Decay bounds for nonlocal evolution equations in Orlicz spaces

Abstract

We show decay bounds of the form ∫Rdϕ(u(x,t))dx≤Ct−μ for integrable and bounded solutions to the nonlocal evolution equation ut(x,t)=∫RdJ(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.