On the study of the existence of nontrivial solutions for a parabolic fractional problem
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Date
2024
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20 August 1955 University of Skikda
Abstract
The fundamental focus of this thesis is to investigate certain type of nonlinear
fractional partial differential equations (FPDEs), both elliptic and parabolic
in nature. Where we interest in this works under certain assumptions on the nonlinear terms to study the existence of weak solutions to five classes of fractional
partial differential equations. We use the technique of the Leray-Schauder degree
theory together with the application of Schauder fixed point theorem for demonstrate this. Then, for the uniqueness of weak solutions, we suggest the Banach
contraction principle theorem. we also use the Galerkin approach to prove the existence and uniqueness results. The first class is the semilinear fractional elliptic
problem involving the distributional Riesz fractional gradient in Bessel potential
spaces. The second class is semilinear fractional system involving a nonlocal operator. Therefor, the third class in this thesis focuses on adding the transport
term in the nonlinear fractional problem involving the distributional Riesz fractional derivative. Then the primary objective in the forth class is time fractional
semilinear equations involving Riemann-Liouville time fractional derivative with
fractional Laplacian. Finally, the the last class is the time fractional semilinear
equation containing the Riemann-Liouville derivative.
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Keywords
parabolic fractional problem, Distributional Riesz fractional gradient