A Keller–Segel system involving mixed local and nonlocal operators with logistic term on ℝN.

In this paper, we investigate a parabolic–elliptic Keller–Segel system with a logistic term on ℝN$$ {\mathrm{\mathbb{R}}}^N $$, involving mixed local and nonlocal operators −Δ+(−Δ)s$$ -\Delta +{\left(-\Delta \right)}^s $$ with s∈(0,1)$$ s\in \left(0,1\right) $$. At first, we prove that the semigroup {T(x)}t>0$$ {\left\{T(x)\right\}}_{t>0} $$ generated by Δ−(−Δ)s−I$$ \Delta -{\left(-\Delta \right)}^s-I $$ is an analytical semigroup and uses blow‐up arguments in combination with the classical Liouville‐type theorem to demonstrate the regularity of weak solutions of the parabolic equation with the mixed operators. Next, the local existence and uniqueness of classical solutions are established by applying the semigroup theory and regularity results in case of s∈N2N+4,1$$ \mathbf{s}\in \left(\frac{N}{2N+4},1\right) $$. Moreover, we obtain the global existence and boundedness of classical solutions under some conditions on given initial values and the asymptotic behavior of the global solutions with strictly positive initial conditions. [ABSTRACT FROM AUTHOR]