# 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 Lie generators for semigroups of transformations on a polish space(Southwest Texas State University, Department of Mathematics, 1993-08-27) Dorroh, J. R.; Neuberger, John W.Let X be a separable complete metric space. We characterize completely the infinitesimal generators of semigroups of linear transformations in Cb(X), the bounded real-valued continuous functions on X, that are induced by strongly continuous semigroups of continuous transformations in X. In order to do this, Cb(X) is equipped with a locally convex topology known as the strict topology.Item A singular perturbation problem in integrodifferential equations(Southwest Texas State University, Department of Mathematics, 1993-09-16) Liu, James H.An optimal order of convergence result, with respect to the error level in the data, is given for a Tikhonov-like method for approximating values of an unbounded operator. It is also shown that if the choice of parameter in the method is made by the discrepancy principle, then the order of convergence of the resulting method is suboptimal. Finally, a modified discrepancy principle leading to an optimal order of convergence is developed.Item Estimates for smooth absolutely minimizing Lipschitz extensions(Southwest Texas State University, Department of Mathematics, 1993-10-12) Evans, Lawrence C.I present some elementary maximum principle arguments, establishing interior gradient bounds and Harnack inequalities for both u and |Du|, where u is a smooth solution of the degenerate elliptic PDE ∆∞u = 0. These calculations in particular extend to higher dimensions G. Aronsson’s assertion [2] that a nonconstant, smooth solution can have no interior critical point.Item The optimal order of convergence for stable evaluation of differential operators(Southwest Texas State University, Department of Mathematics, 1993-10-14) Groetsch, C. W.; Scherzer, O.An optimal order of convergence result, with respect to the error level in the data, is given for a Tikhonov-like method for approximating values of an unbounded operator. It is also shown that if the choice of parameter in the method is made by the discrepancy principle, then the order of convergence of the resulting method is suboptimal. Finally, a modified discrepancy principle leading to an optimal order of convergence is developed.Item Least-energy Solutions to a Non-autonomous Semilinear Problem with Small Diffusion Coefficient(Southwest Texas State University, Department of Mathematics, 1993-10-15) Ren, XiaofengLeast-energy solutions of a non-autonomous semilinear problem with a small diffusion coefficient are studied in this paper. We prove that the solutions will develop single peaks as the diffusion coefficient approaches 0. The location of the peaks is also considered in this paper. It turns out that the location of the peaks is determined by the non-autonomous term of the equation and the type of the boundary condition. Our results are based on fine estimates of the energies of the solutions and some non-existence results for semilinear equations on half spaces with Dirichlet boundary condition and some decay conditions at infinity.Item Analysis for a molten carbonate fuel cell(Southwest Texas State University, Department of Mathematics, 1993-10-19) van Duijn, C. J.; Fehribach, Joseph D.In this paper we analyze a planar model for a molten carbonate electrode of a fuel cell. The model consists of two coupled second-order ordinary differential equations, one for the concentration of the reactant gas and one for the potential. Restricting ourselves to the case of a positive reaction order in the Butler-Volmer equation, we consider existence, uniqueness, various monotonicity properties, and an explicit approximate solution for the model. We also present an iteration scheme to obtain solutions, and we conclude with a few numerical examples.Item The Lazer Mckenna Conjecture for Radial Solutions in the R(N) Ball(Southwest Texas State University, Department of Mathematics, 1993-10-30) Castro, Alfonso; Gadam, SudhasreeWhen the range of the derivative of the nonlinearity contains the first k eigenvalues of the linear part and a certain parameter is large, we establish the existence of 2k radial solutions to a semilinear boundary value problem. This proves the Lazer McKenna conjecture for radial solutions. Our results supplement those in [5], where the existence of k + 1 solutions were proven.Item One-sided Mullins-Sekerka Flow Does Not Preserve Convexity(Southwest Texas State University, Department of Mathematics, 1993-12-13) Mayer, Uwe F.The Mullins-Sekerka model is a nonlocal evolution model for hyper-surfaces, which arises as a singular limit for the Cahn-Hilliard equation. Assuming the existence of sufficiently smooth solutions we will show that the one-sided Mullins-Sekerka flow does not preserve convexity.Item Homoclinic Orbits for a Class of Symmetric Hamiltonian Systems(Southwest Texas State University, Department of Mathematics, 1994-02-15) Korman, Philip; Lazer, Alan C.We study existence of homoclinic orbits for a class of Hamiltonian systems that are symmetric with respect to independent variable (time). For the scalar case we prove existence and uniqueness of a positive homoclinic solution. For the system case we prove existence of symmetric homoclinic orbits. We use variational approach.Item Large Time Behavior of Solutions to a Class of Doubly Nonlinear Parabolic Equations(Southwest Texas State University, Department of Mathematics, 1994-03-15) Manfredi, Juan J.; Vespri, VincenzoWe study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation ut = div |u|m−1|∇u|p−2∇u in a cylinder Ω × R+, with initial condition u(x, 0) = u0(x) in Ω and vanishing on the parabolic boundary ∂Ω × R+. Here Ω is a bounded domain in RN , the exponents m and p satisfy m + p ≥ 3, p > 1, and the initial datum u0 is in L1(Ω).Item On Critical Points of p Harmonic Functions in the Plane(Southwest Texas State University, Department of Mathematics, 1994-07-06) Lewis, John L.We show that if u is a p harmonic function, 1 < p < ∞, in the unit disk and equal to a polynomial P of positive degree on the boundary of this disk, then ∇u has at most deg P − 1 zeros in the unit disk.Item Existence Results for Non-Autonomous Elliptic Boundary Value Problems(Southwest Texas State University, Department of Mathematics, 1994-07-08) Anuradha, V.; Dickens, S.; Shivaji, R.We study solutions to the boundary value problems −∆u(x) = λf (x, u); x ∈ Ω u(x) + α(x) ∂u(x) / ∂n = 0; x ∈ ∂Ω where λ > 0, Ω is a bounded region in ℝN ; N ≥ 1 with smooth boundary ∂Ω, α(x) ≥ 0, n is the outward unit normal, and f is a smooth function such that it has either sublinear or restricted linear growth in u at infinity, uniformly in x. We also consider f such that f (x, u)u ≤ 0 uniformly in x, when |u| is large. Without requiring any sign condition on f (x, 0), thus allowing for both positone as well as semipositone structure, we discuss the existence of at least three solutions for given λ ∈ (λn, λn+1) where λk is the k-th eigenvalue of −∆ subject to the above boundary conditions. In particular, one of the solutions we obtain has non-zero positive part, while another has non-zero negative part. We also discuss the existence of three solutions where one of them is positive, while another is negative, for λ near λ1, and for λ large when f is sublinear. We use the method of sub-super solutions to establish our existence results. We further discuss non-existence results for λ small.Item A Note on the Uniqueness of Entropy Solutions to First Order Quasilinear Equations(Southwest Texas State University, Department of Mathematics, 1994-07-19) Diller, David J.In this note, we consider entropy solutions to scalar conservation laws with unbounded initial data. In particular, we offer an extension of Kruzkhov's uniqueness proof (see [1]).Item Computing Eigenvalues of Regular Sturm-Liouville Problems(Southwest Texas State University, Department of Mathematics, 1994-08-23) Dwyer, H. I.; Zettl, A.An algorithm is presented for computing eigenvalues of regular self-adjoint Sturm-Liouville problems with matrix coefficients and arbitrary coupled boundary conditions.Item On a Class of Elliptic Systems in R(N)(Southwest Texas State University, Department of Mathematics, 1994-09-23) Costa, David G.We consider a class of variational systems in ℝN of the form {−∆u + a(x)u = Fu(x, u, v) −∆v + b(x)v = F_{v}(x, u, v), where a, b : ℝN → ℝ are continuous functions which are coercive; i.e., a(x) and b(x) approach plus infinity as x approaches plus infinity. Under appropriate growth and regularity conditions on the nonlinearities Fu(.) and Fv(.), the (weak) solutions are precisely the critical points of a related functional defined on a Hilbert space of functions u, v in H1(ℝN. By considering a class of potentials F (x, u, v) which are nonquadratic at infinity, we show that a weak version of the Palais-Smale condition holds true and that a nontrivial solution can be obtained by the Generalized Mountain Pass Theorem. Our approach allows situations in which a(.) and b(.) may assume negative values, and the potential F (x, s) may grow either faster of slower than |s|2.Item Quasireversibility Methods for Non-Well-Posed Problems(Southwest Texas State University, Department of Mathematics, 1994-11-29) Clark, Gordon W.; Oppenheimer, Seth F.The final value problem { ut + Au = 0 , 0 < t < T u(T) = ƒ with positive self-adjoint unbounded A is known to be ill-posed. One approach to dealing with this has been the method of quasireversibility, where the operator is perturbed to obtain a well-posed problem which approximates the original problem. In this work, we will use a quasi- boundary-value method, where we perturb the final condition to form an approximate non-local problem depending on a small parameter α. We show that the approximate problems are well posed and that their solutions uα converge on [0,T] if and only if the original problem has a classical solution. We obtain several other results, including some explicit convergence rates.Item A Rado type theorem for p-harmonic functions in the plane(Southwest Texas State University, Department of Mathematics, 1994-12-06) Kilpelainen, TeroWe show that if u ∈ C1(Ω) satisfies the p-Laplace equation div(|∇u|p−2∇u) = 0 in Ω \ {x : u(x) = 0}, then u is a solution to the p-Laplacian in the whole Ω ⊂ ℝ2.Item Multiple Solutions for Semilinear Elliptic Boundary Value Problems at Resonance(Southwest Texas State University, Department of Mathematics, 1995-01-26) Robinson, Steve B.In recent years several nonlinear techniques have been very successful in proving the existence of weak solutions for semilinear elliptic boundary value problems at resonance. One technique involves a variational approach where solutions are characterized as saddle points for a related functional. This argument requires that the Palais-Smale condition and some coercivity conditions are satisfied so that the saddle point theorem of Ambrossetti and Rabinowitz can be applied. A second technique has been to apply the topological ideas of Leray-Schauder degree. This argument typically creates a homotopy with a uniquely solvable linear problem at one end and the nonlinear problem at the other, and then an a priori bound is established so that the homotopy invariance of Leray-Schauder degree can be applied. In this paper we prove that both techniques are applicable in a wide variety of cases, and that having both techniques at our disposal gives more detailed information about solution sets, which leads to improved existence results such as the existence of multiple solutions.Item Interfacial Dynamics for Thermodynamically Consistent Phase-Field Models with Nonconserved Order Parameter(Southwest Texas State University, Department of Mathematics, 1995-02-24) Fife, Paul C.; Penrose, OliverWe study certain approximate solutions of a system of equations formulated in an earlier paper (Physica D 43 44–62 (1990)) which in dimensionless form are ut + γw(ϕ)t = ∇2u, α∊2 ϕt = ∊2∇2 ϕ + F (ϕ, u), where u is (dimensionless) temperature, ϕ is an order parameter, w(ϕ) is the temperature–independent part of the energy density, and F involves the ϕ–derivative of the free-energy density. The constants α and γ are of order 1 or smaller, whereas ∊ could be as small as 10−8. Assuming that a solution has two single–phase regions separated by a moving phase boundary Γ(t), we obtain the differential equations and boundary conditions satisfied by the ‘outer’ solution valid in the sense of formal asymptotics away from Γ and the ‘inner’ solution valid close to Γ. Both first and second order transitions are treated. In the former case, the ‘outer’ solution obeys a free boundary problem for the heat equations with a Stefan–like condition expressing conservation of energy at the interface and another condition relating the velocity of the interface to its curvature, the surface tension and the local temperature. There are O(∊) effects not present in the standard phase–field model, e.g. a correction to the Stefan condition due to stretching of the interface. For second–order transitions, the main new effect is a term proportional to the temperature gradient in the equation for the interfacial velocity. This effect is related to the dependence of surface tension on temperature. We also consider some cases in which the temperature u is very small, and possibly γ or α as well; these lead to further free boundary problems, which have already been noted for the standard phase-field model, but which are now given a different interpretation and derivation. Finally, we consider two cases going beyond the formulation in the above equations. In one, the thermal conductivity is enhanced (to order O(∊−1)) within the interface, leading to an extra term in the Stefan condition proportional (in two dimensions) to the second derivative of curvature with respect to arc length. In the other, the order parameter has m components, leading naturally to anisotropies in the interface conditions.Item Strong Solutions of Quasilinear Integro-Differential Equations with Singular Kernels in Several Space Dimensions(Southwest Texas State University, Department of Mathematics, 1995-02-24) Engler, HansFor quasilinear integro-differential equations of the form ut − a ∗ A(u) = f , where a is a scalar singular integral kernel that behaves like t−α, 1 ≤ α < 1 and A is a second order quasilinear elliptic operator in divergence form, solutions are found for which A(u) is integrable over space and time.