Positive solutions of a singular ordinary differential equation relevant in Mathematical Physics

2004 ◽  
Author(s):  
J. M. Gomes ◽  
L. Sanchez
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Positive Solutions ◽  
Mathematical Physics ◽  
Singular Ordinary Differential Equation

scholarly journals A blow-up dichotomy for semilinear fractional heat equations

2020 ◽  
Author(s):  
Robert Laister ◽  
Mikołaj Sierżęga
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Positive Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Heat Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Whole Space ◽  
Necessary And Sufficient

Abstract We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.

Integrable-square solutions of a singular ordinary differential equation

1981 ◽  
Vol 89 (3-4) ◽  
pp. 267-279 ◽  
Author(s):  
Richard C. Gilbert
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Characteristic Polynomial ◽  
Characteristic Equation ◽  
Complex Plane ◽  
Asymptotic Expansions ◽  
Differential Operators ◽  
Singular Ordinary Differential Equation ◽  
Ordinary Differential Operators

SynopsisAbsolutely square integrable solutions are determined for the equation= λywhere the ζn−r(x) are holomorphic in a sector of the complex plane and have asymptotic expansions asxapproaches infinity. It is shown that the number of such solutions depends upon the roots of the characteristic equation and their multiplicity, and upon the sign of the derivative of the characteristic polynomial. Application is made to formally symmetric ordinary differential operators.

The Reduction of a Type of Singular Partial Differential Equations

2014 ◽  
Vol 602-605 ◽  
pp. 3616-3619
Author(s):  
Ping Li Li ◽  
Tian Tian Sun
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Domain ◽  
Partial Differential ◽  
Singular Ordinary Differential Equation

Using the theory with respect to the existence of holomorphic solutions to a singular ordinary differential equation in the complex domain, this paper shows how to constructs biholomorphic trans-formations in different cases to reduce partial differential equations with singularity at the origin to more simple forms.

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