Blow-up of solutions for discrete semilinear wave equation with the scale-invariant damping

We consider the blow-up problem for discretized scale-invariant nonlinear dissipative wave equations. It is known that the critical exponents for undiscretized equations (continuous equations) are given by Fujita and Strauss exponents depending on the space dimensions. Our purpose is to obtain results for the discretized equations that correspond to those shown for the continuous one. The proof is based on Matsuya [6], who showed the blow-up problem for discrete semilinear wave equations without dissipative terms, and we found that the result is sharp in the case of one and two space dimensions compared to the continuous equations.