that*
*is a convergent sequence and*
*Then*
*in*

*Proof* Let

### 3 Blow-up solutions and their properties

This section contains the main results of the paper. First, we present a lemma which
describes how positive solutions of (1) may behave at the singular point

**Lemma 4***Let*
*and let the function*
*be positive*. *Let**v**be a positive solution of* (1). *Then either*

*or*

*Proof* By Corollary 3.5 in [12], if
*v* satisfies (19). Now, assume that

Then *v* has to satisfy

because otherwise, if (22) was not true, then
*h* is positive, (1) indicates that the function

(i) Let us assume that

Since
*v* is increasing on

which is a contradiction to (22).

(ii) Assume that

Then there exists

Hence, *v* is increasing on

By integration, we obtain from (24)

and, by virtue of

which again is a contradiction.

(iii) Assume that

Then there exists

Hence, *v* is decreasing on

and, since

In addition (25) yields

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

where
*ψ*, *g* satisfy the conditions

(

(
*x* for a.e.

(
*ϕ* is increasing and

Together with (27) we consider the conditions

and

We define a *positive solution to problem* (27), (28) as a function
*v* satisfies the boundary conditions (28), and (27) holds for a.e.

Define a set by

Clearly, for each

**Lemma 5***Let* (
*hold*. *Let*
*Then*
*on*

*Proof* Since

Assume that

which is not possible, since

Moreover, let us define sets

and

It is obvious that

we can rewrite (27) and obtain the form

We now introduce a function *f*,

Under conditions (
*f* satisfies (H_{1})-(H_{3}); see the proof of [[20], Theorem 5.1]. Therefore, the results of Section 2 hold for problem (34), (4). As
in Section 2, we define the sets and

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.

**Theorem 4***Let* (
*hold and*
*Let us assume that* (33) *holds*. *Then*
*if and only if*
*and*

*Proof* (⇒) Let
*u* is bounded on

that

Hence,

We now argue as in the proof of [[20], Lemma 3.3] and arrive at

for
*u* is bounded on

for
*u* can be extended on

(⇐) Let

yields

Since

Hence, note that

Now, we are in the position to provide results on the solvability of problem (27), (28) and formulate the properties of its solutions.

**Theorem 5***Let* (
*hold*. *Then the following statements hold*:

(a) *For each*
*the set*
*is nonempty and there exist*
*such that*
*for*
*and*

(b) *If*
*then*
*for*

(c) *If*
*and*
*for some*
*then either*
*for*
*or there exists*
*such that*
*for*
*and*
*for*

(d)
*is a singleton set for each*
*where*
*is at most countable*.

*Proof* The result follows by combining results from Lemma 1(a), (b), (c), and (e) with those
from Theorem 4. □

If

The next theorem shows that the set

is covered by graphs of the functions from ℛ.

**Theorem 6***Let* (
*hold*. *Then*, *for each*
*and each*
*there exists*
*satisfying*
*In particular*, *if for some*
*and*
*the inequality*
*holds*, *then for each*
*there exist*
*satisfying*

*Proof* Choose

By Corollary 4, there exists

and hence, by (33) and (36) with

Using constants from the interval

**Theorem 7***Let* (
*hold and let*
*be as in* (38). *Then the following statements hold*:

(a) *For each*
*there exists*
*such that*

(b) *For each*
*there exists a unique*
*satisfying* (39).

*Proof* (a) Choose

By Lemma 1(c),

and therefore the second condition in (39) holds.

(b) Choose
*v* satisfies (40) and (41) which results in (39). It remains to prove that *v* is unique. Assume that there exists a function
*w* satisfies (39). Let

Let
*v* by

Then

**Theorem 8***Let* (
*hold and let*
*be from* (38). *Let*
*and*
*Then either*
*for*
*or there exists*
*such that*
*for*
*and*
*for*

*Proof* According to (42), there exist

**Corollary 6***Let*
*be a convergent sequence and let us denote*
*Then*
*in*

*Proof* The proof is indirect. Assume that the statement of the corollary does not hold.
Then there exist *ε*,

Let