J. A. Soriano

GLOBAL EXISTENCE AND ASYMPTOTIC STABILITY FOR THE NONLINEAR AND GENERALIZED DAMPED EXTENSIBLE PLATE EQUATION

The nonlinear and damped extensible plate (or beam) equation is considered where Ω is any bounded or unbounded open set of Rn, α>0 and f, g are power like functions. The existence of global solutions is proved by means of the Fixed Point Theorem and continuity arguments. To this end we avoid handling the nonlinearity M(∫Ω|∇u|2dx) in the a priori estimates of energy. Furthermore, uniform decay rates of the energy are also obtained by making use of the perturbed energy method for domains with finite measure.

Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-A sharp result

This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by u tt - Δ M u + α(x) g(u t ) = 0 on M × ]0, ∞[, where M C R 3 is a smooth oriented embedded compact surface without boundary. Denoting by g the Riemannian metric induced on M by R 3 , we prove that for each ∈ > 0, there exist an open subset V C M and a smooth function f: M→ R such that meas(V) ≥ meas(M) - e, Hessf ≈ g on V and inf Δf(x)| > 0. x∈v In addition, we prove that if α(x) ≥ a o > 0 on an open subset M* C M which contains M\V and if g is a monotonic increasing function such that k|s| ≤ g(s)| ≤ K|s| for all |s| > 1, then uniform and optimal decay rates of the energy hold.

On existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions

Abstract We study the global existence of solutions of the nonlinear degenerate wave equation (ρ⩾0) (∗) ρ(x)y″−Δy=0 in Ω × ]0,∞[, y=0 on Γ 1 × ]0,∞[, ∂y ∂ν +y′+f(y)+g(y′)=0 on Γ 0 × ]0,∞[, y(x,0)=y 0 , ( ρ y′)(x,0)=( ρ y 1 )(x) in Ω, where y′ denotes the derivative of y with respect to parameter t, f(s)=C0|s|δs and g is a nondecreasing C1 function such that k1|s|ξ+2⩽g(s)s⩽k2|s|ξ+2 for some k1,k2>0 with 0 0 if n=1,2. The existence of solutions is proved by means of the Faedo–Galerkin method. Furthermore, when ξ=0 the uniform decay is obtained by making use of the perturbed energy method.

Global Solvability and Asymptotic Stability for the Wave Equation with Nonlinear Boundary Damping and Source Term

We study the global existence and uniform decay rates of solutions of the problem

Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping

\begin{array}{*{20}c} {(P)} & {\left\{ \begin{gathered} u_{tt} - \Delta u = \left| u \right|^\rho uin\Omega \times ]0, + \infty [ \hfill \\ u = 0on\Gamma _0 \times ]0, + \infty [ \hfill \\ \partial _\nu u + g\left( {u_t } \right) = 0on\Gamma _1 \times ]0, + \infty [ \hfill \\ u\left( {x,0} \right) = u^0 \left( x \right),u_t \left( {x,0} \right) = u^1 \left( x \right); \hfill \\ \end{gathered} \right.} \\ \end{array}

Existence and Exponential Decay for a Kirchhoff–Carrier Model with Viscosity

where Ω is a bounded domain of R n, n≥1, with a smooth boundary \( \Gamma = \Gamma _0 \cup \Gamma _1 \) and 0 0, n=1, 2.