Decay estimates for the Cauchy problem for the damped extensible beam equation

作者：Racke Reinhard; Yoshikawa Shuji^*

来源：APPLICABLE ANALYSIS, 2016, 95(5): 1118-1136.

DOI：10.1080/00036811.2015.1053051

• decay estimate
• extensible beam
• Cauchy problem
• Ball's model
• Kelvin-Voigt damping

The extensible beam equation proposed byWoinowsky-Krieger is a fourth order dispersive equation with nonlocal nonlinear terms. In this paper we study the Cauchy problem of the extended model by Ball who proposed the following model with external and structural damping terms:
rho partial derivative(2)(t) u + delta partial derivative(t)u + kappa partial derivative(4)(x)u + eta partial derivative(t)partial derivative(4)(x) u = (alpha + beta integral(R) vertical bar partial derivative(x)u vertical bar(2)dx + gamma eta integral(R) partial derivative(t)partial derivative(x) u partial derivative(x) udx)partial derivative(2)(x) u.
For eta > 0 this represents a Kelvin-Voigt damping. We show the unique global existence of solutions for this problem and give a precise description of the decay of solutions in time.

• 出版日期2016-5-3