Nonexistence of Global Solutions for Coupled System of Pseudoparabolic Equations with Variable Exponents and Weak Memories

Zennir, Khaled; Dridi, Hanni; Alodhaibi, Sultan S.; Alkhalaf, Salem

doi:10.1155/2021/5573959

Cited by 3 publications

(1 citation statement)

References 12 publications

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“…References [15][16][17][18][19][20] are devoted to the study of various local and non-local initialboundary value problems for Sobolev type differential equations and its subclass of pseudoparabolic equations.…”

Section: Introductionmentioning

confidence: 99%

Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator

Aitzhanov,

Bekenayeva,

Abdikalikova

2023

Mathematics

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Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a crucial role in describing plenty of natural processes concerning physics, biology, geology, and so on. In this paper, we studied a loaded equation in relation to a spatial variable for a linear pseudoparabolic equation, with an initial and second boundary value condition (the Neumann condition), and a fractional Caputo derivative. A distinctive feature of the considered problem is that the load at the point is in the higher partial derivatives of the solution. The problem is reduced to a loaded equation with a nonlocal boundary value condition. A way to solve the considered problem is by using the method of energy inequalities, so that a priori estimates of solutions for non-local boundary value problems are obtained. To prove that this nonlocal problem is solvable, we used the method of continuation with parameters. The existence and uniqueness theorems for regular solutions are proven.

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Section: Introductionmentioning

confidence: 99%

Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator

Aitzhanov,

Bekenayeva,

Abdikalikova

2023

Mathematics

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Solvability of pseudoparabolic equation with Caputo fractional derivative

Aitzhanov

Kusherbayeva

Bekenayeva

2022

Chaos, Solitons & Fractals

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Blow-up and global existence of solutions for time-space fractional pseudo-parabolic equation

Li¹,

Yang

2023

MATH

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<abstract>In this article, we consider the Cauchy problem for the following time-space fractional pseudo-parabolic equations <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{l} { }_{0}^{C} D_{t}^{\alpha}(I-m \Delta ) u+\left ( - \Delta \right ) ^{\frac{\beta }{2} } u = |u|^{p-1} u, \quad x \in \mathbb{R}^{N}, \quad t>0, \\ u(0, x) = u_{0}(x), \quad\quad\quad\quad\quad\quad\quad\qquad x \in \mathbb{R}^{N}, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula> where $ 0 < \alpha < 1, \ 0 < \beta < 2, \ p > 1, \ m > 0, \ u_{0} \in L^{q}\left(\mathbb{R}^{N}\right) $. An estimating $ L^p-L^q $ for solution operator of time-space fractional pseudo-parabolic equations is obtained. The critical exponents of this problem are determined when $ u_0\in L^{q}(\mathbb{R}^{N}). $ Moreover, we also obtain global existence of the mild solution when $ u_0\in L^p(\mathbb{R}^{N})\cap L^q(\mathbb{R}^{N}) $ small enough.</abstract>

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Nonexistence of Global Solutions for Coupled System of Pseudoparabolic Equations with Variable Exponents and Weak Memories

Cited by 3 publications

References 12 publications

Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator

Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator

Solvability of pseudoparabolic equation with Caputo fractional derivative

Blow-up and global existence of solutions for time-space fractional pseudo-parabolic equation

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