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The existence of R ‐bounded solution operators of the thermoelastic plate equation with Dirichlet boundary conditions
Author(s) -
. Suma'inna
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4687
Subject(s) - mathematics , thermoelastic damping , bounded function , semigroup , dirichlet boundary condition , domain (mathematical analysis) , mathematical analysis , resolvent , analytic semigroup , boundary (topology) , boundary value problem , dirichlet problem , operator (biology) , pure mathematics , physics , thermal , transcription factor , gene , biochemistry , chemistry , repressor , meteorology
We consider the linearized thermoelastic plate equation with the Dirichlet boundary condition in a general domain Ω, given byu t t + Δ 2 u + Δ θ =f 1 , in Ω × ( 0 , ∞ ) ,θ t − Δ θ − Δ u t =f 2 , in Ω × ( 0 , ∞ ) ,with the initial condition u | ( t =0) = u 0 , u t | ( t =0) = u 1 , and θ | ( t =0) = θ 0 in Ω and the boundary condition u = ∂ ν u = θ =0 on Γ, where u = u ( x , t ) denotes a vertical displacement at time t at the point x =( x 1 ,⋯, x n )∈Ω, while θ = θ ( x , t ) describes the temperature. This work extends the result obtained by Naito and Shibata that studied the problem in the half‐space case. We prove the existence of R ‐bounded solution operators of the corresponding resolvent problem. Then, the generation of C 0 analytic semigroup and the maximal L p ‐ L q ‐regularity of time‐dependent problem are derived.