An existence result for a class of second order evolution equations of mixed type

We give an existence and uniqueness result for a linear abstract evolution equation of second order with some coefficient in front of the second temporal derivative which may degenerate to zero and change sign.     

Solvability of mixed type operator equations

In this paper we prove the existence of monotonic solution of the mixed type operator equations. We discuss also the existence of solutions of nonlinear integral equations of fractional orders. The technique rely on the concept of measure of noncompactness and its associated Darbo fixed point theorem.     

Carleson type estimates for second order elliptic equations with unbounded drift

We derive a Carleson type estimate for positive solutions of non-divergence second order elliptic equations Lu = a _ij D _ij u + b _i D _i u = 0 in a bounded domain Ω ⊂ ℝ ⁿ . We assume that b _i ∈ L ⁿ (Ω) and Ω is a twisted H?lder domain of order α ∈ (1/2, 1] which satisfies a weak regularity condition. We also provide an example which shows that the main result fails in general if α ∈ (0, 1/2]. Bibliography: 18 titles.     

Solvability in Hardy space for second order elliptic equations

We obtain a priori estimates and solvability in Hardy type space in a bounded domain of R ⁿ for second order elliptic equations with coefficients of limited smoothness. Such a result can be served as an endpoint case of the classical L ^p (1 < p < ∞) theory for second order elliptic equations. Our approach is based on a standard technique of perturbation rather than that of integral representation formula.     

Convergence and decay estimates for a class of second order dissipative equations involving a non-negative potential energy

We estimate the rate of decay of the difference between a solution and its limiting equilibrium for the following abstract second order problem

     

Asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order

In this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear, gradient-like evolution equations of mixed first and second order. The proof of convergence is based on the Lojasiewicz-Simon inequality, the construction of an appropriate Lyapunov functional, and some differential inequalities. Applications are given to nonautonomous semilinear wave and heat equations with dissipative, dynamical boundary conditions, a nonlinear hyperbolic-parabolic partial differential equation, a damped wave equation and some coupled system.     

A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation

We study a nonlocal mixed problem for a nonlinear pseudoparabolic equation, which can, for example, model the heat conduction involving a certain thermodynamic temperature and a conductive temperature. We prove the existence, uniqueness and continuous dependence of a strong solution of the posed problem. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of deriving the a priori estimate is based on constructing a suitable multiplicator. From the resulted energy estimate, it is possible to establish the solvability of the linear problem. Then, by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.     

Solvability of the Tricomi problem for second order equations of mixed type with degenerate curve on the sides of an angle

In [1]–[6], the author posed and discussed the Tricomi problem of second order mixed equations, but he only consider some special mixed equations. In [3], the author discussed the uniqueness of solutions of the Tricomi problem for some second order mixed equation with nonsmooth degenerate line. The present paper deals with the Tricomi problem for general second order mixed equations with degenerate curve on the sides of an angle. I first give the formulation of the above problem, and then prove the solvability of the Tricomi problem for the mixed equations with degenerate curve on the sides of an angle, by using the existence of solutions of the mixed problem for the degenerate elliptic equations (see [11]). Here I mention that the used method in this paper is different to those in other papers or books, because I introduce the new notation (2.1) below, such that the second order equation of mixed type can be reduced to the first order complex equation of mixed type with singular coefficients, hence I can use the advantage of complex analytic method. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a class of second order Fuchsian hyperbolic equations

Summary We study a class of second order Fuchsian hyperbolic operators. The well-posedness of the Cauchy problem in a space of regular distributions is proved, together with results on the propagation of singularities of the solution. Moreover we give a representation formula for the distribution solutions of the homogeneous equation.     

Solvability of nonlocal boundary value problems for ordinary differential equations of higher order

In this paper, we prove existence results for a nonlocal boundary value problem concerning a higher order differential equation. Our method is based upon the coincidence degree theory of Mawhin. The interesting point is that the degree of some variables among x₀,x₁,…,x_n−1 in the function f(t,x₀,x₁,…,x_n−1) are allowable to be greater than 1. Some examples are given to illustrate the main results.     

Fractal oscillations for a class of second order linear differential equations of Euler type

The s-dimensional fractal oscillations for continuous and smooth functions defined on an open bounded interval are introduced and studied. The main purpose of the paper is to establish this kind of oscillations for solutions of a class of second order linear differential equations of Euler type. Next, it will be shown that the dimensional number s only depends on a positive real parameter α appearing in a singular term of the main equation. It continues some recent results on the rectifiable and unrectifiable oscillations given in Paši? [M. Paši?, Rectifiable and unrectifiable oscillations for a class of second-order linear differential equations of Euler type, J. Math. Anal. Appl. 335 (2007) 724-738] and Wong [J.S.W. Wong, On rectifiable oscillation of Euler type second order linear differential equations, Electron. J. Qual. Theory Differ. Equ. 20 (2007) 1-12].     

A class of second order differential equations

We study the differential equationf″=N(f)f′ ² +M(f)f′+L(f), whereL, M, N are rational functions, and prove that if the differential equation has a transcendental meromorphic solutionf with order,p(f)>2, then the differential equation must be one of nine forms; and, moreover, we construct examples showing the existence of these nine forms with a transcendental meromorphic solution.     

Weighted estimates of solutions to second order parabolic equations

We evaluate the rate of decay for solutions to second order parabolic equations, which vanish on the boundary, while the right-hand side is allowed to be unbounded. Our approach is based on a special growth lemma, and it works for both divergence and non-divergence equations, in domains satisfying a general “exterior measure condition” (A). The result for elliptic case is published in Cho and Safonov (2007) [2].     

Error estimates for series solutions to a class of nonlinear integral equations of mixed type

In this paper, we prove that the accelerated Adomian polynomials formula suggested by Adomian (Nonlinear Stochastic Systems: Theory and Applications to Physics, Kluwer, Dordrecht, 1989) and the accelerated formula suggested by El-Kalla (Int. J. Differ. Equs. Appl. 10(2):225?C234, 2005; Appl. Math. E-Notes 7:214?C221, 2007) are identically the same. The Kalla-iterates exhibit the same faster convergence exhibited by Adomian??s accelerated iterates with the additional advantage of absence of any derivative terms in the recursion, thereby allowing for ease of computation. Moreover, the formula of El-Kalla is used directly to prove the convergence of the series solution to a class of nonlinear two dimensional integral equations. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the Adomian series solution.     

Solvability of a class of integro-differential equations of first order with variable coefficients

In the present paper, we consider a class of linear integro-differential equations of first order with a stochastic kernel and with variable coefficients on the semiaxis. These equations have important applications in physical kinetics. By combining special factorization methods with methods involving integral Fredholm equations of the second kind, we can construct solutions of such equations in the Sobolev space W ₁ ¹ (?⁺). In certain singular cases, we can also describe the structure of the obtained solutions.     

Decay estimates for second order evolution equations with memory

This paper develops a unified method to derive decay estimates for general second order integro-differential evolution equations with semilinear source terms. Depending on the properties of convolution kernels at infinity, we show that the energy of a mild solution decays exponentially or polynomially as t→+∞. Our approach is based on integral inequalities and multiplier techniques.These decay results can be applied to various partial differential equations. We discuss three examples: a semilinear viscoelastic wave equation, a linear anisotropic elasticity model, and a Petrovsky type system.     

A problem with a nonlocal boundary condition on the characteristic for a class of equations of mixed type

We study the boundary-value problem with a nonlocal boundary condition on the characteristic for a class of equations of mixed type. The unique solvability of the problem is proved.     

Solvability of a boundary value problem for operator-differential equations of mixed type

We study boundary value problems for operator-differential equations of mixed type, prove existence of generalized solutions, and establish their smoothness in weighted Sobolev spaces. The results are applied to odd order forward-backward equations.     

Pointwise estimates in a class of fourth order nonlinear elliptic equations

Summary The subharmonicity of a function, which is defined on solutions of a fourth order nonlinear elliptic differential equation, leads to estimates on the solution and the gradient and Laplacian of the solution at interior points in terms of values on the boundary.

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Zusammenfassung Die subharmonische Eigenschaft von Funktionen, die durch die Lösungen von nichtlinearen elliptischen Differentialgleichungen vierter Ordnung bestimmt sind, führt zu punktweiser Abschätzung der Lösung, deren Gradienten, und Laplace Operatoren, abhängig von den Randwerten sind.