On the Cauchy problem for the reaction-diffusion system with point-interaction in ${\mathbb{R}}^2$

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