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In this paper, we investigate the viscoelastic wave equation with variable coefficients and a delay, which is subject to nonlinear and nonlocal boundary dissipation. The existence of strong and weak solutions is obtained by means of Faedo-Galerkin approximation and denseness argument. By introducing an equivalent energy functional and using the Riemannian geometry method, we show the general decay rate of energy when kernel function satisfies some sufficient small conditions that we provide the range accurately. [ABSTRACT FROM AUTHOR]

In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin–Pipkin type. Aiming at minimal assumptions on the involved memory kernels – which we allow to be weakly singular – we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with a quasilinear wave equation with nonlinear damping and source terms of variable exponents-type acting in a part of the boundary. Under suitable conditions on the exponents and the initial data, we study the blow-up properties. Firstly, by using Faedo-Galerkin method and Banach-Fixed-Point Theorem, we establish the existence of a weak solution, under suitable assumptions on the variable exponents and the initial data. Secondly, we show a finite time blow-up with lower and upper bound as well. Next, an infinite time blow-up is proved under some conditions in the exponents and the initial data. [ABSTRACT FROM AUTHOR]

Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions, provided that the initial data are compactly supported and sufficiently small in Sobolev norm. In this work, Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions. A natural question then arises whether the time-growth is a true phenomenon, despite the possible conservation of basic energy. In the present paper, we establish that the top-order energy of the solutions in Alinhac theorem remains globally bounded in time.

We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term f (⋅) to satisfy ∫ − ∞ t e λ s ∥ f (s) ∥ L 2 2 d s < ∞ for all t ∈ R , we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in H 1 topology and the usual (L 2 , L 2) pullback D λ -attractor indeed can attract in the H 1 -norm, provided that ∫ − ∞ t e λ s ∥ f (s) ∥ H − 1 (O s) 2 d s < ∞ and f ∈ L loc 2 (R , L 2 (O s)) . [ABSTRACT FROM AUTHOR]