论文标题
带有拉普拉斯操作员分数幂的进化方程的测试功能方法
A test function method for evolution equations with fractional powers of the Laplace operator
论文作者
论文摘要
在本文中，我们讨论了一种测试函数方法，以在初始数据的符号条件下获得具有分数衍生物和功率非线性的高阶进化方程的全球时间解决方案的不存在。为了处理拉普拉斯运营商的分数力量，我们引入了合适的测试功能和合适的弱解决方案。通过缩放参数和反例保证了提供的不存在结果的最佳性。特别是，我们的手稿提供了不存在的对应物，对于以下问题全球存在的几个最新结果是：\] u（0，x）= u_0（x），\ u_t（0，x）= u_1（x）\ end {cases} \]，$ f = | u | u |^p $或$ f = | u_t |^p $，其中$θ\ geq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q queq ands and f = spequs。
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: \[ \begin{cases} u_{tt} + (-Δ)^θu_t + (-Δ)^σ u = f(u,u_t),& t>0, \ x\in\mathbb R^n,\\ u(0,x)=u_0(x), \ u_t(0,x)=u_1(x) \end{cases} \] with $f=|u|^p$ or $f=|u_t|^p$, where $θ\geq0$ and $σ>0$ are fractional powers.
