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 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 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 Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

scholarly journals Regularized Asymptotics of the Solution of the Singularly Perturbed First Boundary Value Problem on the Semiaxis for a Parabolic Equation with a Rational “Simple” Turning Point

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 405
Author(s):  
Alexander Yeliseev ◽  
Tatiana Ratnikova ◽  
Daria Shaposhnikova
Keyword(s):  
Parabolic Equation ◽  
Maximum Principle ◽  
Boundary Value Problem ◽  
Regularization Method ◽  
Turning Point ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Asymptotic Convergence ◽  
The Maximum Principle ◽  
Regularized Asymptotics

The aim of this study is to develop a regularization method for boundary value problems for a parabolic equation. A singularly perturbed boundary value problem on the semiaxis is considered in the case of a “simple” rational turning point. To prove the asymptotic convergence of the series, the maximum principle is used.

scholarly journals Uniqueness and stability of solutions for a type of parabolic boundary value problem

1989 ◽  
Vol 12 (4) ◽  
pp. 735-739
Author(s):  
Enrique A. Gonzalez-Velasco
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Nonnegative Solution ◽  
Boundary Value ◽  
Stability Of Solutions ◽  
General Boundary Conditions ◽  
Boundary Values ◽  
Parabolic Boundary ◽  
One Dimensional ◽  
The One

We consider a boundary value problem consisting of the one-dimensional parabolic equationgut=(hux)x+q, where g, h and q are functions of x, subject to some general boundary conditions. By developing a maximum principle for the boundary value problem, rather than the equation, we prove the uniqueness of a nonnegative solution that depends continuously on boundary values.

scholarly journals On the Weak Solution of a Semilinear Boundary Value Problem without the Landesman-Lazer Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Sikiru Adigun Sanni
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Boundary Value

We prove the existence of weak solution to a semilinear boundary value problem without the Landesman-Lazer condition.

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