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

2026/07/16 by Wenhui Chen
#math.AP

paper · pdf

Abstract

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 ∫_ℝv₁(x) dx=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.

Related

• Lifespan estimates for 1d damped wave equation with zero moment initial data
• The sharp lower bound of the lifespan of solutions to semilinear wave equations with low powers in two space dimensions
• On extended lifespan for 1d damped wave equation
• Recent developments on the lifespan estimate for classical solutions of nonlinear wave equations in one space dimension
• Lifespan estimates for wave equations with damping and potential posed on asymptotically Euclidean manifolds