On some contractive properties for the heat equations
For a solution u(x,t) of the initial-boundary value problem for the heat equation \[ (1)\quad \partial u/\partial t=\Delta u,\quad x\in \Omega,\quad t>0; \] \[ u(x,0)=u_ 0,\quad x\in \Omega;\quad u(x,t)=0,\quad x\in \partial \Omega,\quad t>0 \] where \(\Omega\) is a bounded domain in \(R^ N\), the following contraction property of the corresponding semigroup is considered: \[ (2)\quad \| \Delta u(\cdot,t)\|_{L^ p(\Omega)}\leq \| \nabla u(\cdot,s)\|_{L^ p(\Omega)},\quad 0<s\leq t, \] where \(1\leq p<\infty\), and \(\nabla\) is the gradient of u. Three functionals which are equivalent to the functional \(u\mapsto \max \{| u(x)-u(y)|\), (x,y)\(\in \Omega,| x-y| \leq \epsilon \}\), the usual Hölder norm and Lipschitz norm, respectively, and which represent the regularity of u, are introduced. It is shown that the above three functionals have the same type of contractive properties as in (2) under the assumption that \(\Omega\) is convex.
scientific article; zbMATH DE number 4190264