scholarly journals On the Existence of Singular Solutions

2001 ◽  
Vol 8 (4) ◽  
pp. 669-681
Author(s):  
M. Bartušek ◽  
J. Osička
Keyword(s):  
Singular Solution ◽  
Sufficient Conditions ◽  
Singular Solutions

Abstract Sufficient conditions are given, under which the equation 𝑦(𝑛) = ƒ(𝑡, 𝑦, 𝑦′, . . . , 𝑦(𝑙))𝑔(𝑦(𝑛 – 1)) has a singular solution 𝑦[𝑇, τ) → 𝐑, τ < ∞ satisfying , 𝑖 = 0, 1, . . . , 𝑙 and for 𝑗 = 𝑙 + 1, . . . , 𝑛 – 1 where 𝑙 ∈ {0, 1, . . . , 𝑛 – 2}.

Classification of singular solutions of porous media equations with absorption

2005 ◽  
Vol 135 (3) ◽  
pp. 563-584 ◽  
Author(s):  
Xinfu Chen ◽  
Yuanwei Qi ◽  
Mingxin Wang
Keyword(s):  
Porous Media ◽  
Singular Solution ◽  
Singular Solutions ◽  
Constant Independent ◽  
Porous Media Equation ◽  
Porous Media Equations ◽  
Self Similar ◽  
Image Position ◽  
Very Singular Solution

We consider, for m ∈ (0, 1) and q > 1, the porous media equation with absorption We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in Rn × [0, ∞)\{(0, 0)}, and satisfy u(x, 0) = 0 for all x ≠ 0. We prove the following results. When q ≥ m + 2/n, there does not exist any such singular solution. When q < m + 2/n, there exists, for every c > 0, a unique singular solution u = u(c), called the fundamental solution with initial mass c, which satisfies ∫Rnu(·, t) → c as t ↘ 0. Also, there exists a unique singular solution u = u∞, called the very singular solution, which satisfies ∫Rnu∞(·, t) → ∞ as t ↘ 0.In addition, any singular solution is either u∞ or u(c) for some finite positive c, u(c1) < u(c2) when c1 < c2, and u(c) ↗ u∞ as c ↗ ∞.Furthermore, u∞ is self-similar in the sense that u∞(x, t) = t−αw(|x| t−αβ) for α = 1/(q − 1), β = ½(q − m), and some smooth function w defined on [0, ∞), so that is a finite positive constant independent of t > 0.

scholarly journals Very singular solutions to a nonlinear parabolic equation with absorption II. Uniqueness

2004 ◽  
Vol 134 (1) ◽  
pp. 39-54 ◽  
Author(s):  
Saïd Benachour ◽  
Herbert Koch ◽  
Philippe Laurençot
Keyword(s):  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Singular Solution ◽  
Singular Solutions ◽  
Nonlinear Parabolic ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions ◽  
Very Singular Solution

We prove the uniqueness of the very singular solution to when 1 < p < (N + 2)/(N + 1), thus completing the previous result by Qi and Wang, restricted to self-similar solutions.

Discontinuous equilibrium solutions and cavitation in nonlinear elasticity

1982 ◽  
Vol 306 (1496) ◽  
pp. 557-611 ◽  
Keyword(s):  
Singular Solution ◽  
Direct Method ◽  
Homogeneous Solution ◽  
Singular Solutions ◽  
Large Strains ◽  
Critical Surface ◽  
Surface Traction ◽  
Equilibrium Solutions ◽  
Growth Properties ◽  
The Stability

A study is made of a class of singular solutions to the equations of nonlinear elastostatics in which a spherical cavity forms at the centre of a ball of isotropic material placed in tension by means of given surface tractions or displacements. The existence of such solutions depends on the growth properties of the stored-energy function W for large strains and is consistent with strong ellipticity of W . Under appropriate hypotheses it is shown that a singular solution bifurcates from a trivial (homogeneous) solution at a critical value of the surface traction or displacement, at which the trivial solution becomes unstable. For incompressible materials both the singular solution and the critical surface traction are given explicitly, and the stability of all solutions with respect to radial motion is determined. For compressible materials the existence of singular solutions is proved for a class of strongly elliptic materials by means of the direct method of the calculus of variations, an important step in the analysis being to show that the only radial equilibrium solutions without cavities are homogeneous. Work of Gent & Lindley (1958) shows that the critical surface tractions obtained agree with those observed in the internal rupture of rubber.

Singular solutions of semilinear elliptic equations with fractional Laplacian in entire space

2016 ◽  
Vol 18 (01) ◽  
pp. 1550024 ◽  
Author(s):  
Zhuoran Du
Keyword(s):  
Elliptic Equations ◽  
Singular Solution ◽  
Fractional Laplacian ◽  
Semilinear Elliptic Equations ◽  
Singular Solutions ◽  
Entire Space ◽  
Semilinear Elliptic ◽  
Supercritical Exponent ◽  
Extension Result

We consider the existence of positive singular solutions of the problem [Formula: see text] where [Formula: see text] and [Formula: see text]. We obtain the existence of positive singular solutions [Formula: see text] of this problem in space dimensions [Formula: see text] for the case [Formula: see text] under the condition [Formula: see text]. By the well-known Caffarelli–Silvestre extension result, the trace of [Formula: see text] is a positive singular solution for the following equation with a supercritical exponent [Formula: see text] where [Formula: see text] denotes the fractional Laplacian.

ON SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS OF BRIOT–BOUQUET TYPE

2018 ◽  
Vol 98 (1) ◽  
pp. 122-133
Author(s):  
FENGBAI LI
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Type Systems ◽  
Singular Solutions ◽  
Partial Differential ◽  
First Order ◽  
Associated Matrix ◽  
Necessary And Sufficient

We study systems of partial differential equations of Briot–Bouquet type. The existence of holomorphic solutions to such systems largely depends on the eigenvalues of an associated matrix. For the noninteger case, we generalise the well-known result of Gérard and Tahara [‘Holomorphic and singular solutions of nonlinear singular first order partial differential equations’, Publ. Res. Inst. Math. Sci.26 (1990), 979–1000] for Briot–Bouquet type equations to Briot–Bouquet type systems. For the integer case, we introduce a sequence of blow-up like changes of variables and give necessary and sufficient conditions for the existence of holomorphic solutions. We also give some examples to illustrate our results.

scholarly journals PERTURBING SINGULAR SOLUTIONS OF THE GELFAND PROBLEM

2007 ◽  
Vol 09 (05) ◽  
pp. 639-680 ◽  
Author(s):  
J. DÁVILA ◽  
L. DUPAIGNE
Keyword(s):  
Unit Ball ◽  
Singular Solution ◽  
Dirichlet Condition ◽  
Extremal Solution ◽  
Singular Solutions ◽  
The One ◽  
Gelfand Problem ◽  
Small Deformations

The equation -Δu = λeu posed in the unit ball B ⊆ ℝN, with homogeneous Dirichlet condition u|∂B = 0, has the singular solution [Formula: see text] when λ = 2(N - 2). If N ≥ 4 we show that under small deformations of the ball there is a singular solution (u,λ) close to (U,2(N - 2)). In dimension N ≥ 11 it corresponds to the extremal solution — the one associated to the largest λ for which existence holds. In contrast, we prove that if the deformation is sufficiently large then even when N ≥ 10, the extremal solution remains bounded in many cases.

scholarly journals SINGULAR SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR ABSORPTION AND HARDY POTENTIALS

2012 ◽  
Vol 14 (02) ◽  
pp. 1250013 ◽  
Author(s):  
VITALI LISKEVICH ◽  
ANDREY SHISHKOV ◽  
ZEEV SOBOL
Keyword(s):  
Nonlinear Absorption ◽  
Singular Solution ◽  
Complete Classification ◽  
Beltrami Operator ◽  
Singular Solutions ◽  
Heat Equations ◽  
Nonnegative Solutions ◽  
The One ◽  
Very Singular Solution

We study the existence and nonexistence of singular solutions to the equation [Formula: see text], p > 1, in ℝN× [0, ∞), N ≥ 3, with a singularity at the point (0, 0), that is, nonnegative solutions satisfying u(x, 0) = 0 for x ≠ 0, assuming that α > -2 and [Formula: see text]. The problem is transferred to the one for a weighted Laplace–Beltrami operator with a nonlinear absorption, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a source solution with a multiple of the Dirac mass as initial datum, or as a unique very singular solution, leads to a complete classification of singular solutions to the original problem, which exist if and only if [Formula: see text].

Optimal Singular Solutions for Linear Multi-Input Systems

1966 ◽  
Vol 88 (2) ◽  
pp. 323-328 ◽  
Author(s):  
R. A. Rohrer ◽  
M. Sobral
Keyword(s):  
Sufficient Conditions ◽  
Singular Control ◽  
Open Loop ◽  
The State ◽  
Singular Solutions ◽  
Linear Feedback ◽  
State Variables ◽  
Performance Indices ◽  
Explicit Formulae ◽  
Necessary And Sufficient

Linear, stationary systems with multiple inputs when subject to performance indices quadratic in the state variables, but explicitly independent of the control variables, may be optimally governed by singular control. Necessary and sufficient conditions for the optimality of both partially singular control and totally singular control are obtained. Moreover, explicit formulae are presented for both open-loop and linear-feedback implementations of the optimal singular control.

scholarly journals XXIV.—On the p-discriminant of a Differential Equation of the First Order, and on Certain Points in the General Theory of Envelopes connected therewith

1897 ◽  
Vol 38 (4) ◽  
pp. 803-824
Author(s):  
Chrystal
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Differential Equations ◽  
Singular Solution ◽  
Singular Solutions ◽  
English Text ◽  
First Order

The theory of the singular solutions of differential equations of the first order, even in the interesting and suggestive form due to Professor Cayley (Mess. Math., ii., 1872), as given in English text-books, is defective, inasmuch as it gives no indication as to what are normal and what are abnormal phenomena. Moreover, Cayley added an appendix to his theory regarding the circumstances under which a singular solution exists, which is misleading so far as the theory of differential equations is concerned, if not altogether erroneous.The main purpose of the following notes is to throw light on the point last mentioned by means of a number of examples. I have also taken the opportunity to furnish simple demonstrations of several well-known theorems regarding the p-discriminant which do not find a place in the current English text-books.

Classification of singular solutions of porous media equations with absorption

2005 ◽  
Vol 135 (3) ◽  
pp. 563-584
Author(s):  
Xinfu Chen ◽  
Yuanwei Qi ◽  
Mingxin Wang
Keyword(s):  
Porous Media ◽  
Singular Solution ◽  
Singular Solutions ◽  
Constant Independent ◽  
Porous Media Equation ◽  
Porous Media Equations ◽  
Self Similar ◽  
Image Position ◽  
Very Singular Solution

We consider, for m ∈ (0, 1) and q > 1, the porous media equation with absorption We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in ℝn × [0, ∞)\{(0, 0)}, and satisfy u(x, 0) = 0 for all x ≠ = 0. We prove the following results. When q ≥ m + 2/n, there does not exist any such singular solution. When q < m + 2/n, there exists, for every c > 0, a unique singular solution u = u(c), called the fundamental solution with initial mass c, which satisfies ∫Rnu(·, t) → c as t ↘ 0. Also, there exists a unique singular solution u = u∞, called the very singular solution, which satisfies ∫Rnu∞(·, t) → ∞ as t ↘ 0.In addition, any singular solution is either u∞ or u(c) for some finite positive c, u(c1) < u(c2) when c1 < c2, and u(c) ↗ u∞ as c ↗ ∞.Furthermore, u∞ is self-similar in the sense that u∞(x, t) = t−αw(|x| t−αβ) for α = 1/(q − 1), β = ½(q − m), and some smooth function w defined on [0, ∞), so that is a finite positive constant independent of t > 0.

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