Some Nonexistence Theorem for Initial-Boundary Value Problems with Nonlinear Boundary Constraints

1974 ◽  
Vol 46 (2) ◽  
pp. 277 ◽  
Author(s):  
H. A. Levine ◽  
L. E. Payne
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Boundary Constraints ◽  
Nonexistence Theorem ◽  
Initial Boundary Value

scholarly journals TWO INITIAL-BOUNDARY VALUE PROBLEMS WITH NONLINEAL BOUNDARY CONDITIONS FOR ONE-DIMENSION HYPERBOLIC EQUATION

2017 ◽  
Vol 17 (2) ◽  
pp. 46-56
Author(s):  
L.S. Pulkina ◽  
M.V. Strigun
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Nonlinear Boundary Conditions ◽  
One Dimension ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

In this paper, the initial-boundary value problems for hyperbolic equationwith nonlinear boundary conditions are considered. Existence and uniqueness ofgeneralized solution are proved.

scholarly journals Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition

2021 ◽  
Vol 5 (4) ◽  
pp. 134
Author(s):  
Serik E. Aitzhanov ◽  
Abdumauvlen S. Berdyshev ◽  
Kymbat S. Bekenayeva
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Priori Estimates ◽  
Initial Boundary Value

This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the studied problems is that the boundary conditions are set in the form of a nonlinear boundary condition with a fractional differentiation operator. The main result of this work is establishing the local or global solvability of stated problems, depending on the parameters of the equation. The Galerkin method is used to prove the existence of a quasi-linear pseudo-parabolic equation’s weak solution in a bounded domain. Using Sobolev embedding theorems, a priori estimates of the solution are obtained. A priori estimates and the Rellich–Kondrashov theorem are used to prove the existence of the desired solutions to the considered boundary value problems. The uniqueness of the weak generalized solutions of the initial boundary value problems is proved on the basis of the obtained a priori estimates and the application of the generalized Gronwall lemma. The need to consider and study such initial boundary value problems for a quasi-linear pseudo-parabolic equation follows from practical requirements, such as solving fractional differential equations that simulate physical processes that occur during the study of liquid filtration processes, etc.

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