Heat flow for Dirichlet-to-Neumann operator with critical growth
The authors study the heat flow equation for the Dirichlet-to-Neumann operator with critical growth. More precisely, the authors consider the case that the initial value is lower-energy, i.e. \(E(v_0) <\frac{1}{2N}S^N\) where \(S\) is the best Sobolev constant, \[ E(v_0) = \frac12 \int_{\Omega\times(0,\infty)}|\nabla v_0(t)|^2dxdy - \frac{1}{2^\ast}\int_{\Omega \times\{0\}}|v(t)| ^{2^\ast} dx \] and \(2^\ast\) is \( \frac{2N}{N-1}\). Under such an assumption, the authors study the existence, blowup and regularity of the solutions. Lastly they prove also that a concentration phenomenon of the solution when the time goes to infinity occurs.
scientific article; zbMATH DE number 6850682
scientific article