Variable-coefficient viscoelastic wave equation with acoustic boundary conditions: global existence, blowup and energy decay rates

This paper deals with the following initial acoustic boundary value problem for a variable-coefficient wave equation with memory term u(tt) - Delta u - Delta u(t) - Delta u(tt) + integral(t)(0) b(t - s)div(a1(x)del u(s))ds + a(2)(x)u(t) vertical bar u(t)vertical bar(q-2) = u vertical bar u vertical bar(P-2). Firstly, we get the global existence of solutions by energy estimates under some advisable assumptions on the kernel function b and variable coefficients a(1)(x), a(2)(x). In addition, we discuss the global nonexistence of solutions, and then derive the evaluate for the lifespan of weak solutions under some certain conditions. Finally, we establish the polynomial or exponential energy decay rates for global solutions by using perturbed energy method.