Asymptotic stability and blowup of solutions for a class of viscoelastic inverse problem with boundary feedback

Abstract

In this paper, we consider a nonlinear viscoelastic inverse problem with memory in the boundary. Under some suitable conditions on the coefficients, relaxation function, and initial data, we proved stability of solutions when the integral overdetermination tends to zero as time goes to infinity. Furthermore, we show that there are solutions under some conditions on initial data that blow up in finite time.

General decay and blow up of solutions for a class of inverse problem with elasticity term and variable‐exponent nonlinearities

“…where the Young's inequality (6) has been used. Again by using the Young's inequality (7) and Cauchy-Schwarz inequality, we get…”

“…As we mentioned before, existence of variable-exponent nonlinearities makes study of inverse problems difficult. However, we try to extend the previous results 7,8 to the inverse problems with variable-exponent nonlinearities. To the best of our knowledge, this is the first work dealing with inverse source problem subject to the variable-exponent nonlinearities.…”

“…See, in this regard, Shahrouzi. 7,8 On the other hand, it is known that modeling of some physical phenomena such as flows of electro-rheological fluids, nonlinear viscoelasticity, and image processing gives rise to equation with nonstandard growth conditions, that is, equations with variable exponents of nonlinearities. In direct problems, equations with nonlinearities of variable-exponent type have largely been discussed by several authors.…”

“…) and Young's inequality(7) yieldse −𝜆t | ∫ Ω e 𝜆t(r(x)−2) |∇v| r(x)−1 ∇𝜔(x)dx| = e −2𝜆t | ∫ Ω |e 𝜆t ∇v| r(x)−1 ∇𝜔(x)dx| ≤ e −2𝜆t ( ∫ Ω r(x) − 1 r(x) |e 𝜆t ∇v| r(x) dx + ∫ Ω 1 r(x) |∇𝜔| r(x) dx ) ≤ ∫ Ω e 𝜆t(r(x)−2) |∇v| r(x) dx + e −2𝜆t ∫ Ω 1 r(x) |∇𝜔| r(x) dx, (63) be −𝜆t | ∫ Ω e 𝜆t(p(x)−2) |u| p(x)−1 𝜔(x)dx| = e −2𝜆t ∫ Ω |e 𝜆t v| p(x)−1 .b𝜔(x)dx| ≤ e −2𝜆t…”

A Study on the Blow-Up of Solutions for a Lame´ System of Inverse Problem

“…For more information about the concavity argument, we refer the readers to [16][17][18]. In [26] Shahrouzi and Tahamtani by using the same method found conditions on data that guaranteeing the global nonexistence and asymptotic stability results for a class of Petrovsky inverse source problems (see also [22][23][24]27]). Bukhge ǐm et al [7] considered an inverse problem for the stationary elasticity system with constant Lamé coefficients and variable matrix coefficient depending on the spatial variables and frequency.…”