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

• Published in: Boundary value problems Vol. 2014; no. 1; pp. 1 - 23
• Main Authors: Gschwindl, Jürgen, Rachůnková, Irena, Staněk, Svatoslav, Weinmüller, Ewa B
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 16-05-2014; Hindawi Limited; BioMed Central Ltd
• Subjects: A Tribute to Professor Ivan Kiguradze; Analysis; Approximations and Expansions; Boundary conditions; Boundary value problems; Computer simulation; Difference and Functional Equations; Differential equations; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinear dynamics; Nonlinearity; Ordinary Differential Equations; Partial Differential Equations; Research Article; blow-up, positive solutions; singular ordinary differential equation of the second order; time singularities; polynomial collocation; existence of solutions
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/1687-2770-2014-121

In this paper, we investigate the structure and properties of the set of positive blow-up solutions of the differential equation
(
t
k
v
′
(
t
)
)
′
=
t
k
h
(
t
,
v
(
t
)
)
,
t
∈
(
0
,
T
]
⊂
R
, where
k
∈
(
1
,
∞
)
. The differential equation is studied together with the boundary conditions
lim
t
→
0
+
v
(
t
)
=
∞
,
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.