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

TL;DR

This work analyzes a reaction-diffusion system in $\mathbb{R}^2$ with a point-interaction Laplacian $\Delta_\alpha$, using the absolutely continuous projection to manage the point spectrum. By developing sharp semigroup bounds for $e^{t\Delta_\alpha}$ and employing a Duhamel formulation, the authors prove local well-posedness in energy-type spaces $L^\infty((0,T);H^1_\alpha)$ and $L^r((0,T);H^{s+1}_\alpha)$ with $r>2$, $s<2/r$, and small-data/global results via projections and a Lagrange multiplier. They establish global existence for small initial data in $H^1_\alpha\cap L^1$, with $u$ maintaining the same regularity class and a bound on $\rho(t)$, and prove polynomial decay in time for $u$, its gradient, and the multiplier, under suitable exponent conditions. Overall, the paper extends the analysis of nonlinear diffusion with singular perturbations beyond the standard Laplacian, providing a robust framework for diffusion with point interactions and shedding light on long-time behavior.

Abstract

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

Theorems & Definitions (42)

• Remark 1.2.1
• Remark 1.2.2
• Theorem 1.3.1
• Theorem 1.3.2
• Theorem 1.3.3
• Theorem 1.3.4
• Remark 1.3.5
• Proposition 2.1.1
• proof
• Remark 2.1.2