scholarly journals On Spectral and Evolutional Problems Generated by a Sesquilinear Form

2020 ◽  
Vol 66 (2) ◽  
pp. 335-371
Author(s):  
A. R. Yakubova
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Superposition Principle ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Time Interval ◽  
Sesquilinear Form ◽  
Two Parameters ◽  
Initial Boundary Value ◽  
Necessary And Sufficient

On the base of boundary-value, spectral and initial-boundary value problems studied earlier for the case of single domain, we consider corresponding problems generated by sesquilinear form for two domains. Arising operator pencils with corresponding operator coefficients acting in a Hilbert space and depending on two parameters are studied in detail. In the perturbed and unperturbed cases, we consider two situations when one of the parameters is spectral and the other is fixed. In this paper, we use the superposition principle that allow us to present the solution of the original problem as a sum of solutions of auxiliary boundary-value problems containing inhomogeneity either in the equation or in one of the boundary conditions. The necessary and sufficient conditions for the correct solvability of boundary-value problems on given time interval are obtained. The theorems on properties of the spectrum and on the completeness and basicity of the system of root elements are proved.

scholarly journals Blowing-up solutions of the time-fractional dispersive equations

2021 ◽  
Vol 10 (1) ◽  
pp. 952-971
Author(s):  
Ahmed Alsaedi ◽  
Bashir Ahmad ◽  
Mokhtar Kirane ◽  
Berikbol T. Torebek
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Nonlinear Capacity ◽  
Blowing Up ◽  
De Vries ◽  
Initial Boundary Value

Abstract This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.

scholarly journals Matching Spectral and Initial-Boundary Value Problems

2017 ◽  
Vol 63 (2) ◽  
pp. 316-339
Author(s):  
K A Radomirskaya
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Adjoint Operator ◽  
Initial Boundary ◽  
Operator Pencil ◽  
Initial Boundary Value Problems ◽  
Matching Problems ◽  
Two Parameters ◽  
Initial Boundary Value

Based on the approach to abstract matching boundary-value problems introduced in [18], we consider matching spectral problems for one and two domains. We study in detail the arising operator pencil with self-adjoint operator coefficients. This pencil acts in a Hilbert space and depends on two parameters. Both possible cases are considered, where one parameter is spectral and the other is fixed, and properties of solutions are obtained depending on this. Also we study initial-boundary value problems of mathematical physics generating matching problems. We prove theorems on unique solvability of a strong solution ranging in the corresponding Hilbert space.

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