Sharp lifespan estimates and a Huygens-type effect for one-dimensional Nakao's problem

We study the lifespan of small data solutions to the one-dimensional Nakao's problem, which weakly couples a semilinear damped wave equation and a semilinear wave equation.

For compactly supported initial data in a natural energy and integrability class, we establish lower lifespan bounds.

Under the standard integral positivity assumptions, these bounds match the known upper estimates in a large region of the $(p,q)$-plane, including every $p>1$ when $q\geqslant3$.

We further exploit a Huygens-type cancellation effect.

Namely, the condition $\int_{\mathbb{R}}v_1(x)\,\mathrm{d}x=0$ eliminates the constant interior profile of the homogeneous free wave and yields a strictly improved lower bound for the lifespan in a nonempty parameter region.

The proof combines diffusion-type $L^m-L^r$ estimates for the damped component with the one-dimensional d'Alembert formula within a time-dependent continuation framework.