EXISTENCE OF A WEAK SOLUTION IN AN INFINITE VISCOELASTIC STRIP WITH A SEMI-INFINITE CRACK

2004 ◽  
Vol 14 (07) ◽  
pp. 975-986 ◽  
Author(s):  
HIROMICHI ITOU ◽  
ATUSI TANI
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sobolev Type ◽  
Viscoelastic Strip ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
Initial Boundary Value

We study an initial-boundary value problem in an infinite viscoelastic strip with a semi-infinite fixed crack. For this problem we prove the existence and uniqueness of a weak solution which is prescribed on each side of the extended crack in Sobolev-type spaces.

scholarly journals On Initial Boundary Value Problem for Parabolic Differential Operator with Non-coercive Boundary Conditions

2020 ◽  
pp. 547-558
Author(s):  
Alexander N. Polkovnikov
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sobolev Type ◽  
Type Spaces ◽  
Bochner Space ◽  
Initial Boundary Value

We consider initial boundary value problem for uniformly 2-parabolic differential operator of second order in cylinder domain in Rn with non-coercive boundary conditions. In this case there is a loss of smoothness of the solution in Sobolev type spaces compared with the coercive situation. Using by Faedo-Galerkin method we prove that problem has unique solution in special Bochner space

scholarly journals Existence and uniqueness for the nonstationary problem of the electrical heating of a conductor due to the Joule-Thomson effect

1993 ◽  
Vol 16 (1) ◽  
pp. 125-138 ◽  
Author(s):  
Xiangsheng Xu
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonstationary Problem ◽  
Electrical Heating ◽  
Thomson Effect ◽  
Special Cases ◽  
Initial Boundary Value

Existence of a weak solution is established for the initial-boundary value problem for the system∂∂tu−div(θ(u)∇u)+σ(u)α(u)∇u∇v=σ(u)|∇v|2,div(σ(u)∇v)=0. The question of uniqueness is also considered in some special cases.

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.

AN INEQUALITY BY GIANAZZA, SAVARÉ AND TOSCANI AND ITS APPLICATIONS TO THE VISCOUS QUANTUM EULER MODEL

2013 ◽  
Vol 15 (05) ◽  
pp. 1250067 ◽  
Author(s):  
XIANGSHENG XU
Keyword(s):  
Weak Solution ◽  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Fisher Information ◽  
Gradient Flow ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Drift Diffusion ◽  
Euler Model ◽  
Initial Boundary Value

In this paper we present a simplified version of a coercivity inequality due to Gianazza, Savaré, and Toscani [The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation, Arch. Ration. Mech. Anal.194 (2009) 133–220]. Then we use the inequality to construct a weak solution to the initial-boundary value problem for the viscous quantum Euler model.

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