Decay bounds for nonlocal evolution equations in Orlicz spaces

Abstract

We show decay bounds of the form

$${\int }_{{\mathbb{R}}^d}\varphi \left(u\right(x,t\left)\right)dx\le C{t}^{-\mu }$$ for integrable and bounded solutions to the nonlocal evolution equation

$$u_t(x,t)={\int }_{{\mathbb{R}}^d}J(x,y)G\left(u\right(y,t)-u(x,t\left)\right)\left(u\right(y,t)-u(x,t\left)\right)dy+f\left(u\right(x,t\left)\right).$$ Here $G$ is a nonnegative and even function, and $f$ verifies $f\left(\xi \right)\xi \le 0$ for all $\xi \ge 0$. We remark that $G$ is not assumed to be homogeneous. The function $\varphi$ and the exponent $\mu$ depend on $G$ via adequate hypotheses, while $J$ is a nonnegative kernel satisfying suitable assumptions.