On the Cauchy problem for the reaction-diffusion system with point-interaction in mathbb R²

The paper studies the existence of solutions for the reaction-diffusion equation in $\mathbb R^2$ with point-interaction laplacian $\Delta_\alpha$ with $\alpha\in(-\infty,+\infty]$, assuming the functions to remain on the absolute continuous projection space. By semigroup estimates, we get the existence and uniqueness of solutions on $$ L^\infty\left((0,T);H^1_\alpha\left(\mathbb R^2\right)\right)\cap L^r\left((0,T);H^{s+1}_\alpha\left(\mathbb R^2\right)\right), $$ with $r>2$, $s<\frac{2}{r}$ for the Cauchy problem with small $T>0$ or small initial conditions on $H^1_\alpha(\mathbb R^2)$. Finally, we prove decay in time of the functions.