Abstract
Abstract 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.
Highlights
1 Introduction In this paper, we investigate the structure and properties of the set of positive blow-up solutions of the differential equation tkv (t) = tkh t, v(t), t ∈ (, T] ⊂ R, ( )
Models in the form of ( ) arise in many applications. They occur in the study of phase transitions of Van der Waals fluids [ – ], in population genetics, where they characterize the spatial distribution of the genetic composition of a population [, ], 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 [ ]
If we denote such v by vα and define wα(t) = tk– v(t) for t ∈ (, T], and wα( ) = α, we find that the graphs of these functions do not intersect, cf
Summary
We call a function u : [ , T] → R a positive solution of the Dirichlet problem ( ), ( ) if u ∈ AC [ , T], u > on ( , T), u satisfies the boundary conditions ( ), and ( ) holds for a.e. t ∈ [ , T]. The following statements hold: (a) For each c ≥ the set Sc is nonempty and there exist functions uc,min, uc,max ∈ Sc such that uc,min(t) ≤ u(t) ≤ uc,max(t) for t ∈ [ , T] and u ∈ Sc.
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