Positive blow-up solutions of nonlinear models from real world dynamics
© Gschwindl et al.; licensee Springer. 2014
Received: 13 December 2013
Accepted: 5 May 2014
Published: 16 May 2014
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.
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 .
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 and for . Since for each , it is obvious that .
If we denote such v by and define for , and , then we find that the graphs of these functions do not intersect, cf. Theorem 8, and that for each , the set is compact in ; see Theorem 9.
The function is measurable for all .
The function is continuous for a.e. .
- (iii)For each compact set there exists a function such that
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  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
where . For and , (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:
(H2) for a.e. and all ,
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.
We now denote by the set of all positive solutions of problem (3), (4), and , where .
In the following lemma we cite those results from  which will be used in the analysis of problem (1), (2).
For each the set is nonempty and there exist functions such that for and .
If , , , then for .
If , , , and for some , then either for or there exists such that for and for .
For each and each there exists satisfying .
is a one-point set for each , where is at most countable.
For each , the set is compact in .
- (g)If , then if and only if it is a solution of the equation(6)
in the set .
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).
Theorem 1 Let (H1)-(H3) hold. Let us assume that there exists such that . Then for any there exists satisfying .
Proof Step 1. Auxiliary Dirichlet problem.
and taking the limit we obtaina , which together with gives .
Consequently, for , and so = 1 on this interval. Thus, v is a solution of problem (3), (7).
Step 2. Continuation of the solution v.
and either or . Assume that .
which is not possible. The case can be discussed analogously. Hence, (14) holds.
holds. This is a contradiction.
Therefore , and then (14) yields . Consequently , 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 and each there exists satisfying .
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. □
Proof The result follows from Lemma 2, since . □
Corollary 3 Let , . Then .
which together with Corollary 2 gives . □
In particular, .
Proof Inequality (16) follows from Lemma 2 and the fact that for a.e. by (H2). □
Combining the above two equalities yields the result. □
Lemma 3 Let (H1)-(H3) hold. Let and be not a singleton set. Then for each there exists a unique such that . Consequently, .
Since is a compact set in by Lemma 1(f), the set is closed. In fact, let , , where , and let . Then there exist a subsequence of and such that in . In particular, as . Therefore, .
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. □
Lemma 3 implies that functions from can be uniquely determined by the values of their derivatives at the singular point ; see Theorem 2.
Theorem 2 Let (H1)-(H3) hold. Then there exists a unique satisfying if and only if .
for each , where we set if is a singleton set and . In view of Corollary 4, for . Consequently, (18) follows.
Let us now choose . Then there exists satisfying . The uniqueness of u follows from Corollary 3. □
For , we denote by the unique element of satisfying .
Theorem 3 Let (H1)-(H3) hold. Assume that is a convergent sequence and . Then in .
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
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 .
- (i)Let us assume that(23)
- (ii)Assume that
Hence, v is increasing on and there exists .
- (iii)Assume that(25)
In addition (25) yields and (20) follows. □
where and ψ, g satisfy the conditions
() and for a.e. ,
() and is increasing in x for a.e. ,
We define a positive solution to problem (27), (28) as a function such that on , v satisfies the boundary conditions (28), and (27) holds for a.e. .
Clearly, for each there exists such that (29) holds.
Lemma 5 Let ()-() hold. Let . Then on .
which is not possible, since and . □
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.
Theorem 4 Let ()-() hold and . Let us assume that (33) holds. Then if and only if and .
for , and hence, . Now, [, Corollary 1] guarantees that u can be extended on with such that the equality (35) holds for . Consequently, .
Hence, note that , , and so . As a result, we have . □
Now, we are in the position to provide results on the solvability of problem (27), (28) and formulate the properties of its solutions.
For each the set is nonempty and there exist such that for and .
If , , , then for .
If , , , and for some , then either for or there exists such that for and for .
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. □
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 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). □
Using constants from the interval , we can uniquely determine all functions in ℛ.
- (a)For each there exists such that(39)
For each there exists a unique satisfying (39).
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. □
Then . We now specify further properties of functions .
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 , , 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 . □
Corollary 6 Let be a convergent sequence and let us denote . Then in .
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 . □
Theorem 9 Let ()-() hold and let be from (38). For each the set defined by (44) is compact in .
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 MATLAB TM 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. For further information see . This software proved useful for the approximation of numerous singular BVPs important for applications; see e.g. [9, 27–29].
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.
aNote that .
bThe required smoothness of higher derivatives is related to the order of the used collocation method.
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.
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