A test function method for evolution equations with fractional powers of the Laplace operator

TL;DR

This paper develops a test function method to prove the nonexistence of global solutions for certain higher order evolution equations involving fractional Laplacians, extending previous results on global existence for small data.

Contribution

It introduces a novel test function approach for fractional Laplace operators and establishes optimal nonexistence results for a class of fractional evolution equations.

Findings

Proves nonexistence of global solutions under specific conditions.

Provides a framework applicable to equations with fractional powers of Laplacian.

Establishes the optimality of nonexistence results through scaling and counterexamples.

Abstract

In this paper, we discuss a test function method to obtain nonexistence of global-in-time solutions for higher order evolution equations with fractional derivatives and a power nonlinearity, under a sign condition on the initial data. In order to deal with fractional powers of the Laplace operator, we introduce a suitable test function and a suitable class of weak solutions. The optimality of the nonexistence result provided is guaranteed by both scaling arguments and counterexamples. In particular, our manuscript provides the counterpart of nonexistence for several recent results of global existence of small data solutions to the following problem: \[ \begin{cases} u_{tt} + (-\Delta)^{\theta}u_t + (-\Delta)^{\sigma} u = f(u,u_t),& t>0, \ x\in\mathbb R^n,\\ u(0,x)=u_0(x), \ u_t(0,x)=u_1(x) \end{cases} \] with $f=|u|^p$ or $f=|u_t|^p$, where $\theta\geq0$ and $\sigma>0$ are fractional powers.