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arXiv · 2605.07911

Blow-up of solutions to semilinear parabolic equations driven by mixed local-nonlocal operators with large initial data

Stefano Biagi·Fabio Punzo·Eugenio Vecchi
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Abstract

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.

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Stefano Biagi, Fabio Punzo, Eugenio Vecchi. 2026-05-08. Blow-up of solutions to semilinear parabolic equations driven by mixed local-nonlocal operators with large initial data. https://arxiv.org/abs/2605.07911

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