A Benefit from the L ^∞ Smallness of Initial Data for the Semilinear Wave Equation with Structural Damping

Abstract

In this note, we prove the global existence of small data solutions for a semilinear wave equation with structural damping,

$$u_{tt}-\Delta u + \mu(-\Delta)^{\frac{1}{2}} u_t = |u|^p, $$

for any n≥2 and p>1+2/(n−1). The damping term allows us to derive linear $L^{q_{1}}-L^{q_{2}}$ estimates, for 1≤q ₁≤q ₂≤∞, without loss of regularity, in any space dimension. These estimates provide the basic tool to state our result, in which we assume initial data to be small in (L ¹∩H ¹∩L ^∞)×(L ¹∩L ⁿ).