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

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. Furthermore, we derive the linearization of energy-bounded solutions using the methodology introduced in Gérard (J Funct Anal 141:60–98, 1996). 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 of this work lies in overcoming this challenge within the radial framework through the combined application of the Strauss inequality and Strichartz estimates.