# Electronic Journal of Differential Equations

Permanent URI for this collectionhttps://hdl.handle.net/10877/86

The *Electronic Journal of Differential Equations* is hosted by the Department of Mathematics at Texas State University. Since its foundation in 1993, this e-journal has been dedicated to the rapid dissemination of high quality research in mathematics.

**Journal Website: http://ejde.math.txstate.edu/**

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### Recent Submissions

Item Weak asymptotic solution for a non-strictly hyperbolic system of conservation laws-II(Texas State University, Department of Mathematics, 2016-04-07) Sahoo, Manas Ranjan; Singh, HarendraIn this article we introduce a concept of entropy weak asymptotic solution for a system of conservation laws and construct the same for a prolonged system of conservation laws which is highly non-strictly hyperbolic. This is first done for Riemann type initial data by introducing δ, δ′ and δ″ waves along a discontinuity curve and then for general initial data by piecing together the Riemann solutions.Item Nonexistence of global solutions of Emden-Fowler type semilinear wave equations with non-positive energy(Texas State University, Department of Mathematics, 2016-04-07) Li, Meng-RongIn this article we study the blow-up phenomena of solutions to the Emden-Fowler type semilinear wave equation t2utt - uxx = up in [1, T) x (α, b)).Item Bifurcation for elliptic forth-order problems with quasilinear source term(Texas State University, Department of Mathematics, 2016-04-06) Saanouni, Soumaya; Trabelsi, NihedWe study the bifurcations of the semilinear elliptic forth-order problem with Navier boundary conditions ∆2u - div(c(x)∇u) = λƒ(u) in Ω, ∆u = u = 0 on ∂Ω. Where Ω ⊂ ℝn, n ≥ 2 is a smooth bounded domain, ƒ is a positive, increasing and convex source term and c(x) is a smooth positive function on Ω̅ such that the L∞-norm of its gradient is small enough. We prove the existence, uniqueness and stability of positive solutions. We also show the existence of critical value λ* and the uniqueness of its extremal solutions.Item Blow-up phenomena for second-order differential inequalities with shifted arguments(Texas State University, Department of Mathematics, 2016-03-31) Jleli, Mohamed; Kirane, Mokhtar; Samet, BessemWe provide sufficient conditions for the blow-up of solutions to certain second-order differential inequalities and systems with advanced and delayed arguments. Our proofs are based on the test function method.Item A general product measurability theorem with applications to variational inequalities(Texas State University, Department of Mathematics, 2016-03-31) Kuttler, Kenneth; Li, Ji; Shillor, MeirThis work establishes the existence of measurable weak solutions to evolution problems with randomness by proving and applying a novel theorem on product measurability of limits of sequences of functions. The measurability theorem is used to show that many important existence theorems within the abstract theory of evolution inclusions or equations have straightforward generalizations to settings that include random processes or coefficients. Moreover, the convex set where the solutions are sought is not fixed but may depend on the random variables. The importance of adding randomness lies in the fact that real world processes invariably involve randomness and variability. Thus, this work expands substantially the range of applications of models with variational inequalities and differential set-inclusions.Item Existence, boundary behavior and asymptotic behavior of solutions to singular elliptic boundary-value problems(Texas State University, Department of Mathematics, 2016-03-31) Gao, Ge; Yan, BaoqiangIn this article, we consider the singular elliptic boundary-value problem -∆u + ƒ(u) - u-γ = λu in Ω, u > 0 in Ω, u = 0 on ∂Ω. Using the upper-lower solution method, we show the existence and uniqueness of the solution. Also we study the boundary behavior and asymptotic behavior of the positive solutions.Item Infinitely many solutions for fractional Schrödinger equations in ℝN(Texas State University, Department of Mathematics, 2016-03-30) Chen, CaishengUsing variational methods we prove the existence of infinitely many solutions to the fractional Schrödinger equation (-∆)s u + V(x)u = ƒ(x, u), x ∈ ℝN, where N ≥ 2, s ∈ (0, 1). (-∆)s stands for the fractional Laplacian. The potential function satisfies V(x) ≥ V0 > 0. The nonlinearity ƒ(x, u) is superlinear, has subcritical growth in u, and may or may not satisfy the (AR) condition.Item Stability of the basis property of eigenvalue systems of Sturm-Liouville operators with integral boundary condition(Texas State University, Department of Mathematics, 2016-03-30) Imanbaev, NurlanWe study a question on stability and instability of the basis property of a system of eigenfunctions of the Sturm - Liouville operator, with an integral perturbation of anti-periodic type on the boundary conditions.Item Exponential stability of traveling waves for non-monotone delayed reaction-diffusion equations(Texas State University, Department of Mathematics, 2016-03-29) Liu, Yixin; Yu, Zhixian; Xia, JingThis article concerns the exponential stability of non-critical traveling waves (the wave speed is greater than the minimum speed) for non-monotone time-delayed reaction-diffusion equations. With the help of the weighted energy method, we prove that the non-critical travelling waves are exponentially stable when the initial perturbation around the wave is small.Item Regularity criteria for the wave map and related systems(Texas State University, Department of Mathematics, 2016-03-29) Fan, Jishan; Zhou, YongWe obtain some regularity criteria for the wave map, a liquid crystals model, and the Hall-MHD with ion-slip effect.Item Basic existence and a priori bound results for solutions to systems of boundary value problems for fractional differential equations(Texas State University, Department of Mathematics, 2016-03-23) Tisdell, ChristopherThis article examines the qualitative properties of solutions to systems of boundary value problems involving fractional differential equations. Our primary interest is in forming new results that involve sufficient conditions for the existence of solutions. To do this, we formulate some new ideas concerning a priori bounds on solutions, which are then applied to produce the novel existence results. The main techniques of the paper involve the introduction of novel fractional differential inequalities and the application of the fixed-point theorem of Schafer. We conclude the work with several new results that link the number of solutions to our problem with a fractional initial value problem, akin to an abstract shooting method. A YouTube video from the author that is designed to complement this research is available at youtube.com/watch?v=cDUrLsQLGvAItem Asymptotically periodic solutions of Volterra integral equations(Texas State University, Department of Mathematics, 2016-03-23) Islam, Muhammad N.We study the existence of asymptotically periodic solutions of a nonlinear Volterra integral equation. In the process, we obtain the existence of periodic solutions of an associated nonlinear integral equation with infinite delay. Schauder's fixed point theorem is used in the analysis.Item A priori bounds and existence of non-real eigenvalues of fourth-order boundary value problem with indefinite weight function(Texas State University, Department of Mathematics, 2016-03-23) Han, Xiaoling; Gao, TingIn this article, we give a priori bounds on the possible non-real eigenvalue of regular fourth-order boundary value problem with indefinite weight function and obtain a sufficient conditions for such problem to admit non-real eigenvalue.Item Reaction diffusion equations with boundary degeneracy(Texas State University, Department of Mathematics, 2016-03-23) Zhan, HuashuiIn this article, we consider the reaction diffusion equation ∂u/∂t = ∆A(u), (x, t) ∈ Ω x (0, T), with the homogeneous boundary condition. Inspired by the Fichera-Oleĭnik theory, if the equation is not only strongly degenerate in the interior of Ω, but also degenerate on the boundary, we show that the solution of the equation is free from any limitation of the boundary condition.Item Uniform convergence of the spectral expansions in terms of root functions for a spectral problem(Texas State University, Department of Mathematics, 2016-03-18) Kerimov, Nazim; Goktas, Sertac; Maris, Emir A.In this article, we consider the spectral problem -y″ + q(x)y = λy, 0 < x < 1, y′(0) sin β = y(0) cos β, 0 ≤ β < π; y′(1) = (αλ + b)y(1) where λ is a spectral parameter, α and b are real constants and α < 0, q(X) is a real-valued continuous function on the interval [0, 1]. The root function system of this problem can also consist of associated functions. We investigate the uniform convergence of the spectral expansions in terms of root functions.Item Ergodicity of the two-dimensional magnetic Benard problem(Texas State University, Department of Mathematics, 2016-03-16) Yamazaki, KazuoWe study the two-dimensional magnetic Benard problem with noise, white in time. We prove the well-posedness including the path-wise uniqueness of the generalized solution, and the existence of the unique invariant, and consequently ergodic, measure under random perturbation.Item Multiple solutions for p-Laplacian problems involving general subcritical growth in bounded domains(Texas State University, Department of Mathematics, 2016-03-18) Chung, Nguyen Thanh; Minh, Pham Hong; Nga, Tran HongUsing variational methods, we study the existence of multiple solutions for a class of p-Laplacian problems with concave-convex nonlinearities in bounded domains. Our result improves those in [8,9] stated only for subcritical growth.Item Entropy solutions of exterior problems for nonlinear degenerate parabolic equations with nonhomogeneous boundary condition(Texas State University, Department of Mathematics, 2016-03-18) Zhang, Li; Su, NingIn this article, we consider the exterior problem for the nonlinear degenerate parabolic equation ut - ∆b(u) + ∇ ⋅ ɸ(u) = F(u), (t, x) ∈ (0, T) x Ω, Ω is the exterior domain of Ω0 (a closed bounded domain in ℝN with its boundary Γ ∈ C1,1), b is non-decreasing and Lipschitz continuous, ɸ = (φ1,…,φN) is vectorial continuous, and F is Lipschitz continuous. In the nonhomogeneous boundary condition where b(u) = b(ɑ) on (0, T) x Γ, we establish the comparison and uniqueness, the existence using penalized method.Item Quasi-spectral decomposition of the heat potential(Texas State University, Department of Mathematics, 2016-03-17) Kal'menov, Tynysbek; Arepova, GaukharIn this article, by multiplying of the unitary operator (Pƒ) (x,t) = ƒ(x, T - t), 0 ≤ t ≤ T, the heat potential turns into a self-adjoint operator. From the spectral decomposition of this completely continuous self-adjoint operator we obtain a quasi-spectral decomposition of the heat potential operator.Item Semi-classical states for Schrödinger-Poisson systems on R^3(Texas State University, Department of Mathematics, 2016-03-17) Zhu, HongboIn this article, we study the nonlinear Schrödinger-Poisson equation -ε2∆u + V(x)u + φ(x)u = ƒ(u), x ∈ ℝ3, -ε2∆φ = u2, lim |x|→∞ φ(x) = 0. Under suitable assumptions on V(x) and ƒ(x), we prove the existence of ground state solution around local minima of the potential V(x) as ε → 0. Also, we show the exponential decay of ground state solution.