Blow-up of solutions to semilinear parabolic equations driven by mixed local-nonlocal operators with large initial data
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913103633973248 |
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| author | Biagi, Stefano Punzo, Fabio Vecchi, Eugenio |
| author_facet | Biagi, Stefano Punzo, Fabio Vecchi, Eugenio |
| contents | We investigate finite-time blow-up for nonnegative solutions to the Cauchy problem associated with semilinear parabolic equations driven by a mixed local--nonlocal operator. The reaction term is assumed to satisfy suitable structural hypotheses, the prototype being $f(u)=u^p$ with $p>1$. By adapting the Kaplan method to the present framework, we prove that solutions blow up in finite time whenever the initial datum is sufficiently large. In the prototype case $f(u)=u^p$, this conclusion holds for every $p>1$. As a particular case of our operator, we also include the fractional Laplacian; to the best of our knowledge, this type of result is new even in that special case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07911 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Blow-up of solutions to semilinear parabolic equations driven by mixed local-nonlocal operators with large initial data Biagi, Stefano Punzo, Fabio Vecchi, Eugenio Analysis of PDEs We investigate finite-time blow-up for nonnegative solutions to the Cauchy problem associated with semilinear parabolic equations driven by a mixed local--nonlocal operator. The reaction term is assumed to satisfy suitable structural hypotheses, the prototype being $f(u)=u^p$ with $p>1$. By adapting the Kaplan method to the present framework, we prove that solutions blow up in finite time whenever the initial datum is sufficiently large. In the prototype case $f(u)=u^p$, this conclusion holds for every $p>1$. As a particular case of our operator, we also include the fractional Laplacian; to the best of our knowledge, this type of result is new even in that special case. |
| title | Blow-up of solutions to semilinear parabolic equations driven by mixed local-nonlocal operators with large initial data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.07911 |