On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ<Superscript>3</Superscript>

Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Protter in 1952. In contrast to the well-posedness of the Darboux problem in the 2D case, the new 3D problems are strongly ill-posed. A similar statement of 3D problem for Keldysh-type equations is also given. For mixed type equations of Tricomi and Keldysh type, we introduce the notion of generalized or quasi-regular solutions and find sufficient conditions for the uniqueness of such solutions to the Protter’s problems. The dependence of lower order terms is also studied.     

Schröndiger Type and Relaxed Dirichlet Problems for the Subelliptic p-Laplacian

We study at first the solutions of a Schrödinger type problem relative to the subelliptic p-Laplacian: we prove, for potentials that are in the Kato space, an Harnack inequality on enough small intrinsic balls; the continuity of the solutions to the homogeneous Dirichlet problem follows from some estimates in the proof of the Harnack inequality. In the second part of the paper we study a relaxed Dirichlet problem for the subelliptic p-Laplacian and we prove a Wiener type criterion for the regularity of a point (with respect to our problem).     

A Note on the Stability of the Integral-Differential Equation of the Hyperbolic Type in a Hilbert Space

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme approximately solving this problem is presented. The stability estimates for the solution of this difference scheme are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented. The stability estimates for the solutions of these difference schemes are obtained.     

Convergence Rate of Galerkin Method for a Certain Class of Nonlinear Operator-Differential Equations

In this article, we study a Galerkin method for a nonstationary operator equation with a leading self-adjoint operator A(t) and a subordinate nonlinear operator F. The existence of the strong solutions of the Cauchy problem for differential and approximate equations are proved. New error estimates for the approximate solutions and their derivatives are obtained. The developed method is applied to an initial boundary value problem for a partial differential equation of parabolic type.     

A fourth‐order parabolic equation in two space dimensions

In this paper, we consider an initial‐boundary problem for a fourth‐order nonlinear parabolic equations. The problem as a model arises in epitaxial growth of nanoscale thin films. Based on the L^p type estimates and Schauder type estimates, we prove the global existence of classical solutions for the problem in two space dimensions. Copyright © 2007 John Wiley & Sons, Ltd.     

A local hybrid kernel meshless method for numerical solutions of two-dimensional fractional cable equation in neuronal dynamics

This study deals with obtaining numerical solutions of two-dimensional (2D) fractional cable equation in neuronal dynamics by using a recently introduced meshless method. In solution process at first stage, time derivatives that are appeared in the considered problem are discretized by using finite difference method. Then a meshless method based on hybridization of Gaussian and cubic kernels is developed in local fashion. The problem is solved both on regular and irregular domians. L_∞ and RMS error norms are calculated and compared with other numerical methods in literature as well as exact solutions. Also, obtained condition numbers are monitored. Numerical simulations show that local hybrid kernel meshless method is a thriving method for solving 2D fractional cable equation on regular and irregular domians.     

Equivalence and existence of weak Pareto optima for multiobjective optimization problems with cone constraints

In this work, a differentiable multiobjective optimization problem with generalized cone constraints is considered, and the equivalence of weak Pareto solutions for the problem and for its η-approximated problem is established under suitable conditions. Two existence theorems for weak Pareto solutions for this kind of multiobjective optimization problem are proved by using a Karush–Kuhn–Tucker type optimality condition and the F-KKM theorem.     

The steady states and convergence to equilibria for a 1‐D chemotaxis model with volume‐filling effect

We consider a chemotaxis model with volume‐filling effect introduced by Hillen and Painter. They also proved the existence of global solutions for a compact Riemannian manifold without boundary. Moreover, the existence of a global attractor in W^{1, p}(Ω??ⁿ), p>n, p?2, was proved by Wrzosek. He also proved that the ω‐limit set consists of regular stationary solutions. In this paper, we prove that the 1‐D stationary problem has at most an infinitely countable number of regular solutions. Furthermore, we show that as t→∞ the solution of the 1‐D evolution problem converges to an equilibrium in W^{1, p}, p?2. Copyright © 2009 John Wiley & Sons, Ltd.     

Large‐time behavior of solutions to the Rosenau equation with damped term

In this paper, we consider the initial value problem for the Rosenau equation with damped term. The decay structure of the equation is of the regularity‐loss type, which causes the difficulty in high‐frequency region. Under small assumption on the initial value, we obtain the decay estimates of global solutions for n≥1. The proof also shows that the global solutions may be approximated by the solutions to the corresponding linear problem for n≥2. We prove that the global solutions may be approximated by the superposition of nonlinear diffusion wave for n = 1. Copyright © 2016 John Wiley & Sons, Ltd.     

On some abstract variable domain hyperbolic differential equations

The Cauchy problem is studied for a class of linear abstract differential equations of hyperbolic type with variable domain. Existence and uniqueness results are proved for (suitably defined) weak solutions. Some applications to P.D.E. are also given: they concern linear hyperbolic equations either in non-cylindrical regions or with mixed variable lateral conditions.This work was supported in part by the M.U.R.S.T. (Italy), through 60% and 40% research funds, and by the «G.N.AF.A. of the C.N.R.» (Italy).     

The Tricomi problem for second order linear equations of mixed type with parabolic degeneracy

In Bers, 1958, Mathematical Aspects of Subsonic and Transonic Gas Dynamics (New York: Wiley); Bitsadze, 1988, Some Classes of Partial Differential Equations (New York: Gordon and Breach); Rassias, 1990, Lecture Notes on Mixed Type Partial Differential Equations (Singapore: World Scientific); Salakhitdinov and Islomov, 1987, The Tricomi problem for the general linear equation of mixed type with a nonsmooth line of degeneracy. Soviet Math. Dokl., 34, 133–136; Smirnov, 1978, Equations of Mixed type (Providence, RI: American Mathematical Society), the authors posed and discussed the Tricomi problem of second order equations of mixed type with parabolic degeneracy, which possesses important application to gas dynamics. The present article deals with the Tricomi problem for general second order equations of mixed type with parabolic degeneracy. Firstly the formulation of the problem for the equations is given, next the representations and estimates of solutions for the above problem are obtained, finally the existence of solutions for the problem is proved by the successive iteration and the method of parameter extension. In this article, we use the complex method, namely the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used (see Wen, 2002, Linear and Quasilinear Equations of Hyperbolic and Mixed Types (London: Taylor and Francis)).     

Generalized Higher-Order Optimality Conditions for Set-Valued Optimization under Henig Efficiency

In this paper, generalized mth-order contingent epiderivative and generalized mth-order epiderivative of set-valued maps are introduced, respectively. By virtue of the generalized mth-order epiderivatives, generalized necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Generalized Kuhn–Tucker type necessary and sufficient optimality conditions are also obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.     

Optimal <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Control in Coefficients for Dirichlet Elliptic Problems: <Emphasis Type="Italic">W</Emphasis>-Optimal Solutions

In this paper, we study a Dirichlet optimal control problem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The coefficients may degenerate and, therefore, the problems may exhibit the so-called Lavrentieff phenomenon and non-uniqueness of weak solutions. We consider the solvability of this problem in the class of W-variational solutions. Using a concept of variational convergence of constrained minimization problems in variable spaces, we prove the existence of W-solutions to the optimal control problem and provide the way for their approximation. We emphasize that control problems of this type are important in material and topology optimization as well as in damage or life-cycle optimization.     

Approximation of the Navier–Stokes system with variable viscosity by a system of Cauchy–Kowaleska type

In this paper, we study the existence of weak solutions when n?4 of the mixed problem for the Navier–Stokes equations defined in a bounded domain Q using approximation by a system of Cauchy–Kowaleska type. Periodical solutions are also analyzed. Copyright © 2008 John Wiley & Sons, Ltd.     

Redistribution of Damping in Viscoelasticity

Physically admissible, global BV solutions are constructed to the Cauchy problem for the equations of one-dimensional, nonlinear viscoelasticity of the Boltzmann type. The required BV bounds are derived with the help of energy estimates, in conjunction with a change of variables that redistributes the damping on an equitable basis among the equations of the system, thus exposing the dissipative effect of viscosity on the variation of solutions.     

Dual finite element analysis of three-dimensional axisymmetric elliptic problems. Part I

This paper consists of two parts in which we propose two types of pure dual finite element (FE) models of a three-dimensional (3D) axisymmetric elliptic problem with mixed boundary conditions. Using cylindrical coordinates and weighted Sobolev spaces, a dual 3D problem is transformed into a 2D problem and finite element spaces of divergence-free vector functions are constructed with the help of a stream function. In part I of the paper a priori and a posteriori error estimates are derived for the first type of the FE model. © 1993 John Wiley & Sons, Inc.     

A class of second‐order differential operators with eigenparameter‐dependent boundary and transmission conditions

In this study, we investigate a Sturm–Liouville type problem with eigenparameter‐dependent boundary conditions and eigenparameter‐dependent transmission conditions. By establishing a new self‐adjoint operator A associated with the problem, we construct fundamental solutions and obtain asymptotic formulae for its eigenvalues and fundamental solutions. Also we investigate some properties of its spectrum. Copyright © 2013 John Wiley & Sons, Ltd.     

A class of large solution of 2D tropical climate model without thermal diffusion

In this paper, we consider the Cauchy problem of 2D tropical climate model without thermal diffusion and construct global smooth solutions by choosing a class of special initial data whose L^∞ norm can be arbitrarily large.