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.

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.

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.

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.

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.

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