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

TL;DR

This paper proves that solutions to certain semilinear parabolic equations with mixed local and nonlocal operators blow up in finite time if the initial data is large, including cases with fractional Laplacians.

Contribution

The authors adapt the Kaplan method to mixed local-nonlocal operators and establish finite-time blow-up results for large initial data, including the fractional Laplacian case.

Findings

Solutions blow up in finite time for large initial data.

The result applies to equations with reaction term $f(u)=u^p$, $p>1$.

Includes new results for fractional Laplacian operators.

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.