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.

Blow-up of solutions of a quasilinear parabolic equation

2012 ◽  
Vol 142 (2) ◽  
pp. 425-448
Author(s):  
Ryuichi Suzuki ◽  
Noriaki Umeda
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Positive Function ◽  
Quasilinear Parabolic Equations ◽  
The Cauchy Problem ◽  
Image Position ◽  
Quasilinear Parabolic

We consider non-negative solutions of the Cauchy problem for quasilinear parabolic equations ut = Δum + f(u), where m > 1 and f(ξ) is a positive function in ξ > 0 satisfying f(0) = 0 and a blow-up conditionWe show that if ξm+2/N /(−log ξ)β = O(f(ξ)) as ξ ↓ 0 for some 0 < β < 2/(mN + 2), one of the following holds: (i) all non-trivial solutions blow up in finite time; (ii) every non-trivial solution with an initial datum u0 having compact support exists globally in time and grows up to ∞ as t → ∞: limtt→∞ inf|x|<Ru(x, t) = ∞ for any R > 0. Moreover, we give a condition on f such that (i) holds, and show the existence of f such that (ii) holds.

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.

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.

Behavior of blow-up solutions for quasilinear parabolic equations

2020 ◽  
Vol 17 (2) ◽  
pp. 278-295
Author(s):  
Yevgeniia Yevgenieva
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Boundary Function ◽  
Upper Bounds ◽  
Energy Estimates ◽  
Quasilinear Parabolic Equations ◽  
Multidimensional Domain ◽  
Quasilinear Parabolic

We study the quasilinear parabolic equation $(|u|^{q-1}u)_t-\Delta_p\,u=0$ in a multidimensional domain $(0,T)\times\Omega$ under the condition $u(t,x)=f(t,x)$ on $(0,T)\times\partial\Omega$, where the boundary function $f$ blows-up at a finite time $T$, i.e., $f(t,x)\rightarrow\infty$ as $t\rightarrow T$. For $p\geqslant q>0$ and the boundary function $f$ with power-like behavior, the upper bounds of weak solutions of the problem are obtained. The behavior of solutions at the transition from the case where $p>q$ to $p=q$ is investigated. A general approach within the method of energy estimates to such problems is described.

Self-similar boundary blow-up for higher-order quasilinear parabolic equations

2005 ◽  
Vol 135 (6) ◽  
pp. 1195-1227 ◽  
Author(s):  
V. A. Galaktionov ◽  
A. E. Shishkov
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Similarity Solutions ◽  
Energy Estimates ◽  
Quasilinear Parabolic Equations ◽  
Asymptotic Results ◽  
Propagation Of Singularities ◽  
Self Similar ◽  
Similar Solutions ◽  
Quasilinear Parabolic

We study evolution properties of boundary blow-up for 2mth-order quasilinear parabolic equations in the case where, for homogeneous power nonlinearities, the typical asymptotic behaviour is described by exact or approximate self-similar solutions. Existence and asymptotic stability of such similarity solutions are established by energy estimates and contractivity properties of the rescaled flows.Further asymptotic results are proved for more general equations by using energy estimates related to Saint-Venant's principle. The established estimates of propagation of singularities generated by boundary blow-up regimes are shown to be sharp by comparing with various self-similar patterns.

scholarly journals Blow-Up Results for a Class of Quasilinear Parabolic Equation with Power Nonlinearity and Nonlocal Source

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Xiaorong Zhang ◽  
Zhoujin Cui
Keyword(s):  
Parabolic Equation ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Initial Energy ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Smooth Bounded Domain ◽  
Nonlocal Source ◽  
Power Nonlinearity ◽  
Quasilinear Parabolic

This paper deals with a class of quasilinear parabolic equation with power nonlinearity and nonlocal source under homogeneous Dirichlet boundary condition in a smooth bounded domain; we obtain the blow-up condition and blow-up results under the condition of nonpositive initial energy.

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