Well-defined solvability and spectral analysis of abstract hyperbolic integrodifferential equations

The authors study integrodifferential equations in Hilbert space. The coefficients of the equations are unbounded and the principal part is an abstract hyperbolic equation perturbed by terms with Volterra integral operators. Such equations can be regarded as an abstract generalization of the Gurtin–Pipkin integrodifferential equation that describes heat transfer in materials with memory and has a number of other applications. Well-defined solvability of initial boundary value problems for such equations is established in weighted Sobolev spaces on the positive semi-axis. The authors examine spectral problems for operator-valued functions representing the symbols of the said equations and study the spectrum of the abstract Gurtin–Pipkin integrodifferential equation.     

Spectral analysis and solvability of abstract hyperbolic equations with aftereffect

We analyze functional-differential equations with unbounded operator coefficients in a Hilbert space whose leading part is an abstract hyperbolic equation perturbed by terms with a retarded argument and by terms with Volterra integral operators.We consider spectral problems for the operator functions that are the symbols of abovementioned equations in the autonomous case.     

Global solvability for abstract semilinear evolution equations

The Cauchy problem for the abstract semilinear evolution equation u^′(t) = Au (t) + B (u (t)) + C (u (t)) is discussed in a general Banach space X. Here A is the so‐called Hille‐Yosida operator in X, B is a differentiable operator from D (A) into X, and C is a locally Lipschitz continuous operator from D (A) into itself. A vectorvalued functional defined only on X is used and appropriate conditions on the nonlinear operators B and C are imposed so that a vector‐valued functional defined on the domain of the operator A may be constructed in order to specify the growth of a global solution. The advantage of our formulation lies in the fact that it is possible to obtain a global solution by checking some energy inequalities concerning only low order derivatives (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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).     

Stability of an abstract system of coupled hyperbolic and parabolic equations

In this paper, we provide a complete stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$\begin{array}{ll}\;\;u_{tt} = -Au + \gamma A^{\alpha} \theta,\\ \quad \theta_t = -\gamma A^{\alpha}u_t - kA^{\beta}\theta,\\ u(0) = u_0, \quad u_t(0) = v_0, \quad \theta(0) = \theta_0\end{array}$$ where A is a self-adjoint, positive definite operator on a Hilbert space H. For ${(\alpha,\beta) \in [0,1] \times [0,1]}$ , the region of exponential stability had been identified in Ammar-Khodja et al. (ESAIM Control Optim Calc Var 4:577–593,1999). Our contribution is to show that the rest of the region can be classified as region of polynomial stability and region of instability. Moreover, we obtain the optimality of the order of polynomial stability.     

On solvability of non-linear semi-periodic boundary-value problem for system of hyperbolic equations

We study a non-linear semi-periodic boundary-value problem for a system of hyperbolic equations with mixed derivative. At that, the semi-periodic boundary-value problem for a system of hyperbolic equations is reduced to an equivalent problem, consisting of a family of periodic boundary-value problems for ordinary differential equations and functional relation. When solving a family of periodic boundary-value problems of ordinary differential equations we use the method of parameterization. This approach allowed to establish sufficient conditions for the existence of an isolated solution of non-linear semi-periodic boundary-value problem for a system of hyperbolic equations.     

On solvability of boundary value problems for higher order nonlinear hyperbolic equations

In the rectangle Ω=[0,a]×[0,b] for the nonlinear hyperbolic equation

the boundary value problems of the type

are considered, where and are linear bounded functionals.Sufficient conditions of solvability and unique solvability of the general problem and its particular cases (Nicoletti type, Dirichlet, Lidstone and Periodic problems) are established.     

On the spectral and growth bound of semigroups associated with hyperbolic equations

Some new and quite general conditions are presented to ensure equality of the spectral bound and the growth bound for those strongly continuous semigroups governing second-order (in time) equations in a Banach (or Hilbert) space. A counterexample arising from higher-order PDEs is given to show that the growth bound could be arbitrarily large while the spectral bound remains zero.     

Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics

In the present paper, we study integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the equation is an abstract hyperbolic equation perturbed by summands with Volterra integral operators. These equations represent an abstract form of the Gurtin–Pipkin integrodifferential equation describing the process of heat conduction in media with memory and the process of sound conduction in viscoelastic media and arise in averaging problems in perforated media (the Darcy law). The correct solvability of initial-boundary problems for the specified equations is established in weighted Sobolev spaces on a positive semiaxis. Spectral problems for operator-functions are analyzed. Such functions are symbols of these equations. The spectrum of the abstract integrodifferential Gurtin–Pipkin equation is investigated.     

Criteria of unique solvability of nonlocal boundary-value problem for systems of hyperbolic equations with mixed derivatives

We consider nonlocal boundary-value problem for a system of hyperbolic equations with two independent variables. We investigate questions of existence of unique classical solution to problem under consideration. In terms of initial data we propose criteria of unique solvability and suggest algorithms of finding of solutions to nonlocal boundary-value problem. As an application we give conditions of solvability of periodic boundary-value problem for a system of hyperbolic equations.     

Least-squares spectral element method for non-linear hyperbolic differential equations

The least-squares spectral element method has been applied to the one-dimensional inviscid Burgers equation which allows for discontinuous solutions. In order to achieve high order accuracy both in space and in time a space–time formulation has been applied. The Burgers equation has been discretized in three different ways: a non-conservative formulation, a conservative system with two variables and two equations: one first order linear PDE and one linearized algebraic equation, and finally a variant on this conservative formulation applied to a direct minimization with a QR-decomposition at elemental level. For all three formulations an h/p-convergence study has been performed and the results are discussed in this paper.