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 \(\Delta _\alpha \) 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

$$\begin{aligned} L^\infty \left( (0,T);H^1_\alpha \left( \mathbb {R}^2\right) \right) \cap L^r\left( (0,T);H^{s+1}_\alpha \left( \mathbb {R}^2\right) \right) , \end{aligned}$$

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.