On the Cauchy problem for the reaction-diffusion system with point-interaction in $\mathbb R^2$
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| Format: | Preprint |
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2025
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| _version_ | 1866916685687029760 |
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| author | Barbera, Daniele Georgiev, Vladimir Rastrelli, Mario |
| author_facet | Barbera, Daniele Georgiev, Vladimir Rastrelli, Mario |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08460 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Cauchy problem for the reaction-diffusion system with point-interaction in $\mathbb R^2$ Barbera, Daniele Georgiev, Vladimir Rastrelli, Mario Analysis of PDEs Mathematical Physics 35K57, 35K45, 35A21 35K57, 35K45, 35A21 35K57, 35K45, 35A21 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. |
| title | On the Cauchy problem for the reaction-diffusion system with point-interaction in $\mathbb R^2$ |
| topic | Analysis of PDEs Mathematical Physics 35K57, 35K45, 35A21 35K57, 35K45, 35A21 35K57, 35K45, 35A21 |
| url | https://arxiv.org/abs/2504.08460 |