Positive blow-up solutions of nonlinear models from real world dynamics

In this paper, we investigate the structure and properties of the set of positive blow-up solutions of the differential equation {(t^k{v}^{\prime }(t))}^{\prime }=t^{k}h(t,v(t)), t\in (0,T]\subset \mathbb{R}, where k\in (1,\mathrm{\infty }). The differential equation is studied together with the boundary conditions {lim}_{t\to 0+}v(t)=\mathrm{\infty }, v(T)=0. 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.

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

where k\in (1,\mathrm{\infty }).

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 [0,T]\times [0,\mathrm{\infty }). We define a positive solution of (1) as a function v which satisfies (1) for a.e. t\in [0,T], is positive on (0,T), and has absolutely continuous first derivative on each compact subinterval in (0,T].

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 {lim}_{t\to 0+}{v}^{\prime }(t)=0. In contrast to this, some real problems lead to the investigation of unbounded solutions which are characterized by the condition {lim}_{t\to t^{\ast }}v(t)=\mathrm{\infty } for some t^{\ast }\in [0,T] and which are called blow-up solutions. We refer to [14–16]. Here, we are interested in blow-up solutions of (1), where t^{\ast }=0. In particular, (1) will be considered together with the boundary conditions

In this case, we speak about a positive solution of problem ( 1 ), ( 2 ). Let us denote by \mathcal{Z} the set of all positive solutions to (1), (2). Moreover, let \mathcal{R}=\{v\in \mathcal{Z}:t^{k}v\text{ is bounded on }(0,T]\} and {\mathcal{R}}_c=\{v\in \mathcal{R}:v^{\prime }(T)=-c\} for c\ge 0. Since v^{\prime }(T)\le 0 for each v\in \mathcal{Z}, it is obvious that \mathcal{R}={\bigcup }_{c\ge 0}{\mathcal{R}}_c.

Our main goal is to find conditions for the data function h in (1), which guarantee that the set {\mathcal{R}}_c is nonempty for each c\ge 0 and then to investigate the properties of this set. For example, we prove that the difference of any two functions in {\mathcal{R}}_c, c\ge 0, retains its sign on [0,T), and that there exist minimal and maximal solutions v_{c,min},v_{c,max}\in {\mathcal{R}}_c for each c\ge 0, cf. Theorem 5. If the interior of the set \{(t,x)\in {\mathbb{R}}^2:0\le t\le T,v_{c,min}(t)\le x\le v_{c,max}(t)\} is nonempty, we show that this interior is fully covered by ordered graphs of other functions belonging to {\mathcal{R}}_c for each c\ge 0; see Theorem 6. Finally, in Theorem 7, the existence of a positive constant {\alpha }_0 such that {\alpha }_0={lim}_{t\to 0+}{t}^{k-1}{v}_{0,min}(t) is shown and all functions v from ℛ are uniquely characterized by the condition

If we denote such v by v_{\alpha } and define w_{\alpha }(t)=t^{k-1}v(t) for t\in (0,T], and w_{\alpha }(0)=\alpha, then we find that the graphs of these functions do not intersect, cf. Theorem 8, and that for each {\alpha }^{\ast }>{\alpha }_0, the set \{w_{\alpha }:{\alpha }_0\le \alpha \le {\alpha }^{\ast }\} is compact in C[0,T]; see Theorem 9.

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 C[0,T] the Banach space of functions continuous on [0,T] equipped with the maximum norm

Similarly, C^1[0,T] means the Banach space of functions having a continuous first derivative on [0,T] with the corresponding maximum norm {\parallel x\parallel }_{\mathrm{\infty }}+{\parallel x^{\prime }\parallel }_{\mathrm{\infty }}. By L^1[0,T] we denote the set of functions which are Lebesgue integrable on [0,T]. Moreover, A{C}^1[0,T] is the set of functions whose first derivative is absolutely continuous on [0,T], while A{C}_{\mathrm{loc}}^1(0,T] is the set of functions having absolutely continuous first derivative on each compact subinterval of (0,T]. We say that h(t,x) satisfies the Carathéodory conditions on [0,T]\times [0,\mathrm{\infty }), if the following three conditions hold:

1. (i)

   The function h(\cdot ,x):[0,T]\to \mathbb{R} is measurable for all x\in [0,\mathrm{\infty }).

2. (ii)

   The function h(t,\cdot ):[0,\mathrm{\infty })\to \mathbb{R} is continuous for a.e. t\in [0,T].

3. (iii)

   For each compact set \mathcal{U}\subset [0,\mathrm{\infty }) there exists a function m_{\mathcal{U}}\in L^1[0,T] such that

   |h(t,x)|\le m_{\mathcal{U}}(t)\text{for a.e. }t\in [0,T]\text{ and all }x\in \mathcal{U}.

For functions satisfying above conditions, we use the notation h\in Car([0,T]\times [0,\mathrm{\infty })).

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.

In this section we consider the auxiliary singular differential equation,

where . For k:=-a and v(t):=t^{-k}u(t), (3) becomes a special case of (1) and therefore, results obtained for (3) apply for (1). For the further analysis, we assume that f satisfies the following conditions:

(H₁) f\in Car([0,T]\times [0,\mathrm{\infty })),

(H₂) for a.e. t\in [0,T] and all x\in [0,\mathrm{\infty }),

(H₃) f(t,x) is increasing in x for a.e. t\in [0,T] and

We now study (3) subject to the boundary conditions

and require that

holds. We call a function u:[0,T]\to \mathbb{R} a positive solution of the Dirichlet problem (3), (4) if u\in A{C}^1[0,T], u>0 on (0,T), u satisfies the boundary conditions (4), and (3) holds for a.e. t\in [0,T]. Clearly, for each positive solution u of (3), (4), there exists c\ge 0 such that (5) is satisfied.

We now denote by \mathcal{S} the set of all positive solutions of problem (3), (4), and {\mathcal{S}}_c=\{u\in \mathcal{S}:u^{\prime }(T)=-c\}, where c\ge 0.

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

Lemma 1 Let (H₁)-(H₃) hold. Then the following statements hold:

1. (a)

   For each c\ge 0 the set {\mathcal{S}}_c is nonempty and there exist functions u_{c,min},u_{c,max}\in {\mathcal{S}}_c such that u_{c,min}(t)\le u(t)\le u_{c,max}(t) for t\in [0,T] and u\in {\mathcal{S}}_c.

2. (b)

   If c_1>c_2\ge 0, u_i\in {\mathcal{S}}_{c_i}, i=1,2, then u_1(t)>u_2(t) for t\in (0,T).

3. (c)

   If c\ge 0, u_i\in {\mathcal{S}}_c, i=1,2, and u_1(t_0)>u_2(t_0) for some t_0\in (0,T), then either u_1(t)>u_2(t) for t\in (0,T) or there exists t_{\ast }\in (t_0,T) such that u_1(t)>u_2(t) for t\in (0,t_{\ast }) and u_1(t)=u_2(t) for t\in [t_{\ast },T].

4. (d)

   For each t_0\in (0,T) and each A>u_{0,max}(t_0) there exists u\in \mathcal{S} satisfying u(t_0)=A.

5. (e)

   {\mathcal{S}}_c is a one-point set for each c\in [0,\mathrm{\infty })\setminus \mathrm{\Gamma }, where \mathrm{\Gamma }\subset [0,\mathrm{\infty }) is at most countable.

6. (f)

   For each , the set {\bigcup }_{K\le c\le Q}{\mathcal{S}}_c is compact in C^1[0,T].

7. (g)

   If c\ge 0, then u\in {\mathcal{S}}_c if and only if it is a solution of the equation

   u(t)=t\frac{c{T}^{a+1}}{|a+1|}(T^{-a-1}-t^{-a-1})+t{\int }_t^T{s}^{-a-2}\left({\int }_s^T{\xi }^{a+1}f(\xi ,u(\xi ))\mathrm{d}\xi \right)\mathrm{d}s

   (6)

in the set C^1[0,T].

We now formulate new results which complete those from [20]. We first establish a relation between {\mathcal{S}}_0 and the set \{(t,x)\in {\mathbb{R}}^2:0\le t\le T,u_{0,min}(t)\le x\le u_{0,max}(t)\} if its interior is nonempty. This question was a short time ago an open problem [[20], Remark 4.4]. We note that the relation between {\mathcal{S}}_c and the set \{(t,x)\in {\mathbb{R}}^2:0\le t\le T,u_{c,min}(t)\le x\le u_{c,max}(t)\} with c>0 having nonempty interior is described in Lemma 1(d).

Theorem 1 Let (H₁)-(H₃) hold. Let us assume that there exists t_0\in (0,T) such that . Then for any Q\in (u_{0,min}(t_0),u_{0,max}(t_0)) there exists u\in {\mathcal{S}}_0 satisfying u(t_0)=Q.

Proof Step 1. Auxiliary Dirichlet problem.

Choose Q\in (u_{0,min}(t_0),u_{0,max}(t_0)). Consider (3) subject to the Dirichlet conditions

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 (H₁) that there exists \phi \in L^1[t_0,T] such that

Let

and let \chi :\mathbb{R}\to [0,1] be given as

We now consider the auxiliary differential equation

It is not difficult to verify that \chi (y)|y|\le S for y\in \mathbb{R}. Hence, cf. (9),

for a.e. t\in [t_0,T] and all u_{0,min}(t)\le x\le u_{0,max}(t), y\in \mathbb{R}. Since \chi (u_{0,min}^{\prime }(t))=1 and \chi (u_{0,max}^{\prime }(t))=1 for t\in [t_0,T], u_{0,min}(T)=u_{0,max}(T)=0, , and u_{0,min}, u_{0,max} solve (3) on [0,T], we conclude that u_{0,min} and u_{0,max} are lower and upper functions of problem (3), (7) (see 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 t\to T we obtain^a u_{0,min}^{\prime }(T)\ge v^{\prime }(T)\ge u_{0,max}^{\prime }(T), which together with u_{0,min}^{\prime }(T)=u_{0,max}^{\prime }(T)=0 gives v^{\prime }(T)=0.

We now prove that on [t_0,T]. The proof is indirect. Let us assume that there exists \xi \in [t_0,T) such that |v^{\prime }(\xi )|=S and for t\in (\xi ,T]. Then \chi (v^{\prime }(t))=1 on this interval, and therefore

From (8) and (9) we conclude that

Since

we have

In particular,

By the Gronwall lemma,

Therefore

which contradicts

Consequently, for t\in [t_0,T], and so \chi (v^{\prime }) = 1 on this interval. Thus, 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 [t_0,T], v^{\prime }(T)=0 and because v(t_0)=Q\in (u_{0,min}(t_0),u_{0,max}(t_0)). It is easy to verify that the equality

is satisfied for a.e. t\in [t_0,T]. We now integrate the last equality two times over [t,T]\subset [t_0,T] and have (note that v(T)=v^{\prime }(T)=0)

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 \gamma =inf\{t:t\in J\}. Then and (12) with v replaced by u holds for a.e. t\in J. The integration now yields

and we claim that

We show inequality (14) indirectly. Let us assume that (14) does not hold. Then there exists \nu \in (\gamma ,t_0) such that

and either u(\nu )=u_{0,max}(\nu ) or u(\nu )=u_{0,min}(\nu ). Assume that u(\nu )=u_{0,max}(\nu ).

Since, by (H₃), for a.e. t\in [\nu ,t_0] and f(t,u(t))\le f(t,u_{0,max}(t)) for a.e. t\in [t_0,T], we have

which is not possible. The case u(\nu )=u_{0,min}(\nu ) can be discussed analogously. Hence, (14) holds.

Suppose that \gamma >0. Then J=(\gamma ,T] and since u is bounded on (\gamma ,T], it follows that {lim\hspace{0.17em}sup}_{t\to {\gamma }^{+}}|u^{\prime }(t)|=\mathrm{\infty }. The integration of the equality

over [t,T]\subset (\gamma ,T] gives

Since |\frac{a}{t^2}u(t)+f(t,u(t))|\le p(t) for a.e. t\in [\gamma ,T], where p\in L^1[\gamma ,T], we have

Using the Gronwall lemma, we deduce that for t\in (\gamma ,T],

holds. This is a contradiction.

Therefore \gamma =0, and then (14) yields {lim}_{t\to 0^{+}}u(t)=0. Consequently J=[0,T], and the assertion of the theorem follows from (13). □

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

Corollary 1 For each t_0\in (0,T) and each A\ge u_{0,min}(t_0) there exists u\in \mathcal{S} satisfying u(t_0)=A.

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

Remark 1 Corollary 1 says that the set \mathcal{U}=\{(t,x)\in {\mathbb{R}}^2:t\in [0,T],x\ge u_{0,min}(t)\} is covered by the graphs of functions from \mathcal{S}, that is,

By Lemma 1(b), (c) we know that functions from \mathcal{S} are uniquely determined by the values −c of their derivatives at the right end point t=T only in the case that {\mathcal{S}}_c is a singleton set for each c\ge 0. Since we cannot uniquely determine all functions from \mathcal{S} via their derivatives at t=T, see Lemma 1(e), we discuss their derivatives at the singular point t=0.

Lemma 2 Let (H₁)-(H₃) hold. Let u\in {\mathcal{S}}_c, c\ge 0. Then

Proof It follows from Lemma 1(g) that (6) holds for t\in [0,T]. Since u(0)=0, we have (note that )

and (15) follows. □

Corollary 2 Let u\in \mathcal{S}. Then

Proof The result follows from Lemma 2, since u\in {\mathcal{S}}_{-u^{\prime }(T)}. □

Corollary 3 Let u,v\in \mathcal{S}, u\ne v. Then u^{\prime }(0)\ne v^{\prime }(0).

Proof Since u\ne v, there exists t_0\in (0,T) such that u(t_0)\ne v(t_0). We can assume that for instance . Then on (0,t_0] and u\le v on (t_0,T] by Lemma 1(b)(c). Hence, u^{\prime }(T)\ge v^{\prime }(T) and

which together with Corollary 2 gives . □

Corollary 4 Let u\in {\mathcal{S}}_c, c\ge 0. Then

In particular, u_{0,min}^{\prime }(0)>0.

Proof Inequality (16) follows from Lemma 2 and the fact that f(t,u(t))>0 for a.e. t\in [0,T] by (H₂). □

Corollary 5 Let u\in \mathcal{S}. Then

Proof The integration of (3) over [\epsilon ,T]\subset (0,T] gives

Using integration by parts we obtain

and, therefore,

Taking the limit \epsilon \to 0, we have

On the other hand, it follows from Corollary 2 that

Combining the above two equalities yields the result. □

Lemma 3 Let (H₁)-(H₃) hold. Let c\ge 0 and {\mathcal{S}}_c be not a singleton set. Then for each \alpha \in [u_{c,min}^{\prime }(0),u_{c,max}^{\prime }(0)] there exists a unique u\in {\mathcal{S}}_c such that u^{\prime }(0)=\alpha. Consequently, [u_{c,min}^{\prime }(0),u_{c,max}^{\prime }(0)]=\{u^{\prime }(0):u\in {\mathcal{S}}_c\}.

Proof Let u\in {\mathcal{S}}_c, u_{c,min}\ne u\ne u_{c,max}. Then it follows from Lemma 1(a) and (H₃) that

Hence, by Lemma 2,

Since {\mathcal{S}}_c is a compact set in C^1[0,T] by Lemma 1(f), the set I=\{u^{\prime }(0):u\in {\mathcal{S}}_c\}\subset [u_{c,min}^{\prime }(0),u_{c,max}^{\prime }(0)] is closed. In fact, let \{{\beta }_n\}\subset I, {\beta }_n=u_n^{\prime }(0), where u_n\in {\mathcal{S}}_c, and let {lim}_{n\to \mathrm{\infty }}{\beta }_n=\beta. Then there exist a subsequence \{{\beta }_{k_n}\} of \{{\beta }_n\} and u\in {\mathcal{S}}_c such that {lim}_{n\to \mathrm{\infty }}{u}_{k_n}=u in C^1[0,T]. In particular, {\beta }_{k_n}=u_{k_n}^{\prime }(0)\to u^{\prime }(0) as n\to \mathrm{\infty }. Therefore, \beta =u^{\prime }(0)\in I.

It remains to prove that I=[u_{c,min}^{\prime }(0),u_{c,max}^{\prime }(0)]. Assume that the equality does not hold. Then, from the structure of bounded and closed subsets of ℝ the existence of an open interval (\alpha ,\beta )\subset [u_{c,min}^{\prime }(0),u_{c,max}^{\prime }(0)]\setminus I, \alpha ,\beta \in I, follows. Let \alpha =u_{\alpha }^{\prime }(0) and \beta =u_{\beta }^{\prime }(0), where u_{\alpha },u_{\beta }\in {\mathcal{S}}_c. Due to Lemma 1(c), there exists t_0\in (0,T) such that on (0,t_0], u_{\alpha }\le u_{\beta } on (t_0,T]. Choose \tau \in (u_{\alpha }(t_0),u_{\beta }(t_0)). By Corollary 1, there exists v\in {\mathcal{S}}_c such that v(t_0)=\tau. Then on (0,t_0] and u_{\alpha }\le v\le u_{\beta } on (t_0,T]. Consequently, on (t_0,T], that is, v^{\prime }(0)\in (\alpha ,\beta ), which is not possible. □

Since functions from \mathcal{S} belong to A{C}^1[0,T], we have for each u\in \mathcal{S}. Corollary 4 yields u_{0,min}^{\prime }(0)>0. Let us denote

Lemma 3 implies that functions from \mathcal{S} can be uniquely determined by the values of their derivatives at the singular point t=0; see Theorem 2.

Theorem 2 Let (H₁)-(H₃) hold. Then there exists a unique u\in \mathcal{S} satisfying u^{\prime }(0)=\alpha if and only if \alpha \in J.

Proof We first show

It follows from Lemma 3 that

for each K\ge 0, where we set u_{c,min}=u=u_{c,max} if {\mathcal{S}}_c is a singleton set and {\mathcal{S}}_c=\{u\}. In view of Corollary 4, u^{\prime }(0)>c/|a+1| for u\in {\mathcal{S}}_c. Consequently, (18) follows.

Let us now choose \alpha \in J. Then there exists u\in \mathcal{S} satisfying u^{\prime }(0)=\alpha. The uniqueness of u follows from Corollary 3. □

For \alpha \in J, we denote by u_{\alpha } the unique element of \mathcal{S} satisfying u_{\alpha }^{\prime }(0)=\alpha.

Theorem 3 Let (H₁)-(H₃) hold. Assume that \{{\alpha }_n\}\subset J is a convergent sequence and {lim}_{n\to \mathrm{\infty }}{\alpha }_n=\alpha. Then {lim}_{n\to \mathrm{\infty }}{u}_{{\alpha }_n}=u_{\alpha } in C^1[0,T].

Proof Let K=sup\{-u_{{\alpha }_n}^{\prime }(T):n\in \mathbb{N}\}. Then because {\alpha }_n=u_{{\alpha }_n}^{\prime }(0)>-u_{{\alpha }_n}^{\prime }(T)/|a+1| for n\in \mathbb{N} by Corollary 4. Since \{u_{{\alpha }_n}\}\subset {\bigcup }_{0\le c\le K}{\mathcal{S}}_c and {\bigcup }_{0\le c\le K}{\mathcal{S}}_c is compact in C^1[0,T] by Lemma 1(f), the sequence \{u_{{\alpha }_n}\} is relatively compact in C^1[0,T]. Let \{u_{{\alpha }_{{\ell }_n}}\} be a subsequence of \{u_{{\alpha }_n}\} which is convergent in C^1[0,T], and let {lim}_{n\to \mathrm{\infty }}{u}_{{\alpha }_{{\ell }_n}}=u. Then u\in {\mathcal{S}}_{c_0} for a c_0\in [0,K] and u^{\prime }(0)={lim}_{n\to \mathrm{\infty }}{u}_{{\alpha }_{{\ell }_n}}^{\prime }(0)={lim}_{n\to \mathrm{\infty }}{\alpha }_{{\ell }_n}=\alpha. Therefore, u=u_{\alpha } and hence any subsequence \{u_{{\alpha }_{{\ell }_n}}\} of \{u_{{\alpha }_n}\} converging in C^1[0,T] has the same limit u_{\alpha }. Consequently, \{u_{{\alpha }_n}\} is convergent in C^1[0,T] and u_{\alpha } is its limit. □

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 t=0.

Lemma 4 Let k\in (1,\mathrm{\infty }) and let the function h\in Car([0,T]\times (0,\mathrm{\infty })) be positive. Let v be a positive solution of (1). Then either

or

Proof By Corollary 3.5 in [12], 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 t^k{v}^{\prime }(t) is increasing on (0,T] and hence there exists {lim}_{t\to 0+}{t}^k{v}^{\prime }(t). The next part of the proof is divided into three cases.

1. (i)

   Let us assume that

   \underset{t\to 0+}{lim}{t}^k{v}^{\prime }(t)=0.

   (23)

Since t^k{v}^{\prime }(t) is increasing on (0,T], we deduce t^k{v}^{\prime }(t)>0 and v^{\prime }(t)>0 for t\in (0,T]. Consequently, v is increasing on (0,T]. This together with v>0 and v\in A{C}_{\mathrm{loc}}^1(0,T] yields . Therefore, h(t,v(t))\in L^1[0,T]. Integrating (1) and using (23), we obtain

which is a contradiction to (22).

1. (ii)

   Assume that

   \underset{t\to 0+}{lim}{t}^k{v}^{\prime }(t)=\ell >0.

Then there exists t_0\in (0,T) such that

Hence, v is increasing on (0,t_0] and there exists {lim}_{t\to 0+}v(t)\ge 0.

By integration, we obtain from (24)

and, by virtue of k>1 and \ell >0, we arrive at

which again is a contradiction.

1. (iii)

   Assume that

   (25)

Then there exists t_0\in (0,T) such that

Hence, v is decreasing on (0,t_0] and there exists {lim}_{t\to 0+}v(t)\in (0,\mathrm{\infty }]. By integration, we obtain from (26)

and, since k>1 and , we have

In addition (25) yields {lim}_{t\to 0+}{v}^{\prime }(t)=-\mathrm{\infty } and (20) follows. □

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 k\in (1,\mathrm{\infty }) and ψ, g satisfy the conditions

({\mathrm{H}}_1^0) \psi \in L^1[0,T] and \psi >0 for a.e. t\in [0,T],

({\mathrm{H}}_2^0) g\in Car([0,T]\times [0,\mathrm{\infty })) and g(t,x) is increasing in x for a.e. t\in [0,T],

({\mathrm{H}}_3^0) 0\le g(t,x)\le \varphi (x) for a.e. t\in [0,T] and all x\in [0,\mathrm{\infty }), where \varphi \in C[0,\mathrm{\infty }), 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\in A{C}_{\mathrm{loc}}^1(0,T] such that v>0 on (0,T), v satisfies the boundary conditions (28), and (27) holds for a.e. t\in [0,T].

Define a set \mathcal{Z} by