Exponential Growth of Solutions for a Variable-Exponent Fourth-Order Viscoelastic Equation With Nonlinear Boundary Feedback

Abstract

In this paper we study a variable-exponent fourth-order viscoelastic equation of the form$$|u_{t}|^{\rho(x)}u_{tt}+\Delta[(a+b|\Delta u|^{m(x)-2})\Delta u]-\int_{0}^{t}g(t-s)\Delta^{2}u(s)ds=|u|^{p(x)-2}u,$$in a bounded domain of $R^{n}$. Under suitable conditions on variable exponents and initial data, we prove that the solutions will grow up as an exponential function with positive initial energy level. Our result improves and extends many earlier results in the literature such as the on by Mahdi and Hakem (Ser. Math. Inform. 2020, https://doi.org/10.22190/FUMI2003647M).

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