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Full-Text Articles in Physical Sciences and Mathematics
On The Qualitative Theory Of The Nonlinear Parabolic P-Laplacian Type Reaction-Diffusion Equations, Roqia Abdullah Jeli
On The Qualitative Theory Of The Nonlinear Parabolic P-Laplacian Type Reaction-Diffusion Equations, Roqia Abdullah Jeli
Theses and Dissertations
This dissertation presents full classification of the evolution of the interfaces and asymptotics of the local solution near the interfaces and at infinity for the nonlinear second order parabolic p-Laplacian type reaction-diffusion equation of non-Newtonian elastic filtration ut − ( |ux| p−2 ux ) x +buβ = 0, p > 1, β > 0. (1) Nonlinear partial differential equation (1) is a key model example expressing competition between nonlinear diffusion with gradient dependent diffusivity in either slow (p > 2) or fast (1 < p < 2) regime and nonlinear state dependent reaction (b > 0) or absorption (b < 0) forces. If interface is finite, it may expand, shrink, or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters p, β,sign b, and asymptotics of the initial function near its support. In the fast diffusion regime strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The methods of the proof are generaliii ization of the methods developed in U.G. Abdulla & J. King, SIAM J. Math. Anal., 32, 2(2000), 235-260; U.G. Abdulla, Nonlinear Analysis, 50, 4(2002), 541-560 and based on rescaling laws for the nonlinear PDE and blow-up techniques for the identification of the asymptotics of the solution near the interfaces, construction of barriers using special comparison theorems in irregular domains with characteristic boundary curves.
Lower Order Perturbations Of Critical Fractional Laplacian Equations, Khalid Fanoukh Al Oweidi
Lower Order Perturbations Of Critical Fractional Laplacian Equations, Khalid Fanoukh Al Oweidi
Theses and Dissertations
We give sufficient conditions for the existence of nontrivial solutions to a class of critical nonlocal problems of the Brezis-Nirenberg type. Our result extends some results in the literature for the local case to the nonlocal setting. It also complements the known results for the nonlocal case.
Boundary Value Problems In A Multidimensional Box For Higher Order Linear And Quasi-Linear Hyperbolic Equations, Noha Aljaber
Boundary Value Problems In A Multidimensional Box For Higher Order Linear And Quasi-Linear Hyperbolic Equations, Noha Aljaber
Theses and Dissertations
Boundary value problems in a multidimensional box for higher order linear hyperbolic equations are considered. The concept of associated problems are introduced. For general boundary value problems there are established: (i) Necessary and sufficient conditions for a linear problem to have the Fredholm property in two–dimensional case; (ii) Necessary and sufficient conditions of well–posedness in two–dimensional case; (iii) Unimprovable sufficient conditions for a linear problem to have the Fredholm property; (iv) Unimprovable sufficient conditions of well–posedness and α–well–posedness; (v) Effective sufficient conditions of unqie solvability of two–point, periodic and Dirichlet type problems. (iv) Unimprovable conditions of unique solvability of two …
On The Qualitative Theory Of The Nonlinear Degenerate Second Order Parabolic Equations Modeling Reaction-Diffusion-Convection Processes, Habeeb Abed Kadhim Aal-Rkhais
On The Qualitative Theory Of The Nonlinear Degenerate Second Order Parabolic Equations Modeling Reaction-Diffusion-Convection Processes, Habeeb Abed Kadhim Aal-Rkhais
Theses and Dissertations
We consider nonlinear second order degenerate or singular parabolic equation ut − a(um)xx + buβ + c(up)x = 0, a, m, β, p > 0, b, c ∈ R describing reaction-diffusion-convection processes arising in many areas of science and engineering, such as filtration of oil or gas in porous media, transport of thermal energy in plasma physics, flow of chemically reacting fluid, evolution of populations in mathematical biology etc. We apply the methods developed in U.G. Abdulla, Journal of Differential Equations, 164, 2(2000), 321-354 for the reaction-diffusion equation (c = 0) and prove the existence, uniqueness, boundary regularity and comparison theorems …