The asymptotic behaviour of the solutions of the Kassoy problem with a modified source term

1989 ◽  
Vol 113 (3-4) ◽  
pp. 347-356 ◽  
Author(s):  
C. Budd ◽  
Y. Qi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Asymptotic Behaviour ◽  
Blow Up ◽  
Similarity Solutions ◽  
Parabolic Partial Differential Equation ◽  
Equation Problem ◽  
All Solutions ◽  
Image Position ◽  
Semilinear Parabolic

SynopsisWe study the asymptotic behaviour as x →∞ of the solutions of the ordinary differential equation problemThis equation generalises the ordinary differential equation obtained by studying the blow-up of the similarity solutions of the semilinear parabolic partial differential equation vt=vxx = ev. We show that if λ≦1, all solutions of (*) tend to —∞ as rapidly as the function —exp (x2/4) (E- solutions). However, if λ>1, then there also exists a solution which tends to –∞, like 2λlog(x) (L-solutions). Thus, the case λ = 1, for which (*) reduces tothe Kassoy equation, is the borderline between two quite different forms of asymptotic behaviour of the function u(x).

Subexponential solutions of linear integro-differential equations and transient renewal equations

2002 ◽  
Vol 132 (3) ◽  
pp. 521-543 ◽  
Author(s):  
John A. D. Appleby ◽  
David W. Reynolds
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Renewal Equation ◽  
Asymptotically Stable ◽  
Renewal Equations ◽  
Integro Differential Equation ◽  
All Solutions ◽  
Image Position

This paper studies the asymptotic behaviour of the solutions of the scalar integro-differential equation The kernel k is assumed to be positive, continuous and integrable.If it is known that all solutions x are integrable and x(t) → 0 as t → ∞, but also that x = 0 cannot be exponentially asymptotically stable unless there is some γ > 0 such that Here, we restrict the kernel to be in a class of subexponential functions in which k(t) → 0 as t → ∞ so slowly that the above condition is violated. It is proved here that the rate of convergence of x(t) → 0 as t → ∞ is given by The result is proved by determining the asymptotic behaviour of the solution of the transient renewal equation If the kernel h is subexponential, then

scholarly journals Critical exponents for a semilinear parabolic equation with variable reaction

2012 ◽  
Vol 142 (5) ◽  
pp. 1027-1042 ◽  
Author(s):  
R. Ferreira ◽  
A. de Pablo ◽  
M. Pérez-LLanos ◽  
J. D. Rossi
Keyword(s):  
Critical Exponents ◽  
Finite Time ◽  
Existence And Uniqueness ◽  
Parabolic Problem ◽  
Blow Up ◽  
Bounded Function ◽  
Semilinear Parabolic Equation ◽  
All Solutions ◽  
Image Position ◽  
Semilinear Parabolic

We study the blow-up phenomenon for non-negative solutions to the following parabolic problem:where 0 < p− = min p ≤ p(x) ≤ max p = p+ is a smooth bounded function. After discussing existence and uniqueness, we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p+ > 1.When Ω = ℝN we show that if p− > 1 + 2/N, then there are global non-trivial solutions, while if 1 < p− ≤ p+ ≤ 1 + 2/N, then all solutions to the problem blow up in finite time. Moreover, in the case when p− < 1 + 2/N < p+, there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global non-trivial solutions.When Ω is a bounded domain we prove that there are functions p(x) and domains Ω such that all solutions to the problem blow up in finite time. On the other hand, if Ω is small enough, then the problem possesses global non-trivial solutions regardless of the size of p(x).

On the linearised theory of buckling for a compressed viscoelastic rod

1985 ◽  
Vol 99 (3-4) ◽  
pp. 371-386 ◽  
Author(s):  
David W. Reynolds
Keyword(s):  
Differential Equation ◽  
Asymptotic Behaviour ◽  
Blow Up ◽  
Static Theory ◽  
Integro Differential Equation ◽  
Image Position ◽  
Viscoelastic Rod

SynopsisResults are obtained on the existence, multiplicity, blow-up and asymptotic behaviour of the solutions to the integro-differential equationwhich arises in the linearised, quasi-static theory of buckling for a viscoelastic rod.

Fast diffusion with loss at infinity

1995 ◽  
Vol 36 (4) ◽  
pp. 438-459 ◽  
Author(s):  
J. R. Philip
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fast Diffusion ◽  
Solution Properties ◽  
Positive Power ◽  
Negative Power ◽  
Exponential Decrease ◽  
Class 1 ◽  
Image Position ◽  
Instantaneous Source

AbstractWe study the equationHere s is not necessarily integral; m is initially unrestricted. Material-conserving instantaneous source solutions of A are reviewed as an entrée to material-losing solutions. Simple physical arguments show that solutions for a finite slug losing material at infinity at a finite nonzero rate can exist only for the following m-ranges: 0 < s < 2, −2s−1 < m ≤ −1; s > 2, −1 < m < −2s−1. The result for s = 1 was known previously. The case s = 2, m = −1, needs further investigation. Three different similarity schemes all lead to the same ordinary differential equation. For 0 < s < 2, parameter γ (0 < γ < ∞) in that equation discriminates between the three classes of solution: class 1 gives the concentration scale decreasing as a negative power of (1 + t/T); 2 gives exponential decrease; and 3 gives decrease as a positive power of (1 − t/T), the solution vanishing at t = T < ∞. Solutions for s = 1, are presented graphically. The variation of concentration and flux profiles with increasing γ is physically explicable in terms of increasing flux at infinity. An indefinitely large number of exact solutions are found for s = 1,γ = 1. These demonstrate the systematic variation of solution properties as m decreases from −1 toward −2 at fixed γ.

Solution Space Decompositions for nth Order Linear Differential Equations

1975 ◽  
Vol 27 (3) ◽  
pp. 508-512
Author(s):  
G. B. Gustafson ◽  
S. Sedziwy
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Solution Space ◽  
Decomposition Theorem ◽  
Linear Differential Equations ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Linear Differential ◽  
Image Position

Consider the wth order scalar ordinary differential equationwith pr ∈ C([0, ∞) → R ) . The purpose of this paper is to establish the following:DECOMPOSITION THEOREM. The solution space X of (1.1) has a direct sum Decompositionwhere M1 and M2 are subspaces of X such that(1) each solution in M1\﹛0﹜ is nonzero for sufficiently large t ﹛nono sdilatory) ;(2) each solution in M2 has infinitely many zeros ﹛oscillatory).

scholarly journals Determination of a control parameter in a parabolic partial differential equation

1991 ◽  
Vol 33 (2) ◽  
pp. 149-163 ◽  
Author(s):  
J. R. Cannon ◽  
Yanping Lin ◽  
Shingmin Wang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Global Solution ◽  
Existence And Uniqueness ◽  
Control Parameter ◽  
Parabolic Partial Differential Equation ◽  
Solution Pair ◽  
Image Position

AbstractThe authors consider in this paper the inverse problem of finding a pair of functions (u, p) such thatwhere F, f, E, s, αi, βi, γi, gi, i = 1, 2, are given functions.The existence and uniqueness of a smooth global solution pair (u, p) which depends continuously upon the data are demonstrated under certain assumptions on the data.

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.

scholarly journals Iterative criteria for bounds on the growth of positive solutions of a delay differential equation

1978 ◽  
Vol 25 (2) ◽  
pp. 195-200
Author(s):  
Raymond D. Terry
Keyword(s):  
Differential Equation ◽  
Positive Solutions ◽  
Delay Differential Equation ◽  
Sufficient Condition ◽  
All Solutions ◽  
Image Position ◽  
Delay Differential

AbstractFollowing Terry (Pacific J. Math. 52 (1974), 269–282), the positive solutions of eauqtion (E): are classified according to types Bj. We denote A neccessary condition is given for a Bk-solution y(t) of (E) to satisfy y2k(t) ≥ m(t) > 0. In the case m(t) = C > 0, we obtain a sufficient condition for all solutions of (E) to be oscillatory.

scholarly journals On a nonlinear boundary value problem involving a small parameter

1962 ◽  
Vol 2 (4) ◽  
pp. 425-439 ◽  
Author(s):  
A. Erdéyi
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Small Parameter ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Nonlinear Ordinary Differential Equation ◽  
Nonlinear Boundary Value Problem ◽  
Image Position

In this paper we shall discuss the boundary value problem consisting of the nonlinear ordinary differential equation of the second order, and the boundary conditions.

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