On a time nonlocal problem for inhomogeneous evolution equations

Under consideration is some problem for inhomogeneous differential evolution equation in Banach space with an operator that generates a C ₀-continuous semigroup and a nonlocal integral condition in the sense of Stieltjes. In case the operator has continuous inhomogeneity in the graph norm. We give the necessary and sufficient conditions for existence of a generalized solution for the problem of whether the nonlocal data belong to the generator domain. Estimates on solution stability are given, and some conditions are obtained for existence of the classical solution of the nonlocal problem. All results are extended to a Sobolev-type linear equation, the equation in Banach space with a degenerate operator at the derivative. The time nonlocal problem for the partial differential equation, modeling a filtrating liquid free surface, illustrates the general statements.     

On blow-up results for solutions of inhomogeneous evolution equations and inequalities

The main purpose of this work is to obtain blow-up results for solutions of the inequality

     

The Fredholm alternative for parabolic evolution equations with inhomogeneous boundary conditions

We study the Fredholm properties of parabolic evolution equations on R with inhomogeneous boundary values. These problems are transformed into evolution equations with inhomogeneities taking values in certain extrapolation spaces. Assuming that the underlying homogeneous problem is asymptotically hyperbolic, we show the Fredholm alternative for these equations. The results are applied to parabolic partial differential equations.     

Wronskian-type formula for inhomogeneous $$TQ$$ equations

Theoretical and Mathematical Physics - It is known that the transfer-matrix eigenvalues of the isotropic open Heisenberg quantum spin-1/2 chain with nondiagonal boundary magnetic fields satisfy a...     

Coupled second order semilinear evolution equations indirectly damped via memory effects

In this paper, we study a system of coupled second order semilinear evolution equations in a Hilbert space, which is partially damped through memory effects. A global existence and uniqueness theorem regarding the solutions to its Cauchy problem is given. Following this, we analyze stability of the system energy, and obtain various decay rates corresponding to the integral kernel, some of them being optimal. Moreover, we apply our abstract theory to concrete systems, including that of Timoshenko type, for which we essentially improve the previous related results.     

Eikonal equations for an inhomogeneous anisotropic medium

This paper studies the wave propagation method in a many-dimensional medium in order to find the cases where the eiconal equation is integrated in an explicit compact form.     

Complete integrability,gauge equivalence and Lax representation of inhomogeneous nonlinear evolution equations

The gauge equivalence between the inhomogeneous versions of the nonlinear Schrödinger and the Heisenberg ferromagnet equations is studied. An unexplicit criterion for integrability is proposed. Examples of gauge equivalent inhomogeneous nonlinear evolution equations are presented. It is shown that in the nonintegrable cases the M-operators in their Lax representations possess unremovable pole singularities lying on the spectrum of the L-operators.To the memory of Michail Constantinovich PolivanovInstitute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 374–386, September, 1992.     

Thermoelastic state of inhomogeneous plates under axially symmetric heating

Under the assumption that the Kirchhoff-Love hypothesis for nonuniform plates is satisfied, the author introduces relations for thermal stresses due to forces and momenta. In particular, the case is studied in which the plate contains cylindrical completely penetrating inclusions.Translated from Matematicheskie Metody i Fiziko-Mekhanicheski Polya, No. 29, pp. 31–34, 1989.     

Coupled spring equations

Coupled spring equations for modelling the motion of two springs with weights attached, hung in series from the ceiling are described. For the linear model using Hooke's Law, the motion of each weight is described by a fourth-order linear differential equation. A nonlinear model is also described and damping and external forcing are considered. The model has many features that permit the meaningful introduction of many concepts including: accuracy of numerical algorithms, dependence on parameters and initial conditions, phase and synchronization, periodicity, beats, linear and nonlinear resonance, limit cycles, harmonic and subharmonic solutions. These solutions produce a wide variety of interesting motions and the model is suitable for study as a computer laboratory project in a beginning course on differential equations or as an individual or a small-group undergraduate research project.     

Weak convergence results for inhomogeneous rotating fluid equations

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vectorB(x); this is a generalization of the usual rotating fluid model (whereB is constant). In the case n whichB has non-degenerate critical points, we prove the weak convergence of Leray-type solutions towards a vector field which satisfies a heat equation as the rotation rate tends to infinity. The method of proof uses weak compactness arguments, which also enable us to recover the usual 2D Navier-Stokes limit in the case whenB is constant.     

Weak convergence results for inhomogeneous rotating fluid equations

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak convergence of Leray-type solutions towards a vector field which satisfies the usual 2D Navier–Stokes equation in the regions of space where B is constant, with Dirichlet boundary conditions, and a heat–type equation elsewhere. The method of proof uses weak compactness arguments. To cite this article: I. Gallagher, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Coupled second order evolution equations with fading memory: Optimal energy decay rate

Of concern is the Cauchy problem for coupled second order semilinear evolution equations in a Hilbert space, with indirect memory-damping. We find an approach to obtain successfully an optimal rate of uniform decay for the system energy, only under basic conditions on the memory kernels. Simultaneously, the same rate is also obtained (with less difficulty) for the corresponding single memory-dissipative second order evolution equations. As can be seen, our results essentially improve the previously related ones in the literature. The abstract results are then applied to several concrete problems in the real world.     

Boundary integral equations for a body with inhomogeneous inclusions

On the basis of the partially singular differential equations of the stationary problem of heat conduction and the quasi-static problem of thermoelasticity, written taking account of conditions of nonideal thermomechanical contact, we derive boundary integral equations for a body with inhomogeneous inclusions. We propose a method of solving these equations taking account of the order of the principal term of the asymptotics of the solution in neighborhoods of the corners of the contact surfaces. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 37–41.     

Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain Ω ⊂ R ² with smooth boundary. By the method of the Green formula, the theory of extensions of differential operators, and the theory of L-traces (i.e., traces associated with the differential operation L), we establish necessary and sufficient (for elliptic operators) conditions of the solvability of each of these problems in the space H ^m (Ω), m ≥ 4.     

Dimension splitting for evolution equations

In this paper, we are concerned with splitting methods for the time integration of abstract evolution equations. We introduce an analytic framework which allows us to prove optimal convergence orders for various splitting methods, including the Lie and Peaceman–Rachford splittings. Our setting is applicable for a wide variety of linear equations and their dimension splittings. In particular, we analyze parabolic problems with Dirichlet boundary conditions, as well as degenerate equations on bounded domains. We further illustrate our theoretical results with a set of numerical experiments. This work was supported by the Austrian Science Fund under grant M961-N13.