2015/03/05 by Joseph L. Shomberg, Shomberg, Joseph L.
Computer Science · Engineering · Mathematics · #35B25 #35B41 #35L52 #35Q72 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Dynamical Systems (math.DS) #FOS: Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1503.01821
openalex publication_date 2015/03/05 · openalex created_date 2022/10/05 · openalex updated_date 2026/07/28
Under consideration is the damped semilinear wave equation \νtt+ut-
Delta u + u + f(u)=0 on a bounded domain \Ω in\n\ℝ3 with a perturbation parameter \ε>0 occurring in an\nacoustic boundary condition, limiting (\ε=0) to a Robin boundary\ncondition. With minimal assumptions on the nonlinear term f, the existence\nand uniqueness of global weak solutions is shown for each\n\ε\∈[0,1]. Also, the existence of a family of global attractors is\nshown to exist. After proving a general result concerning the\nupper-semicontinuity of a one-parameter family of sets, the result is applied\nto the family of global attractors. Because of the complicated boundary\nconditions for the perturbed problem, fractional powers of the Laplacian are\nnot well-defined; moreover, because of the restrictive growth assumptions on\nf, the family of global attractors is obtained from the asymptotic\ncompactness method developed by J. Ball for generalized semiflows. With more\nrelaxed assumptions on the nonlinear term f, we are able to show the global\nattractors possess optimal regularity and prove the existence of an exponential\nattractor, for each \ε\∈[0,1]. This result insures that the\ncorresponding global attractor inherits finite (fractal) dimension, however,\nthe dimension is emnot necessarily uniform in \ε.\n