Abstract
We consider a one-dimensional linear thermoelastic Bresse system with delay term, forcing, and infinity history acting on the shear angle displacement. Under an appropriate assumption between the weight of the delay and the weight of the damping, we prove the well-posedness of the problem using the semigroup method, where an asymptotic stability result of global solution is obtained.
Highlights
IntroductionWe considered with the following problem:. >>>>>>>>>>: ρ1wtt − k0ðwx − lφÞx + ρ3θt + κqx + γψtx = 0, αqt + βq + κθx = 0, klðφx lw ψÞ
In this work, we considered with the following problem: >>>>>>>>>>< ρ1 ρ2 φtt ψtt − − kðφx bψxx + + lw + kðφx ψÞx − k0lðwx − lφÞ ð∞ +μ1 φt ðx, tÞ μ2 φt ðx
A one-dimensional linear thermoelastic Bresse system with delay term, forcing, and infinity history acting on the shear angle displacement is considered
Summary
We considered with the following problem:. >>>>>>>>>>: ρ1wtt − k0ðwx − lφÞx + ρ3θt + κqx + γψtx = 0, αqt + βq + κθx = 0, klðφx lw ψÞ. Αkρ3ρ1 ρ1 k ρ2 − b γ2α , b ð9Þ k = k0: It is well known that, in the single wave equation, if μ2 = 0 , that is, in the absence of a delay, the energy of system exponentially decays (see, e.g., [1–22]). Under the condition of equal speeds of wave propagation, the authors proved that the system is exponentially stable. Otherwise, they show that Bresse system is not exponentially stable. There are several works dedicated to the mathematical analysis of the Bresse system They are mainly concerned with decay rates of solutions of the linear system. + 0, lw γθ1x ð13Þ together with initial and specific boundary conditions and proved an exponential and only polynomial-type decay stabilities results
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