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

The lifespan estimates of classical solutions of one dimensional semilinear wave equations with characteristic weights

“…We compare all the lifespan estimates when a = −1 in our results as for F = t − x −(1+b) with those for F = x −(1+b) by Kitamura, Morisawa and Takamura [7]. If the total integral of g does not vanish, they coincide with each other.…”

“…✷ Proofs of Theorem 1.1 and Theorem 1.3. We shall employ the same argument as in the proof of Theorem 2.2 of [7]. We consider an integral equation:…”

“…We next mention the results to the case where F in (1.3) is a spatial weight only as F = x −(1+b) with b ∈ R. Kitamura, Morisawa and Takamura [7] have obtained the lifespan estimates,…”

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

“…This is almost trivial if one takes a look on the representation of u 0 in (2.4) and the support condition on the data in (2.1). But one can see also Proposition 2.2 in Kitamura, Morisawa and Takamura [7] for its details. So, our unknown functions are U := u − εu 0 and W := w − εu 0 t in (2.14).…”

“…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].…”