Existence of solutions for degenerate quasilinear parabolic equations of higher order

1997 ◽  
Vol 13 (4) ◽  
pp. 465-472 ◽  
Author(s):  
Liu Zhenhai
Keyword(s):  
Parabolic Equations ◽  
Existence Of Solutions ◽  
Higher Order ◽  
Quasilinear Parabolic Equations ◽  
Quasilinear Parabolic

scholarly journals Existence of Solutions for a Class of Quasilinear Parabolic Equations with Superlinear Nonlinearities

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zhong-Xiang Wang ◽  
Gao Jia ◽  
Xiao-Juan Zhang
Keyword(s):  
Sobolev Space ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Existence Of Solutions ◽  
Weighted Sobolev Space ◽  
Boundary Value ◽  
Growth Conditions ◽  
Quasilinear Parabolic Equations ◽  
Superlinear Growth ◽  
Quasilinear Parabolic

Working in a weighted Sobolev space, this paper is devoted to the study of the boundary value problem for the quasilinear parabolic equations with superlinear growth conditions in a domain of RN. Some conditions which guarantee the solvability of the problem are given.

scholarly journals Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent

2021 ◽  
Vol 41 (1) ◽  
pp. 25-53
Author(s):  
Abderrahim Charkaoui ◽  
Houda Fahim ◽  
Nour Eddine Alaa
Keyword(s):  
Fixed Point ◽  
Parabolic Equations ◽  
Variable Exponent ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Quasilinear Parabolic Equations ◽  
Nonstandard Growth ◽  
Nonlinear Parabolic ◽  
Nonstandard Growth Condition ◽  
Quasilinear Parabolic

We are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer's fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems.

scholarly journals Optimal control of quasilinear parabolic equations

1995 ◽  
Vol 125 (3) ◽  
pp. 545-565 ◽  
Author(s):  
Eduardo Casas ◽  
Luis A. Fernández ◽  
Jiongmin Yong
Keyword(s):  
Optimal Control ◽  
Parabolic Equations ◽  
Optimal Control Problems ◽  
Existence Of Solutions ◽  
State Equation ◽  
Divergence Form ◽  
Control Problems ◽  
Quasilinear Parabolic Equations ◽  
Differentiability Property ◽  
Quasilinear Parabolic

This paper deals with optimal control problems governed by quasilinear parabolic equations in divergence form, whose cost functional is of Lagrangian type. Our aim is to prove the existence of solutions and derive some optimality conditions. To attain this second objective, we accomplish the sensitivity analysis of the state equation with respect to the control, proving that, under some assumptions, this relation is Gâteaux differentiable. Finally, a regularising procedure along with Ekeland's variational principle allow us to treat some other problems for which this differentiability property cannot be stated.

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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