scholarly journals Self-similar solutions in a sector for a quasilinear parabolic equation

2012 ◽  
Vol 7 (4) ◽  
pp. 857-879
Author(s):  
Bendong Lou ◽  
Keyword(s):  
Parabolic Equation ◽  
Quasilinear Parabolic Equation ◽  
Self Similar ◽  
Similar Solutions ◽  
Quasilinear Parabolic

Critical diffusion exponents for self-similar blow-up solutions of a quasilinear parabolic equation with an exponential source

1996 ◽  
Vol 126 (2) ◽  
pp. 413-441 ◽  
Author(s):  
Chris J. Budd ◽  
Victor A. Galaktionov
Keyword(s):  
Parabolic Equation ◽  
Blow Up ◽  
Infinite Sequence ◽  
Quasilinear Parabolic Equation ◽  
Homoclinic Bifurcation ◽  
The Self ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions ◽  
Quasilinear Parabolic

We study the self-similar solutions of the quasilinear parabolic equationWe show that there is an exponentsuch that if σ> then the equation admits a countable set {uk(x, t)} of self-similar blow-up solutions. These solutions have the formwhere T> 0 is a finite blow-up time, θ(ξ) solves a nonlinear ODE and each function uk(x, t) is nonconstant in a neighbourhood of the origin and has exactly k maxima and minima for x ≧ 0. There is a further critical exponent σ = ф such that if σ > ф there is a second set of self-similar solutions which are constant (in x) in a neighbourhood of the origin. We conjecture (and provide formal arguments and numerical evidence for) the existence of an infinite sequence σk→σ∞ of critical values, such that σ1 = 0 and uk exists only in the range σ>σk (when σ> 0 the equation has no nontrivial self-similar solutions). The proof of existence when σ>σ∞(σ>ф) is obtained by a combination of comparison and dynamical systems arguments and relates the existence of the self-similar solutions to a homoclinic bifurcation in an appropriate phase-space.

scholarly journals ON SOLUTIONS OF TRAVELING WAVE TYPE FOR A NONLINEAR PARABOLIC EQUATION

2017 ◽  
Vol 21 (6) ◽  
pp. 110-116 ◽  
Author(s):  
S.V. Pikulin
Keyword(s):  
Parabolic Equation ◽  
Traveling Wave ◽  
Nonlinear Term ◽  
Quasilinear Parabolic Equation ◽  
Biological Processes ◽  
Positive Power ◽  
Wave Type ◽  
Nonlinear Parabolic ◽  
Self Similar ◽  
Similar Solutions

We consider the Kolmogorov — Petrovsky — Piskunov equation which isa quasilinear parabolic equation of second order appearing in the flame propagationtheory and in modeling of certain biological processes. An analyticalconstruction of self-similar solutions of traveling wave kind is presented for thespecial case when the nonlinear term of the equation is the product of theargument and a linear function of a positive power of the argument. The approachto the construction of solutions is based on the study of singular pointsof analytic continuation of the solution to the complex domain and on applyingthe Fuchs — Kovalevskaya — Painlev´e test. The resulting representation of thesolution allows an efficient numerical implementation.

scholarly journals Similarity solutions for a quasilinear parabolic equation

1995 ◽  
Vol 37 (2) ◽  
pp. 253-266 ◽  
Author(s):  
Jong-Sheng Guo
Keyword(s):  
Diffusion Equation ◽  
Quasilinear Parabolic Equation ◽  
Similarity Solutions ◽  
Fast Diffusion ◽  
Equation Approach ◽  
Fast Diffusion Equation ◽  
Self Similar ◽  
Similar Solutions ◽  
Quasilinear Parabolic ◽  
Slow Diffusion Equation

AbstractIn this paper, we use an ordinary differential equation approach to study the existence of similarity solutions for the equation u1 = Δ(uα) + θu–β in Rn × (0, ∞) where β > 0, θ ∈ [0, 1}, and n ≥ 1. This includes the slow diffusion equation when α > = 1, and the standard heat equation when α = 1, and the fast diffusion equation when 0 < α < 1. We prove that there are forward self-similar solutions for this equation with initial data of the form c|x|p, where p = 2/(α + β) if θ = 1; p ≥ 0 and 2 + (1 – α)p > 0 if θ = 0, for some positive constant c.

scholarly journals Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Blow Up Analysis ◽  
Inner Absorption ◽  
Quasilinear Parabolic

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.

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