An elementary proof of the exponential blow‐up for semi‐linear wave equations

This paper deals with the upper bound of the life span of classical solutions to □u = ∣u∣^p, u∣_{t = 0} = εφ(x), u_t∣_t=0 = εψ(x) with the critical power of p in two or three space dimensions. Zhou has proved that the rate of the upper bound of this life span is exp(ε^−p(p−1)). But his proof, especially the two‐dimensional case, requires many properties of special functions. Here we shall give simple proofs in each space dimension which are produced by pointwise estimates of the fundamental solution of □. We claim that both proofs are done in almost the same way.