Sun-Hye Park

Energy decay for a von Karman equation with time-varying delay

Abstract We consider a von Karman equation with time-varying delay of the form u t t ( x , t ) + Δ 2 u ( x , t ) + a 0 h 1 ( u t ( x , t ) ) + a 1 h 2 ( u t ( x , t − τ ( t ) ) ) = [ u ( x , t ) , F ( u ( x , t ) ) ] . By introducing suitable energy and Lyapunov functionals, we establish decay estimates for the energy, which depends on the behavior of h 1 .

General decay of a von Karman plate equation with memory on the boundary

Abstract We investigate the influence of boundary dissipation on the decay property of the solutions for a von Karman plate equation with a memory condition on one part of the boundary. Dropping the condition u 0 = 0 on one part of the boundary, we show a general stability result for the equation via setting modified energy functionals which are equivalent to the energy of the equation and using some properties of convex functions. This result allows a wider class of relaxation functions and improve earlier results of Mustafa and Abusharkh (2015) and Park and Park (2005).

Blow-up results for viscoelastic wave equations with weak damping

Abstract In this work we consider a viscoelastic wave equation of the form u t t − Δ u + ∫ 0 t g ( t − s ) Δ u ( s ) d s + h ( u t ) = | u | p − 2 u with Dirichlet boundary condition. There are much literature on the blow-up result of solutions for the wave equation with damping term having polynomial growth near zero. However, to my knowledge, there is no blow-up result of solutions for the viscoelastic wave equation without polynomial growth near zero assumption on the damping term. This work is devoted to study a finite time blow-up result of solution with nonpositive initial energy as well as positive initial energy without imposing any restrictive growth near zero assumption on the damping term.

Stability of a von Karman equation with infinite memory

Abstract In this paper, we consider a von Karman equation with infinite memory. For von Karman equations with finite memory, there is a lot of literature concerning on existence of the solutions, decay of the energy, and existence of the attractors. However, there are few results on existence and energy decay rate of the solutions for von Karman equations with infinite memory. The main goal of the present paper is to generalize previous results by treating infinite history instead of finite history.