Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions

作者：Graber P Jameson^*; Shomberg Joseph L^*

来源：Nonlinearity, 2016, 29(4): 1171-U310.

DOI：10.1088/0951-7715/29/4/1171

• hyperbolic dynamic boundary condition
• semilinear hyperbolic equation
• damped wave equation
• global attractor
• weak exponential attractor

We establish the well-posedness of a strongly damped semilinear wave equation equipped with nonlinear hyperbolic dynamic boundary conditions. Results are carried out with the presence of a parameter distinguishing whether the underlying operator is analytic, alpha > 0, or only of Gevrey class, alpha = 0. We establish the existence of a global attractor for each alpha is an element of [0,1], and we show that the family of global attractors is upper-semicontinuous as alpha -> 0. Furthermore, for each alpha is an element of [0, 1], we show the existence of a weak exponential attractor. A weak exponential attractor is a finite dimensional compact set in the weak topology of the phase space. This result ensures the corresponding global attractor also possesses finite fractal dimension in the weak topology; moreover, the dimension is independent of the perturbation parameter a. In both settings, attractors are found under minimal assumptions on the nonlinear terms.

• 出版日期2016-4