In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation \(\partial _t u+t^\beta \mathcal {L} u= - h(t)u^p\) posed on \(\mathbb {R}^N\) , driven by the mixed local-nonlocal operator \(\mathcal {L}=-\Delta +(-\Delta )^{\alpha /2}\) , \(\alpha \in (0,2)\) , and supplemented with a nonnegative integrable initial data, where \(p>1\) , \(\beta \ge 0\) , and \(h:(0,\infty )\rightarrow (0,\infty )\) is a locally integrable function. We study the large time behavior of non-negative solutions and show that the nonlinear term determines the large time asymptotic for \(p\le 1+{\alpha }/{N(\beta +1)},\) while the classical/anomalous diffusion effects win if \(p>1+{\alpha }/{N(\beta +1)}.\)
