论文标题
liouville定理，用于半线性热方程的古代解决方案及其椭圆形的对应物
A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart
论文作者
论文摘要
我们建立了非线性远古解决方案对非线性热方程的不存在的$ u_t =ΔU+| u |^{p-1} u $，其绝对值比自相似的径向奇异稳态较小，前提是指数$ p $在Serrrin的指数和Joseph和Lundgren的指数之间严格存在。该结果先前是由Fila和Yanagida [Tohoku Math建立的。 J.（2011）]通过将正向自相似解决方案用作障碍。相比之下，我们与一个独立的独立弱超级措施一起运用了一个广泛的论点。我们的方法自然会借给自己在更高维度中稳态问题产生类似的liouville型结果。实际上，在关键的Sobolev指数的情况下，我们显示了与“ Delaunay”型单数溶液相比，绝对值小的解决方案的有效性。
We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation $u_t=Δu+|u|^{p-1}u$ which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent $p$ is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution.
