scholarly journals Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations

2017 ◽  
Vol 86 (306) ◽  
pp. 1527-1552 ◽  
Author(s):  
Georgios Akrivis ◽  
Buyang Li ◽  
Christian Lubich
Keyword(s):  
Parabolic Equations ◽  
Maximal Regularity ◽  
Energy Estimates ◽  
Quasilinear Parabolic Equations ◽  
Quasilinear Parabolic

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.

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.

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