scholarly journals The applications of Sobolev inequalities in proving the existence of solution of the quasilinear parabolic equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yuanfei Li ◽  
Lianhong Guo ◽  
Peng Zeng
Keyword(s):  
Parabolic Equation ◽  
Partial Differential Equations ◽  
Lower Bound ◽  
Parabolic Equations ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Existence Of Solution ◽  
Sobolev Inequalities ◽  
Quasilinear Parabolic Equations ◽  
Quasilinear Parabolic

Abstract The aim of this paper is to show some applications of Sobolev inequalities in partial differential equations. With the aid of some well-known inequalities, we derive the existence of global solution for the quasilinear parabolic equations. When the blow-up occurs, we derive the lower bound of the blow-up solution.

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.

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.

Focusing blow-up for quasilinear parabolic equations

1998 ◽  
Vol 128 (5) ◽  
pp. 965-992 ◽  
Author(s):  
C. J. Budd ◽  
V. A. Galaktionov ◽  
Jianping Chen
Keyword(s):  
Blow Up ◽  
Numerical Study ◽  
Quasilinear Parabolic Equation ◽  
Reaction Diffusion ◽  
Comparison Method ◽  
Dirichlet Boundary Conditions ◽  
Quasilinear Parabolic Equations ◽  
General Behaviour ◽  
Blowing Up ◽  
Quasilinear Parabolic

We study the behaviour of the non-negative blowing up solutions to the quasilinear parabolic equation with a typical reaction–diffusion right-hand side and with a singularity in the space variable which takes the formwhere m ≧ 1, p > 1 are arbitrary constants, in the critical exponent case q = (p–1)/m > 0. We impose zero Dirichlet boundary conditions at the singular point x = 0 and at x = 1, and take large initial data. For a class of ‘concave’ initial functions, we prove focusing at the origin of the solutions as t approaches the blow-up time T in the sense that x = 0 belongs to the blow-up set. The proof is based on an application of the intersection comparison method with an explicit ‘separable’ solution which has the same blow-up time as u. The method has a natural generalisation to the case of more general nonlinearities in the equation. A description of different fine structures of blow-up patterns in the semilinear case m = 1 and in the quasilinear one m > 1 is also presented. A numerical study of the semilinear equation is also made using an adaptive collocation method. This is shown to give very close agreement with the fine structure predicted and allows us to make some conjectures about the general behaviour.

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