Stability result for a variable-exponent viscoelastic double-Kirchhoff type inverse source problem with nonlocal degenerate damping term

This paper aims to study the stability of solutions for a double-Kirchhoff type viscoelastic inverse source problem with nonlocal degenerate damping term and variable-exponent nonlinearities. Proving of existence and stability of solutions to inverse problems is of high importance because inverse problems are nonlinear and improperly posed, and the presence of unknown source functions with nonstandard growth conditions causes the nonexistence and blow-up of solutions. Therefore, in this work by using the suitable auxiliary functionals and by introducing a suitable Lyapunov functional, we shall prove that the solutions of a double-Kirchhoff type viscoelastic inverse source problem are asymptotically stable in the appropriate range of variable exponents.