Decay bounds for nonlocal evolution equations in Orlicz spaces

Abstract

We show decay bounds of the form ∫Rd φ(u (x, t)) dx ≤ Ct−μ for integrable and bounded solutions to the nonlocal evolution equation ut(x, t) = ∫Rd J(x, y)G(u(y, t) − u(x, t))(u(y, t) − u(x, t)) dy + f (u(x, t)). Here G is a nonnegative and even function and f verifies f (ξ)ξ ≤ 0 for all ξ ≥ 0. We remark that G is not assumed to be homogeneous. The function φ and the exponent μ depend on G via adequate hypotheses, while J is a nonnegative kernel satisfying suitable assumptions.