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].

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 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}.

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.

Casimir energy of non-flat bordered membrane

2014 ◽  
Vol 29 (24) ◽  
pp. 1450127 ◽  
Author(s):  
B. Droguett ◽  
J. C. Rojas
Keyword(s):  
Perturbation Theory ◽  
Scalar Field ◽  
Zeta Function ◽  
Coordinate Transformation ◽  
Dirichlet Condition ◽  
Casimir Energy ◽  
Beltrami Operator ◽  
Wkb Method ◽  
The One ◽  
Dimensional Surface

We compute the Casimir energy which arises in a bi-dimensional surface due to the quantum fluctuations of a scalar field. We assume that the boundaries are non-flat and the field obeys Dirichlet condition. We re-parametrize the problem to one which has flat boundary conditions and the irregularity is treated as a perturbation in the Laplace–Beltrami operator associated to the coordinate transformation. Later, to compute the Casimir energy, we use zeta function regularization. It is compared with the results coming from perturbation theory with the one from Wentzel–Kramers–Brillouin (WKB) method.

scholarly journals A MULTIDIMENSIONAL SUBDIFFUSION MODEL WITH CHEMOTAXIS

2019 ◽  
Vol 11 (2) ◽  
pp. 1
Author(s):  
Bambang Hendriya Guswanto
Keyword(s):  
Mathematical Model ◽  
Probability Measure ◽  
Random Walks ◽  
Unit Ball ◽  
Dimensional Case ◽  
One Dimensional ◽  
The One ◽  
The Mathematical Model

The mathematical model for subdiffusion process with chemotaxis proposed by Langlands and Henry [1] for the one-dimensional case is extended to the multi-dimensional case. The model is derived from random walks process using a probability measure on a n-multidimensional unit ball $S^{n-1}$.

scholarly journals Hankel kernels of higher weight for the ball

1993 ◽  
Vol 130 ◽  
pp. 183-192 ◽  
Author(s):  
Jaak Peetre
Keyword(s):  
Unit Ball ◽  
Unit Disk ◽  
Dimensional Case ◽  
One Dimensional ◽  
The One

The purpose of this note is to write down the general form of Hankel kernels for the complex unit ball B in Cd. In the one dimensional case (unit disk Δ in C) this was done in [JP] and our treatment below has been guided by the insights gained there, and later, in a slightly different context, in [P]. We begin by summarizing the relevant facts in the case of the disk in a form convenient for us.

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