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We are concerned with the stabilization of the wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic and frictional effects. Here, one of the novelties is: the viscoelastic and frictional damping together effect only in a part of domain, not in entire domain, which is only assumed to meet the piecewise multiplier geometric condition that their summed interior and boundary measures can be arbitrarily small. Furthermore, there is no other additional restriction for the location of the viscoelastic-effect region. That is, it is dropped that the viscoelastic-effect region includes a part of the system boundary, which is the fundamental condition in almost all previous literature even if when two types of damping together cover the entire system domain. The other distinct novelty is: in this article we remove the fundamental condition that the derivative of the relaxation function is controlled by relaxation function itself, which is a necessity in the previous literature to obtain the optimal uniform decay rate. Under such weak conditions, we successfully establish a series of decay theorems, which generalize and extend essentially the previous related stability results for viscoelastic model regardless of local damping case, entire damping case and mixed-type damping case.

In this paper, we study the global existence of solutions to some semilinear integro-differential evolution equations in Hilbert spaces with sign-varying kernels. By treating the problem through fixed point theory and energy method, we obtain the global existence theorem on [0, +∞) of mild and strong solution to the evolution equations with memory effects for oscillating and sign-varying kernels. This theorem generalizes and improves some previous existence results.

In this paper, we investigate the stability of a string/point mass system with the interior complementary memory and frictional damping. Without the previously fundamental assumptions that the memory-effect region must include at least one of the ends of the interval and the active region of the frictional damping must cover the dynamic boundary, we eventually establish the uniform stabilization of the system by constructing some subtle multiplier functions. We also present the polynomial decay results under the classical assumption that the integral of the kernel is controlled by the kernel. [ABSTRACT FROM AUTHOR]

In this paper, we consider a wave equation with memory term and time‐varying delayed velocity term. By some subtle analysis techniques, assuming memory kernel function with a minimal and general condition, we establish a general uniform energy decay result for the system. The condition here substantially weakens the conditions in earlier literature. From the result, we can give the explicit energy decay rates directly if the memory kernel is expressed explicitly. The result is new even for the case that memory kernel decays polynomially. [ABSTRACT FROM AUTHOR]

This paper concerns the asymptotics of the linear wave equation with frictional damping only on Wentzell boundary in the square. After reformulating the model into an abstract Cauchy problem, we show that the spectrum for the corresponding operator matrix has no purely imaginary values. Moreover, by analyzing a family of eigenvalues for the operator matrix, we prove that there exists a solution of the system, whose energy decay rate can be arbitrarily slow. [ABSTRACT FROM AUTHOR]