Theoretical and computational results of a wave equation with variable exponent and time-dependent nonlinear damping

Abstract We study the following wave equation $$u_{tt}-\Delta u+\alpha (t)\left| u_{t}\right| ^{m(\cdot )-2}u_{t}=0$$ u tt - Δ + α ( t ) m · 2 = 0 with a nonlinear damping having variable exponent m ( x ) and time-dependent coefficient $$\alpha (t)$$ . use multiplier method to establish energy decay results depending on both $$ also give four numerical tests illustrate our theoretical using conservative Lax–Wendroff scheme.