Abstract

In this thesis, our main purpose is to study the qualitative properties of a class of parabolic and stationary problems, as well as establish new versions of discrete Picone\'s inequalities, associated to nonlinear fractional operators. \\\\[4pt] We divided our work into four chapters :\\\\[4pt]
$ \\bullet $ In the first \\textbf{chapter}, we present the comprehensive state of the art and mathematical tools, then included the main results with a glimpse of the proof.\\\\[4pt]
$ \\bullet $ In the second \\textbf{chapter}, we study the existence, uniqueness, and other qualitative properties of the weak solution to a doubly nonlinear parabolic equation involving a nonlinear fractional Laplace operator. First, by using the semi-discretization in time method, we prove the local existence, as well as using fractional Picone inequality, leads to a new comparison principle, hence the uniqueness of weak solutions. Finally, we show that global solutions converge to the unique non-trivial stationary solution by semi-group theory.\\\\[4pt]
$ \\bullet $ In the third \\textbf{chapter}, we firstly established new versions of Picone inequalities to include a large class of fractional and non-homogeneous operators. Second, we give several applications to these inequalities as non-existence, existence, and uniqueness of weak solutions for non-local and non-homogeneous problems. We also obtain comparison principles, a Sturmian comparison principle, and a Hardy-type inequality with weight for this class of operators, as well as some qualitative results to nonlinear elliptic systems with sub-homogeneous growth.\\\\[4pt]
$ \\bullet $ In the last \\textbf{chapter}, we study singular systems involving nonlinear and non-local operators. We first show the non-existence of positive classical solutions. Next, Schauder’s Fixed Point Theorem guaranteed the existence of a positive weak solutions pair in the suitable conical shell, and then H\\\"{o}lder regularity results. Finally, we prove the uniqueness by applying a well-known Krasnoselski\\v{i}\'s argument.\\\\
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\\textbf{key-words :} Fractional $p$-Laplacian operator, doubly nonlinear evolution equation, Picone inequalities, stabilization, nonlinear semi-group theory, positive solutions, non-existence, uniqueness, regularity results, comparison principles, quasilinear singular systems, sub and super-solutions, sub-homogeneous problems, Schauder’s fixed point Theorem.

Information

Item Type:	Thesis
Divisions:
ePrint ID:	5248
Date Deposited:	2024-12-17
Further Information:	Google Scholar
URI:	https://www.univ-soukahras.dz/en/publication/article/5248

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