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 $L=-\mathrm{\Delta }+(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}$, where $0<\alpha <2$. The Cauchy problem under consideration is \begin{equation*} \partial_t u + t^βL 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 $L^{\mathrm{\infty }}({\mathbb{R}}^N)$ 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 $h(t)$.