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Open Access Research Article

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

Jürgen Gschwindl1, Irena Rachůnková2*, Svatoslav Staněk2 and Ewa B Weinmüller1

Author Affiliations

1 Department for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, Wien, A-1040, Austria

2 Department of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, Olomouc, 77146, Czech Republic

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Boundary Value Problems 2014, 2014:121  doi:10.1186/1687-2770-2014-121


Dedicated to Professor Ivan Kiguradze for his merits in mathematical sciences

Published: 16 May 2014

Abstract

In this paper, we investigate the structure and properties of the set of positive blow-up solutions of the differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M2">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M3">View MathML</a>. The differential equation is studied together with the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/121/mathml/M5">View MathML</a>. 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