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

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.