Well-posedness and linearization for a semilinear wave equation with spatially growing nonlinearity

TL;DR

This paper proves global well-posedness and derives linearization for a defocusing semilinear wave equation with spatially growing nonlinearity, overcoming challenges posed by the nonlinearity's growth at infinity.

Contribution

It introduces a novel approach combining Strauss inequality and Strichartz estimates to handle spatially growing nonlinearities in wave equations.

Findings

Established global well-posedness in the energy space for radial data.

Derived the linearization of energy-bounded solutions.

Overcame difficulties due to nonlinearity growth at infinity.

Abstract

We study the initial value problem for a defocusing semi-linear wave equation with spatially growing nonlinearity. By employing Moser-Trudinger type inequalities and Strichartz estimates, we establish global well-posedness in the energy space for radially symmetric initial data. Additionally, we derive the linearization of energy-bounded solutions. The main challenge in our analysis arises from the spatial growth of the nonlinearity at infinity, which prevents the direct application of Sobolev embeddings or Hardy inequalities to control the potential energy. The main novelty in this work lies in overcoming this challenge within the radial framework through the combined application of the Strauss inequality and Strichartz estimates.