Summary: We obtain bounds for the decay rate in the \(L^r(\mathbb R^d)\)-norm for the solutions of a nonlocal and nonlinear evolution equation, namely, \[ u_t(x,t)=\int_{\mathbb R^d}K(x,y)|u(y,t)-u(x,t)|^{p-2}(u(y,t)-u(x,t))dy,\quad x\in\mathbb R^d,\quad t>0. \] We consider a kernel of the form \(K(x,y)=\psi(y-a(x))+\psi(x-a(y))\), where \(\psi\) is a bounded, nonnegative function supported in the unit ball and \(a\) is a linear function \(a(x)=Ax\). To obtain the decay rates, we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form \[ T(u)=-\int_{\mathbb R^d}K(x,y)|u(y)-u(x)|^{p-2}(u(y)-u(x))dy,\quad 1\leqslant p<\infty. \] The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole space \(\mathbb R^d\): \[ \lambda _{1,p}(\mathbb R^d)=2\left(\int_{\mathbb R^d}\psi(z)dz\right)\left|\frac{1}{|\det A|^{1/p}}-1\right|^p. \] Moreover, we deal with the \(p=\infty\) eigenvalue problem, studying the limit of \(\lambda_{1,p}^{1/p}\) as \(p\to\infty\).