On the initial and terminal value problem for a class of semilinear strongly material damped plate equations

We study the semilinear strongly damped plate equation by considering its two different problems. For initial value problem, we prove the local well-posedness and blow-up results of solution for the problem with polynomial nonlinear source terms. For terminal value problem, given the ill-posedness in the sense of Hadamard we propose a regularization method for the problem with logarithmic nonlinearities. Under the a priori assumption on the exact solution belonging to a Gevrey space, we propose the Fourier truncation method to stabilize the ill-posed problem, also by establishing some stability estimates of logarithmic type in Lq space.