Semilinear wave equations of derivative type with spatial weights in one space dimension

The lifespan of classical solutions of semilinear wave equations with spatial weights and compactly supported data in one space dimension

“…For q = 0, the upper bound in this estimate was obtained by Zhou [37], and the lower bound is due to Kitamura, Morisawa and Takamura [14].…”

“…For this equation, one may expect that the result is not so interesting as |x| ∼ t in dealing with the nonlinear term |u t | p . But the result is Theorem 4 (Zhou [37], Kitamura, Morisawa and Takamura [14]). Assume (28) with q = 0.…”

The combined effect in one space dimension beyond the general theory for nonlinear wave equations

“…Sections 5, 6 and 7 are devoted to the proof of the existence part of (1.3). Their main strategy is the iteration method in the weighted L ∞ space due to Kitamura, Morisawa and Takamura [7,8], which is originally introduced by John [4]. Finally, we prove the blow-up part of (1.2) and (1.3), which is different from the functional method by Han and Zhou [2].…”

“…We call [9,10] "general theory" for nonlinear wave equations in one dimension. Recently, Kitamura, Morisawa and Takamura [8] have verified the lower bound for all p ≥ 1 including the case that nonlinear term has spatial weights, in which only the C 1 solution of the associated integral equation is considered for 1 < p < 2. But it can be also the classical solution by trivial modifications on estimating the nonlinear term with Hölder continuity.…”

“…in estimating the difference of the approximation sequence to construct a classical solution for p, q ∈ (1, 2). This fact was overlooked in Kitamura, Morisawa and Takamura [8] as stated in Introduction.…”