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- Title
On the decay of solutions of a viscoelastic wave equation with variable sources.
- Authors
Messaoudi, Salim A.; Al‐Gharabli, Mohammad M.; Al‐Mahdi, Adel M.
- Abstract
In this paper, we consider the following viscoelastic problem with variable exponent nonlinearities: utt−Δu+∫0tg(t−s)Δu(s)ds+a|ut|m(·)−2ut=|u|q(·)−2u,where m(.) and q(.) are two functions satisfying specific conditions. This type of problems appears in fluid dynamics, the electrorheological fluids (smart fluids), which show changing (often dramatically) in the viscosity when an electrical field is applied. The Lebesgue and Sobolev spaces with variable exponents are efficient tools to analyze such problems. In this work, we prove a global existence result using the well‐depth method and establish explicit and general decay results under a very general assumption on the relaxation function. Our results extend and generalize many results in the literature.
- Subjects
ELECTRORHEOLOGICAL fluids; ELECTRORHEOLOGY; SOBOLEV spaces; FLUID dynamics; EXPONENTS; VISCOSITY
- Publication
Mathematical Methods in the Applied Sciences, 2022, Vol 45, Issue 14, p8389
- ISSN
0170-4214
- Publication type
Article
- DOI
10.1002/mma.7141