Structure of boundary blow-up for higher-order quasilinear parabolic equations

2004 ◽  
Vol 460 (2051) ◽  
pp. 3299-3325 ◽  
Author(s):  
V. A. Galaktionov ◽  
A. E. Shishkov
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Higher Order ◽  
Quasilinear Parabolic Equations ◽  
Quasilinear Parabolic

Saint-Venant's principle in blow-up for higher-order quasilinear parabolic equations

2003 ◽  
Vol 133 (5) ◽  
pp. 1075-1119 ◽  
Author(s):  
V. A. Galaktionov ◽  
A. E. Shishkov
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Theory Of Elasticity ◽  
Higher Order ◽  
Energy Estimates ◽  
Quasilinear Parabolic Equations ◽  
Group Invariant ◽  
Blowing Up ◽  
Jacobi Equations ◽  
Quasilinear Parabolic

We prove localization estimates for general 2mth-order quasilinear parabolic equations with boundary data blowing up in finite time, as t → T−. The analysis is based on energy estimates obtained from a system of functional inequalities expressing a version of Saint-Venant's principle from the theory of elasticity. We consider a special class of parabolic operators including those having fixed orders of algebraic homogenuity p > 0. This class includes the second-order heat equation and linear 2mth-order parabolic equations (p = 1), as well as many other higher-order quasilinear ones with p ≠ 1. Such homogeneous equations can be invariant under a group of scaling transformations, but the corresponding least-localized regional blow-up regimes are not group invariant and exhibit typical exponential singularities ~ e(T−t)−γ → ∞ as t → T−, with the optimal constant γ = 1/[m(p + 1) − 1] > 0. For some particular equations, we study the asymptotic blow-up behaviour described by perturbed first-order Hamilton–Jacobi equations, which shows that general estimates of exponential type are sharp.

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.

Sign in / Sign up

Export Citation Format

Share Document