DSpace at Texas State University https://digital.library.txstate.edu:443 The University Scholarship digital repository system captures, stores, indexes, preserves, and distributes digital research material. 2023-05-27T04:12:05Z Inverse problems of periodic spatial distributions for a time fractional diffusion equation https://digital.library.txstate.edu/handle/10877/16886 Inverse problems of periodic spatial distributions for a time fractional diffusion equation Lopushanska, Halyna; Lopushansky, Andrzej; Myaus, Olga For a time fractional diffusion equation and a diffusion-wave equation with Caputo partial derivative we prove the inverse problem is well posed. This problem consists in the restoration of the initial data of a classical solution in time and with values in a space of periodic spatial distributions. A time integral over-determination condition is used. 2016-01-07T00:00:00Z On the Schrödinger equations with isotropic and anisotropic fourth-order dispersion https://digital.library.txstate.edu/handle/10877/16885 On the Schrödinger equations with isotropic and anisotropic fourth-order dispersion Villamizar-Roa, Elder J.; Banquet, Carlos This article concerns the Cauchy problem associated with the nonlinear fourth-order Schrodinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation i∂tu + ε∆u + δAu + λ|u|αu = 0, x ∈ ℝn, t ∈ℝ, where A is either the operator ∆2 (isotropic dispersion) or ∑di=1 ∂xixixixi, 1 ≤ d < n (anisotropic dispersion), and α, ε, λ are real parameters. We obtain local and global well-posedness results in spaces of initial data with low regularity, based on weak- Lp spaces. Our analysis also includes the biharmonic and anisotropic biharmonic equation (ε = 0); in this case, we obtain the existence of self-similar solutions because of their scaling invariance property. In a second part, we analyze the convergence of solutions for the nonlinear fourth-order Schrödinger equation. i∂tu + ε∆u + δ∆2u + λ|u|αu = 0, x ∈ ℝn, t ∈ ℝ, as ε approaches zero, in the H2-norm, to the solutions of the corresponding biharmonic equation i∂tu + δ∆2u + λ|u|αu = 0. 2016-01-07T00:00:00Z Semilinear elliptic problems involving Hardy-Sobolev-Maz'ya potential and Hardy-Sobolev critical exponents https://digital.library.txstate.edu/handle/10877/16884 Semilinear elliptic problems involving Hardy-Sobolev-Maz'ya potential and Hardy-Sobolev critical exponents Jiang, Rui-Ting; Tang, Chun-Lei In this article, we study a class of semilinear elliptic equations involving Hardy-Sobolev critical exponents and Hardy-Sobolev-Maz'ya potential in a bounded domain. We obtain the existence of positive solutions using the Mountain Pass Lemma. 2016-01-07T00:00:00Z Inverse spectral problems for energy-dependent Sturm-Liouville equations with finitely many point delta-interactions https://digital.library.txstate.edu/handle/10877/16883 Inverse spectral problems for energy-dependent Sturm-Liouville equations with finitely many point delta-interactions Manafov, Manaf Dzh. In this study, inverse spectral problems for a energy-dependent Sturm-Liouville equations with finitely many point δ-interactions. The uniqueness theorems for the inverse problems of reconstruction of the boundary value problem from the Weyl function, from the spectral data and from two spectra are proved and a constructive procedure for finding its solution are obtained. 2016-01-06T00:00:00Z