We consider an initial-boundary value problem for the Maxwell's system in a bounded domain with a linear inhomogeneous anisotropic instantaneous material law subject to a nonlinear Silver-Müller-type boundary feedback mechanism incorporating both an instantaneous damping and a time-localized delay effect. By proving the maximal monotonicity property of the underlying nonlinear generator, we establish the global well-posedness in an appropriate Hilbert space. Further, under suitable assumptions and geometric conditions, we show the system is exponentially stable.while Faraday's law of induction and Ampère's circuital law mandate(1.2) Typically, J : [0, ∞) × G → R 3 is a (given) total current density.Since the system (1.1)-(1.2) is underdetermined, two more equations relating the four unknown vector fields E, D, H, B need to be postulated. Letting ε, µ : G → R 3×3 be symmetric,

An initial-boundary value problem for a viscoelastic wave equation subject to a strong timelocalized delay in a Kelvin & Voigt-type material law is considered. Transforming the equation to an abstract Cauchy problem on the extended phase space, a global well-posedness theory is established using the operator semigroup theory both in Sobolev-valued C 0-and BV-spaces. Under appropriate assumptions on the coefficients, a global exponential decay rate is obtained and the stability region in the parameter space is further explored using the Lyapunov's indirect method. The singular limit τ → 0 is further studied with the aid of the energy method. Finally, a numerical example from a real-world application in biomechanics is presented.

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An initial-boundary value problem for the multidimensional type III thermoelaticity for a nonsimple material with a center of symmetry is considered. In the linear case, the well-posedness with and without Kelvin-Voigt and/or frictional damping in the elastic part as well as the lack of exponential stability in the elastically undamped case is proved. Further, a frictional damping for the elastic component is shown to lead to the exponential stability. A Cattaneo-type hyperbolic relaxation for the thermal part is introduced and the well-posedness and uniform stability under a nonlinear frictional damping are obtained using a compactness-uniquenesstype argument. Additionally, a connection between the exponential stability and exact observability for unitary C 0 -groups is established.

We propose an approach to the investigation of generalized solutions of linear operators that satisfy weakened a priori inequalities. This approach generalizes several well-known definitions of generalized solutions of operator equations. We prove existence and uniqueness theorems for a generalized solution.