scholarly journals On the Boundary Value Condition of an Isotropic Parabolic Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Qitong Ou ◽  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Strong Solutions ◽  
Well Posedness ◽  
Boundary Value Condition ◽  
Anisotropic Parabolic Equation ◽  
The Stability ◽  
First Time

The well-posedness problem of anisotropic parabolic equation with variable exponents is studied in this paper. The weak solutions and the strong solutions are introduced, respectively. By a generalized Gronwall inequality, the stability of strong solutions to this equation is established, and the uniqueness of weak solutions is proved. Compared with the related works, a new boundary value condition, ∏ i = 1 N a i x , t = 0 , x , t ∈ ∂ Ω × 0 , T , is introduced the first time and has been proved that it can take place of the Dirichlet boundary value condition in some way.

scholarly journals A New Method to Deal with the Stability of the Weak Solutions for a Nonlinear Parabolic Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
New Method ◽  
Boundary Value Condition ◽  
Partial Boundary Value Condition ◽  
Nonlinear Parabolic ◽  
The Stability

Consider the nonlinear parabolic equation ∂u/∂t-div(a(x)|∇u|p-2∇u)=f(x,t,u,∇u) with axx∈Ω>0 and a(x)x∈∂Ω=0. Though it is well known that the degeneracy of a(x) may cause the usual Dirichlet boundary value condition to be overdetermined, and only a partial boundary value condition is needed, since the nonlinearity, this partial boundary can not be depicted out by Fichera function as in the linear case. A new method is introduced in the paper; accordingly, the stability of the weak solutions can be proved independent of the boundary value condition.

scholarly journals On an Anisotropic Parabolic Equation on the Domain with a Disjoint Boundary

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Boundary Value ◽  
Variable Exponents ◽  
Boundary Value Condition ◽  
Partial Boundary Value Condition ◽  
Anisotropic Parabolic Equation ◽  
The Stability

Consider the anisotropic parabolic equation with the variable exponentsvt=∑i=1n(bi(x)vxiqix-2vxi)xi,wherebi(x),qi(x)∈C1(Ω¯),qi(x)>1, andbi(x)≥0. If{bi(x)}is not degenerate onΣp⊂∂Ω, a part of the boundary, but is degenerate on the remained part∂Ω∖Σp, then the boundary value condition is imposed onΣp, but there is no boundary value condition required on∂Ω∖Σp. The stability of the weak solutions can be proved based on the partial boundary value conditionvx∈Σp=0.

scholarly journals The weak solutions of a nonlinear parabolic equation from two-phase problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhisheng Huang
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Nonlinear Parabolic Equation ◽  
Boundary Value ◽  
Phase Problem ◽  
Two Phase ◽  
Boundary Value Condition ◽  
Partial Boundary Value Condition ◽  
Nonlinear Parabolic ◽  
The Stability

AbstractA nonlinear parabolic equation from a two-phase problem is considered in this paper. The existence of weak solutions is proved by the standard parabolically regularized method. Different from the related papers, one of diffusion coefficients in the equation, $b(x)$ b ( x ) , is degenerate on the boundary. Then the Dirichlet boundary value condition may be overdetermined. In order to study the stability of weak solution, how to find a suitable partial boundary value condition is the foremost work. Once such a partial boundary value condition is found, the stability of weak solutions will naturally follow.

scholarly journals The Partial Second Boundary Value Problem of an Anisotropic Parabolic Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Huashui Zhan
Keyword(s):  
Weak Solution ◽  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Variable Exponents ◽  
Boundary Value Condition ◽  
Anisotropic Parabolic Equation

Consider an anisotropic parabolic equation with the variable exponents vt=∑i=1n(bi(x,t)vxipi(x)-2vxi)xi+f(v,x,t), where bi(x,t)∈C1(QT¯), pi(x)∈C1(Ω¯), pi(x)>1, bi(x,t)≥0, f(v,x,t)≥0. If {bi(x,t)} is degenerate on Γ2⊂∂Ω, then the second boundary value condition is imposed on the remaining part ∂Ω∖Γ2. The uniqueness of weak solution can be proved without the boundary value condition on Γ2.

scholarly journals On Solutions of a Parabolic Equation with Nonstandard Growth Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Growth Condition ◽  
Boundary Value ◽  
Growth Conditions ◽  
Boundary Value Condition ◽  
Nonstandard Growth ◽  
Partial Boundary Value Condition ◽  
Nonstandard Growth Condition ◽  
Nonstandard Growth Conditions ◽  
The Stability

A parabolic equation with nonstandard growth condition is considered. A kind of weak solution and a kind of strong solution are introduced, respectively; the existence of solutions is proved by a parabolically regularized method. The stability of weak solutions is based on a natural partial boundary value condition. Two novelty elements of the paper are both the dependence of diffusion coefficient bx,t on the time variable t, and the partial boundary value condition based on a submanifold of ∂Ω×0,T. How to overcome the difficulties arising from the nonstandard growth conditions is another technological novelty of this paper.

scholarly journals The Evolutionary p(x)-Laplacian Equation with a Partial Boundary Value Condition

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Huashui Zhan ◽  
Zhen Zhou
Keyword(s):  
Diffusion Coefficient ◽  
Explicit Formula ◽  
Boundary Value ◽  
Electrorheological Fluids ◽  
Well Posedness ◽  
Laplacian Equation ◽  
Boundary Value Condition ◽  
Partial Boundary Value Condition ◽  
The Stability ◽  
Diffusion Convection

Consider a diffusion convection equation coming from the electrorheological fluids. If the diffusion coefficient of the equation is degenerate on the boundary, generally, we can only impose a partial boundary value condition to ensure the well-posedness of the solutions. Since the equation is nonlinear, the partial boundary value condition cannot be depicted by Fichera function. In this paper, when α<p--1, an explicit formula of the partial boundary on which we should impose the boundary value is firstly depicted. The stability of the solutions, dependent on this partial boundary value condition, is obtained. While α>p+-1, the stability of the solutions is obtained without the boundary value condition. At the same time, only if α>0 and p->1 can the uniqueness of the solutions be proved without any boundary value condition.

scholarly journals On a degenerate parabolic equation from double phase convection

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Degenerate Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Value Condition ◽  
Double Phase ◽  
Degenerate Parabolic ◽  
Initial Boundary Value

AbstractThe initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let $a(x)$ a ( x ) and $b(x)$ b ( x ) be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x ∈ Ω ‾ and the boundary value condition should be imposed. In this paper, the condition $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x ∈ Ω ‾ is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution u is proved by parabolically regularized method, and $u_{t}\in L^{2}(Q_{T})$ u t ∈ L 2 ( Q T ) is shown. The stability of weak solutions is studied according to the different integrable conditions of $a(x)$ a ( x ) and $b(x)$ b ( x ) . To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by $a(x)b(x)|_{x\in \partial \Omega }=0$ a ( x ) b ( x ) | x ∈ ∂ Ω = 0 is found for the first time.

scholarly journals Diffusion Convection Equation with Variable Nonlinearities

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Huashui Zhan
Keyword(s):  
Diffusion Coefficient ◽  
Weak Solution ◽  
Local Stability ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Boundary Value Condition ◽  
Partial Boundary Value Condition ◽  
Convection Equation ◽  
The Stability ◽  
Diffusion Convection

The paper studies diffusion convection equation with variable nonlinearities and degeneracy on the boundary. Unlike the usual Dirichlet boundary value, only a partial boundary value condition is imposed. If there are some restrictions in the diffusion coefficient, the stability of the weak solution based on the partial boundary value condition is obtained. In general, we may obtain a local stability of the weak solutions without any boundary value condition.

scholarly journals The weak solutions of a doubly nonlinear parabolic equation related to the $p(x)$-Laplacian

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Parabolic Equation ◽  
The Stability ◽  
Initial Boundary Value

Abstract A nonlinear degenerate parabolic equation related to the $p(x)$p(x)-Laplacian $$ {u_{t}}= \operatorname{div} \bigl({b(x)} { \bigl\vert {\nabla a(u)} \bigr\vert ^{p(x) - 2}}\nabla a(u) \bigr)+\sum _{i=1}^{N}\frac{\partial b_{i}(u)}{ \partial x_{i}}+c(x,t) -b_{0}a(u) $$ut=div(b(x)|∇a(u)|p(x)−2∇a(u))+∑i=1N∂bi(u)∂xi+c(x,t)−b0a(u) is considered in this paper, where $b(x)|_{x\in \varOmega }>0$b(x)|x∈Ω>0, $b(x)|_{x \in \partial \varOmega }=0$b(x)|x∈∂Ω=0, $a(s)\geq 0$a(s)≥0 is a strictly increasing function with $a(0)=0$a(0)=0, $c(x,t)\geq 0$c(x,t)≥0 and $b_{0}>0$b0>0. If $\int _{\varOmega }b(x)^{-\frac{1}{p ^{-}-1}}\,dx\leq c$∫Ωb(x)−1p−−1dx≤c and $\vert \sum_{i=1}^{N}b_{i}'(s) \vert \leq c a'(s)$|∑i=1Nbi′(s)|≤ca′(s), then the solutions of the initial-boundary value problem is well-posedness. When $\int _{\varOmega }b(x)^{-(p(x)-1)}\,dx<\infty $∫Ωb(x)−(p(x)−1)dx<∞, without the boundary value condition, the stability of weak solutions can be proved.

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