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

Abstract

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