Decay bounds for nonlocal evolution equations in Orlicz spaces

ROSSI, JULIO; VIDAL, RAÚL; KAUFMANN, URIEL

Annals of Functional Analysis

Duke University Press

Año: 2016 vol. 7 p. 261 - 269

We show decay bounds of the form∫_{R^{d}}φ(u(x,t))dx≤Ct^{-μ}for integrable and bounded solutions to the nonlocal evolution equationu_{t}(x,t)=∫_{R^{d}}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.