论文标题
某些半线性热方程的解决方案不存在
Nonexistence of solutions of certain semilinear heat equations
论文作者
论文摘要
我们考虑一个半线性热方程，涉及仅取决于空间变量的强迫项。首先，通过应用Banach定理的应用证明了局部温和解决方案的存在。借助精心定义的测试功能，我们证明了全球弱解决方案的不存在。最关键的步骤是找到我们的证明中使用的功能$ d（x）$，这似乎仅取决于所考虑的矢量字段。这导致了可能的关键富士型型指数的下限。相同的功能$ d（x）$可能会导致潜在的标准函数，这在使用这些向量字段时最适合。第4节是本文的吸引力，在该论文中，我们将方法应用于Biagi，Bonfiglioli和Bramanti讨论的所有矢量领域，从而引起了Grushin型和Engel型PDOS等。在每种情况下，还提供了局部解决方案爆破时间的上限。
We consider a semilinear heat equation involving a forcing term which depends only on the space variable. To start with, the existence of a local mild solution is proved through an application of the Banach fixed-point theorem. With the help of carefully defined test functions, we then prove the nonexistence of global weak solutions. The most crucial step is to find the function $d(x)$ used in our proofs, which seems to depends only upon the considered vector fields. This leads to lower bounds for a possible critical Fujita-type exponent. The same function $d(x)$ could lead to a potential norm function which would be most suitable while working with these vector fields. Section 4 is the attraction of this paper in which we apply our approach to all of the vector fields discussed by Biagi, Bonfiglioli and Bramanti, giving rise to Grushin-type and Engel-type PDOs, and more. An upper bound for the blow-up time of local solutions is also provided in each of these cases.
