Semilinear Equations Including the Mixed Operator

We study the local and global existence of solutions to a semilinear evolution equation driven by a mixed local-nonlocal operator of the form $L = -\Delta + (-\Delta)^{\alpha/2}$, where $0 < \alpha < 2$. The Cauchy problem under consideration is \begin{equation*} \partial_t u + t^\beta L u = -h(t) u^p, \quad x \in \mathbb{R}^N, \quad t > 0, \end{equation*} with nonnegative initial data $u(x, 0) = u_0(x)$. We establish the existence and uniqueness of local solutions in $L^\infty(\mathbb{R}^N)$ using a contraction mapping argument. Furthermore, we analyze conditions for global existence, proving that solutions remain globally bounded in time under appropriate assumptions on the parameters $\beta$, $p$, and the function $h(t)$.