Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A test function method for evolution equations with fractional powers of
  the Laplace operator   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.

25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th