Marta Sanz-Solé

Global solutions to stochastic wave equations with superlinear coefficients

We consider the nonlinear stochastic wave equation on ℝᵈ, d=1,2,3,
∂ₜ² u(t,x) − Δₓ u(t,x) = b(u(t,x)) + σ(u(t,x)) ∂ₜW(t,x), (t,x) ∈ (0,T] × ℝᵈ,
with given initial conditions. The process ∂ₜW is a space−time white noise if d=1, while for d=2,3 it is white in time and coloured in space. For coefficients b and σ satisfying |b(z)| ≤ b₁ + b₂ |z| (ln |z|)ᵟ¹, |σ(z)| ≤ σ₁ + σ₂ |z| (ln |z|)ᵟ², when |z|→∞, we find conditions on the superlinear exponents and also on the noise when d=2,3, ensuring global existence (and uniqueness) of a random field solution. Examples of relevant noises are also provided. Recent results by M. Foondun and E. Nualart (2021) provide some partial information on the critical values of the growth exponents. The proof relies on sharp moment estimates of the solution, and of increments in time and space of them, of a sequence of stochastic wave equations closely related to the above equation, with globally Lipschitz continuous coefficients. The research is joint work with A. Millet. We were motivated by a paper from R. Dalang, D. Khoshnevisan and T. Zhang (2019) where a similar question is addressed for a nonlinear stochastic heat equation on [0,1].