EXPONENTIAL GROWTH OF SOLUTIONS FOR A VARIABLE-EXPONENT FOURTH-ORDER VISCOELASTIC EQUATION WITH NONLINEAR BOUNDARY FEEDBACK

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)

On the Petrovsky inverse problem with memory term and nonlinear boundary feedback

Abstract In this paper we consider a Petrovsky viscoelastic inverse source problem with memory term in the boundary condition. We obtain sufficient conditions on relaxation function and initial data for which the solutions of problem are asymptotically stable when the integral overdetermination tends to zero as time goes to infinity

The existence of global weak solutions for a weakly dissipative Camassa-Holm equation in H1(R)

The existence of global weak solutions to the Cauchy problem for a weakly dissipative Camassa-Holm equation is established in the space C([0,∞)×R)nL∞([0,∞);H1(R)) under the assumption that the initial value u 0 (x) only belongs to the space H 1 (R) . The limit of viscous approximations, a one-sided super bound estimate and a space-time higher-norm estimate for the equation are established to prove the existence of the global weak solution