Global existence and uniform decay for a nonlinear viscoelastic equation with damping

In this paper we investigate a nonlinear viscoelastic equation with linear damping. Global existence of weak solutions and the uniform decay estimates for the energy have been established.     

On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution

In this paper, we consider a nonlinear viscoelastic wave equation with nonlinear boundary damping and source terms. Under some appropriate assumptions on the relaxation function $g$ and with certain initial data, the global existence of solutions and a general decay for the energy have been established.     

Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay

In this paper, we consider initial-boundary value problem of viscoelastic wave equation with a delay term in the interior feedback. Namely, we study the following equation $$u_{tt}(x, t) - \Delta {u}(x, t) + \int_{0}^{t} g(t - s)\,\Delta {u}(x, s){\rm d}s + \mu_{1} u_{t}(x, t) + \mu_{2} u_{t}(x, t -\tau) = 0$$ together with initial-boundary conditions of Dirichlet type in Ω × (0, + ∞) and prove that for arbitrary real numbers  μ ₁ and μ ₂, the above-mentioned problem has a unique global solution under suitable assumptions on the kernel g. This improve the results of the previous literature such as Nicaise and Pignotti (SIAM J. Control Optim 45:1561–1585, 2006) and Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011) by removing the restriction imposed on μ ₁ and μ ₂. Furthermore, we also get an exponential decay results for the energy of the concerned problem in the case μ ₁ = 0 which solves an open problem proposed by Kirane and Said-Houari (Z. Angew. Math. Phys. 62:1065–1082, 2011).     

Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term

The wave equation with a source term is considered

     

Global smooth solutions for the quasilinear wave equation with boundary dissipation

We consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input-output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system. Our main tool is the geometrical analysis. The main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system. We show that the existence of global solutions depends not only on this dissipation structure but also on a Riemannian metric, given by the coefficients and the equilibrium of the system. Some geometrical conditions on this Riemannian metric are presented to guarantee the existence of global solutions. In particular, we prove that the norm of the state of the system decays exponentially if the input stops after a finite time, which implies the exponential stabilization of the system by boundary feedback.     

On transmission problem for viscoelastic wave equation with a localized a nonlinear dissipation

In this article, we consider the global existence and decay rates of solutions for the transmission problem of Kirchhoff type wave equations consisting of two physically different types of materials, one component being a Kirchhoff type wave equation with time dependent localized dissipation which is effective only on a neighborhood of certain part of boundary, while the other being a Kirchhoff type viscoelastic wave equation with nonlinear memory.     

Global existence and decay for nonlinear dissipative wave equations with a derivative nonlinearity

We prove the global existence of the so-called H² solutions for a nonlinear wave equation with a nonlinear dissipative term and a derivative type nonlinear perturbation. To show the boundedness of the second order derivatives we need a precise energy decay estimate and for this we employ a ‘loan’ method.     

Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

This paper is devoted to studying a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under the suitable conditions, we prove that any weak solutions with negative initial energy will blow up in finite time. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.     

Global existence and blow-up of solutionsfor a higher-order kirchhoff-type equation with nonlinear dissipation

This paper deals with the higher-order Kirchhoff-type equation with nonlinear dissipationu_tt+(∫_Ω׀D^mu׀²dx)^q(−Δ)^mu+u_t׀u_t׀^r=׀u׀^pu,xΩ,t>0,in a bounded domain, where m < 1 is a positive integer, q, p, r < 0 arepositive constants. We obtain that the solution exists globally if p ≤ r, while ifp > max r, 2q , then for any initial data with negative initial energy, the solution blowsup at finite time in L^p+2 norm.     

General decay and blow-up of solutions for a viscoelastic equation with nonlinear boundary damping-source interactions

In this paper, a viscoelastic equation with nonlinear boundary damping and source terms of the form $$\begin{array}{llll}u_{tt}(t)-\Delta u(t)+\displaystyle\int\limits_{0}^{t}g(t-s)\Delta u(s){\rm d}s=a\left\vert u\right\vert^{p-1}u,\quad{\rm in}\,\Omega\times(0,\infty), \\ \qquad\qquad\qquad\qquad\qquad u=0,\,{\rm on}\,\Gamma_{0} \times(0,\infty),\\ \dfrac{\partial u}{\partial\nu}-\displaystyle\int\limits_{0}^{t}g(t-s)\frac{\partial}{\partial\nu}u(s){\rm d}s+h(u_{t})=b\left\vert u\right\vert ^{k-1}u,\quad{\rm on} \ \Gamma_{1} \times(0,\infty) \\ \qquad\qquad\qquad\qquad u(0)=u^{0},u_{t}(0)=u^{1},\quad x\in\Omega, \end{array}$$ is considered in a bounded domain ??. Under appropriate assumptions imposed on the source and the damping, we establish both existence of solutions and uniform decay rate of the solution energy in terms of the behavior of the nonlinear feedback and the relaxation function g, without setting any restrictive growth assumptions on the damping at the origin and weakening the usual assumptions on the relaxation function g. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited.     

Global existence of solutions for the heat equation with a nonlinear boundary condition

We consider the initial-boundary value problem for the heat equation with a nonlinear boundary condition:

     

Global existence for rough solutions of a fourth-order nonlinear wave equation

In this paper, we prove that the cubic fourth-order wave equation is globally well-posed in Hs(Rn) for by following the Bourgain's Fourier truncation idea in Bourgain (1998) [2]. To avoid some troubles, we technically make use of the Strichartz estimate for low frequency part and high frequency part, respectively. As far as we know, this is the first result on the low regularity behavior of the fourth-order wave equation.     

Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction

We establish, subject to some natural additional assumptions imposed on the relation between the source and the damping, both well-posedness and effective optimal decay rates for the solutions of a semilinear model of the wave equation. The theory presented allows to consider both superlinear and sublinear behaviours of the dissipation in the presence of unstructured sources.     

A linear wave equation with a nonlinear boundary condition of viscoelastic type

The paper is devoted to the study of a linear wave equation with a nonlinear boundary condition of viscoelastic type. The existence of a weak solution is proved by using the Faedo–Galerkin method. The uniqueness, regularity and asymptotic expansion of the solution are also discussed.     

Global existence and uniqueness for the magnetic Hartree equation

In this paper, we consider