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

Abstract

This paper is devoted to the initial value problems for semilinear wave equations of derivative type with spatial weights in one space dimension. The lifespan estimates of classical solutions are quite different from those for nonlinearity of unknown function itself as the global-in-time existence can be established by spatial decay.

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

“…so that the desired contradiction can be obtained by the same arguments to the proof of Theorem 2.2 in Kitamura, Morisawa and Takamura [8]. In fact, if we assume that there exists a point…”

“…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] by "general theory" for nonlinear wave equations in one dimension. Recently, Kitamura, Morisawa and Takamura [8] have verified the lower bound for all p ≥ 2 including the case that nonlinear term has spatial weights as well as the one of C 1 solutions of associated integral equations for 1 < p < 2. On the other hand, Zhou [15] obtained…”

“…with suitably small ε, where C is a positive constant independent of ε, by Theorem 1.1 in Zhou [16] for the classical solution of (1.1). This estimate is also available for the continuous solution of (2.14) if one employs the proof of Theorem 2.2 with a = −1 in Kitamura, Morisawa and Takamura [8], which is applied to w ≥ εu 0 t + AL ′ (|w| p ). Therefore we obtain the contradiction if T satisfies T ≥ min{Cε −(p−1) , Cε −(q−1)/2 } with suitably small ε, where C is a positive constant independent of ε, which asserts the statement of Theorem 2.3.…”

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

“…u(x, 0) = εf (x), u t (x, 0) = εg(x), x ∈ R, which replaces the weights of (1.1) with the spatial weights, has the following results according to Kitamura, Morisawa and Takamura [6]: 1) for a = 0, Cε −(p−1)/(−a) for a < 0.…”