Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.14614

The Strauss conjecture for the wave equation with scattering damping

Elliot Adamson·Jason Metcalfe·Chengbo Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

The focus of this article is to establish the existence portion of the Strauss conjecture for wave equations with spatially dependent scattering damping. In the current setting, the damping is not sufficiently strong to alter the critical Strauss exponent. In a subsequent study, the authors will consider the case of damping coefficients with critical decay where a shift of the Strauss exponent is expected. While several recent studies of the Strauss conjecture have relied upon weighted Strichartz estimates, here, instead, an approach that is based only on integrated local energy estimates, in particular those with both $r^p$ and ghost weights, is developed. This technique also applies to the problem for the free equation. Space-time Klainerman-Sobolev estimates provide the necessary decay to close the argument.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Elliot Adamson, Jason Metcalfe, Chengbo Wang. 2026-10-10. The Strauss conjecture for the wave equation with scattering damping. https://arxiv.org/abs/2610.14614

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sharp Poincaré inequalities for the logarithmic smoothness of the Gauss map on hypersurfaces

We establish sharp lower bounds for two geometric functionals which evaluate the global variation of the Gauss map on the boundary of bounded open sets $Ω\subset\mathbb{R}^d$. These functionals arise as the endpoint case $s=0$ of the Gagliardo seminorm in $W^{s,p}(\partialΩ)$ of the normal vector field, for $p\in\{1,2\}$. We also show that balls are the unique minimizers. In addition, some applications to the fractional setting $s>0$ are given.

math.AP↗
arXiv2026

Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent

In this paper, we investigate the Cauchy problem of the fractional wave equation with time-dependent damping and the source nonlinearity $f(u)\approx |u|^p$: $$ \begin{cases} \partial_t^2u(t,x)+(-Δ)^{σ/2} u(t,x)+b(t) \partial_t u(t,x) =f(u(t,x)),\ &(t,x)\ \in [0,T)\times \mathbb{R}^N,\\ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x),\ &x\ \in\ \mathbb{R}^N, \end{cases} $$ where the damping coefficient behaves as $b(t)\approx (1+t)^{-β}$. We first establish local existence and uniqueness of mild and strong solutions in weighted Sobolev spaces by deriving weighted Sobolev estimates for the linear propagators and overcoming the spatial-weight coupling in the nonlinearity through the Bessel potential estimates. Then, in the subcritical and critical cases $1<p\leq p_c:=1+\frac σN$, we derive the upper estimates of the lifespan for the fractional Laplacian with $0<σ<2$ and time-dependent damping $β\in [-1, 1)$ via the framework of ordinary differential inequalities. The blow-up results, together with the global existence in the supercritical case obtained in \cite{LI}, show that the critical exponent for the fractional wave equation is $p_c=1+\fracσ{N}$ for $0<σ<2$. Moreover, combining with the lower estimate of lifespan derived in \cite{LI}, we conclude that the estimate in this paper is sharp. Note that our result for the critical case is completely new even in the classical case $b(t)=1$. We also consider the case of $β=1$, and obtain the upper estimate of the lifespan.

math.AP↗
arXiv2026

On the stationary Navier-Stokes equations in distorted pipes under energy-stable outflow boundary conditions

The steady motion of a viscous incompressible fluid in distorted pipes, of finite length, is modeled through the Navier-Stokes equations with mixed boundary conditions: the inflow is given by an arbitrary member of the Lions-Magenes class with positive influx, and the fluid motion is subject to a directional do-nothing boundary condition on the outlet, together with the standard no-slip assumption on the remaining walls of the domain. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data (that is, inlet velocity and external force) by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument that employs Bernoulli's law for solutions of the stationary Euler equations, as well as some properties of harmonic divergence-free vector fields. Under a suitable smallness assumption on the data, we also prove the unique solvability of the boundary-value problem.

math.AP↗