# 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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Item A bidimensional bi-layer shallow-water model(Texas State University, Department of Mathematics, 2017-07-05) Roamba, Brahima; Zabsonre, Jean de DieuThe existence of global weak solutions in a periodic domain for a non-linear viscous bi-layer shallow-water model with capillarity effects and extra friction terms in a two-dimensional space has been proved in [21]. The main contribution of this article is to show the existence of global weak solutions without friction term or capillary effect following the ideas of [20] for the two dimensional case.Item A blow-up result for a viscoelastic system in ℝn(Texas State University-San Marcos, Department of Mathematics, 2007-08-18) Kafini, Mohammad; Messaoudi, Salim A.In this paper we consider a coupled system of nonlinear viscoelastic equations. Under suitable conditions on the initial data and the relaxation functions, we prove a finite-time blow-up result.Item A boundary blow-up for sub-linear elliptic problems with a nonlinear gradient term(Texas State University-San Marcos, Department of Mathematics, 2006-05-20) Zhang, ZhijunBy a perturbation method and constructing comparison functions, we show the exact asymptotic behaviour of solutions to the semilinear elliptic problem Δu - |∇u|q = b(x)g(u), u > 0 in Ω, u|∂Ω = +∞, where Ω is a bounded domain in ℝN with smooth boundary, q ∈ (1, 2], g ∈ C[0, ∞) ∩ C1 (0, ∞), g(0) = 0, g is increasing on [0, ∞), and b is non-negative non-trivial in Ω, which may be singular or vanishing on the boundary.Item A Brezis-Nirenberg problem on hyperbolic spaces(Texas State University, Department of Mathematics, 2019-05-13) Carriao, Paulo Cesar; Lehrer, Raquel; Miyagaki, Oliimpio Hiroshi; Vicente, AndreWe consider a Brezis-Nirenberg problem on the hyperbolic space ℍn. By using the stereographic projection, the problem becomes a singular problem on the boundary of the open ball B1(0) ⊂ ℝn. Thanks to the Hardy inequality, in a version due to Brezis-Marcus, the difficulty involving singularities can be overcame. We use the mountain pass theorem due to Ambrosetti-Rabinowitz and Brezis-Nirenberg arguments to obtain a nontrivial solution.Item A characterisation of infinity-harmonic and p-harmonic maps via affine variations in L-infinity(Texas State University, Department of Mathematics, 2017-01-26) Katzourakis, NikosLet u : Ω ⊆ ℝn → ℝN be a smooth map and n, N ∈ ℕ. The ∞-Laplacian is the PDE system ∆∞u ≔ (Du ⊗ Du + |Du|2 [Du]⊥ ⊗ I) : D2u = 0, where [Du]⊥ ≔ ProjR(Du)⊥. This system constitutes the fundamental equation of vectorial Calculus of Variations in L∞, associated with the model functional E∞(u, Ω′) = ∥|Du|2∥L∞(Ω′), Ω′ ⋐ Ω. We show that generalised solutions to the system can be characterised in terms of the functional via a set of designated affine variations. For the scalar case N = 1, we utilise the theory of viscosity solutions by Crandall-Ishii-Lions. For the vectorial case N ≥ 2, we utilise the recently proposed by the author theory of D-solutions. Moreover, we extend the result described above to the p-Laplacian, 1 < p < ∞.Item A characterization of balls using the domain derivative(Texas State University-San Marcos, Department of Mathematics, 2006-12-14) Didenko, Andriy; Emamizadeh, BehrouzIn this note we give a characterization of balls in ℝN using the domain derivative. As a byproduct we will show that an overdetermined Stekloff eigenvalue problem is solvable if and only if the domain of interest is a ball.Item A class of nonlinear differential equations on the space of symmetric matrices(Southwest Texas State University, Department of Mathematics, 2004-08-06) Dragan, Vasile; Freiling, Gerhard; Hochhaus, Andreas; Morozan, ToaderIn the first part of this paper we analyze the properties of the evolution operators of linear differential equations generating a positive evolution and provide a set of conditions which characterize the exponential stability of the zero solution, which extend the classical theory of Lyapunov. In the main part of this work we prove a monotonicity and a comparison theorem for the solutions of a class of time-varying rational matrix differential equations arising from stochastic control and derive existence and (in the periodic case) convergence results for the solutions. The results obtained are similar to those known for matrix Riccati differential equations. Moreover we provide necessary and sufficient conditions which guarantee the existence of some special solutions for the considered nonlinear differential equations as: maximal solution, stabilizing solution, minimal positive semi-definite solution. In particular it turns out that under the assumption that the underlying system satisfies adequate generalized stabilizability, detectability and definiteness conditions there exists a unique stabilizing solution.Item A Class of Nonlinear Elliptic Variational Inequalities: Qualitative Properties and Existence of Solutions(Southwest Texas State University, Department of Mathematics, 2002-02-09) Korkut, Luka; Pasic, Mervan; Zubrinic, DarkoWe study a class of nonlinear elliptic variational inequalities in divergence form. In the recent paper [6], we obtained results on the local control of essential infimum and supremum of solutions of quasilinear elliptic equations, and here we extend this point of view to the case of variational inequalities. It implies a new qualitative property of solutions in W1,p(Ω) which we call ``jumping over the control obstacle.'' Using the Schwarz symmetrization technique, we give an existence and symmetrization theorems in W1,p0(Ω) ∩ L∞(Ω) which agree completely with previous qualitative results. Also we consider generating singularities of weak solutions in W1,p(Ω) of variational inequalities.Item A classical solution weakened on the axis for a mixed problem of inhomogeneous hyperbolic equations(Southwest Texas State University, Department of Mathematics, 2003-10-09) Al-Momani, RaidThe main purpose of this paper is to present sufficient conditions, on the forcing term of a mixed problem for three dimensional hyperbolic equations of any even order, for the existence of axially weakened classical solutions.Item A comparison principle for a class of subparabolic equations in Grushin-type spaces(Texas State University-San Marcos, Department of Mathematics, 2007-02-14) Bieske, ThomasWe define two notions of viscosity solutions to subparabolic equations in Grushin-type spaces, depending on whether the test functions concern only the past or both the past and the future. We then prove a comparison principle for a class of subparabolic equations and show the sufficiency of considering the test functions that concern only the past.Item A comparison principle for an American option on several assets: Index and spread options(Southwest Texas State University, Department of Mathematics, 2003-07-07) Laurence, Peter; Stredulinsky, EdwardUsing the method of symmetrization, we compare the price of the American option on an index or spread to that of the solution of a parabolic variational inequality in one spatial variable. This comparison principle is established for a broad class of diffusion operators with time and state dependent coefficients. The purpose is to take a first step towards deriving symmmetrized problems whose solutions bound solutions of multidimensional American option problems with variable coefficients when the computation of the latter lies beyond the scope of the most powerful numerical methods.Item A counterexample to an endpoint bilinear Strichartz inequality(Texas State University-San Marcos, Department of Mathematics, 2006-12-05) Tao, TerenceThe endpoint Strichartz estimate ∥eit∆ƒ∥L2t L∞x(ℝxℝ2 ≲ ∥ƒ∥L2x(ℝ2) is known to be false by the work of Montgomery-Smith [2], despite being only “logarithmically far” from being true in some sense. In this short note we show that (in sharp contrast to the Lpt,x Strichartz estimates) the situation is not improved by passing to a bilinear setting; more precisely, if P, P′ are non-trivial smooth Fourier cutoff multipliers then we show that the bilinear estimate ∥(eit∆Pƒ) (eit∆P′g∥ L1t L∞x (ℝxℝ2) ≲ ∥ƒ∥L2x(ℝ2)∥g∥L2x(ℝ2) fails even when P, P′ have widely separated supports.Item A Coupled Cahn-Hilliard Particle System(Southwest Texas State University, Department of Mathematics, 2002-08-13) Shardlow, TonyA Cahn-Hilliard equation is coupled to a system of stochastic differential equations to model a random growth process. We show the model is well posed and analyze the model asymptotically (in the limit of the interfacial distance becoming small), to recover a free boundary problem. A numerical method together with example solutions is presented.Item A Dirichlet Problem in the Strip(Southwest Texas State University, Department of Mathematics, 1996-10-26) Montefusco, EugenioIn this paper we investigate a Dirichlet problem in a strip and, using the sliding method, we prove monotonicity for positive and bounded solutions. We obtain uniqueness of the solution and show that this solution is a function of only one variable. From these qualitative properties we deduce existence of a classical solution for this problem.Item A discontinuous problem involving the p-Laplacian operator and critical exponent in ℝN(Southwest Texas State University, Department of Mathematics, 2003-04-16) Alves, Claudianor; Bertone, Ana MariaUsing convex analysis, we establish the existence of at least two nonnegative solutions for the quasilinear problem -Δpu = H(u - α)up*-1 + λh(x) in ℝN where Δpu is the p-Laplacian operator, H is the Heaviside function, p* is the Sobolev critical exponent, and h is a positive function.Item A double eigenvalue problem for Schrödinger equations involving sublinear nonlinearities at infinity(Texas State University-San Marcos, Department of Mathematics, 2007-03-09) Kristaly, AlexandruWe present some multiplicity results concerning parameterized Schrödinger type equations which involve nonlinearities with sublinear growth at infinity. Some stability properties of solutions with respect to the parameters are also established in an appropriate Sobolev space.Item A fibering map approach to a semilinear elliptic boundary value problem(Texas State University-San Marcos, Department of Mathematics, 2007-05-10) Brown, Kenneth J.; Wu, Tsung-fangWe prove the existence of at least two positive solutions for the semilinear elliptic boundary-value problem -Δu(x) = λα(x)uq + b(x)up for x ∈ Ω; u(x) = 0 for x ∈ ∂Ω on a bounded region Ω by using the Nehari manifold and the fibering maps associated with the Euler functional for the problem. We show how knowledge of the fibering maps for the problem leads to very easy existence proofs.Item A fractional Gronwall inequality and the asymptotic behaviour of global solutions of Caputo fractional problems(Texas State University, Department of Mathematics, 2021-09-20) Webb, JeffreyWe study the asymptotic behaviour of global solutions of some nonlinear integral equations related to some Caputo fractional initial value problems. We consider problems of fractional order between 0 and 1 and of order between 1 and 2, each in two cases: when the nonlinearity depends only on the function, and when the nonlinearity also depends on fractional derivatives of lower order. Our main tool is a new Gronwall inequality for integrals with singular kernels, which we prove here, and a related boundedness property of a fractional integral of an L1[0, ∞) function.Item A Free Boundary Problem for the p-Laplacian: Uniqueness, Convexity, and Successive Approximation of Solutions(Southwest Texas State University, Department of Mathematics, 1995-06-21) Acker, Andrew F.; Meyer, R.We prove convergence of a trial free boundary method to a classical solution of a Bernoulli-type free boundary problem for the p-Laplace equation, 1 < p < ∞. In addition, we prove the existence of a classical solution in N dimensions when p = 2 and, for 1 < p < ∞, results on uniqueness and starlikeness of the free boundary and continuous dependence on the fixed boundary and on the free boundary data. Finally, as an application of the trial free boundary method, we prove (also for 1 < p < ∞) that the free boundary is convex when the fixed boundary is convex.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.