I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on Semigroups of Linear operators I found on many places properties of the Neumann Laplacean.

In W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, Handbook of Differential Equations: Evolutionary Equations. Vol. 1. North-Holland, 2002, pages 1-85 (it can be seen here),

on page 69 we found an assertion that the Neumann Laplacean generates an ultracontractive semigroup. This means that the below property is satisfied only for any $1\geq t>0$ (see the definition from page 65).

Here is my question: How can we prove that the property is true for any $t\in (0,\infty)$? And why the definition of ultracontractivity is only for $t\in (0,1]$?

For an open, bounded, connected and with an uniform Lipschitz boundary $\Omega\subseteq\mathbb{R}^2$ consider the semigroup of linear operators $S(t)_{t\geq 0}$ generated by the Neumann Laplacean:

\begin{equation} \Delta_N:D(\Delta_N)\to L^2(\Omega),\ D(\Delta_N)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}=0, \ \mathcal{H}^{1}\text{- a.e. on}\ \partial\Omega\right\}. \end{equation}

Then for any $1\leq p\leq q\leq +\infty$ there is a constant $c=c(\Omega,p,q)$ that possess the following property (called ultracontractivity):

\begin{equation} \Vert S(t)\phi\Vert_{L^q(\Omega)}\leq c t^{-\frac{N}{2}\left (\frac{1}{p}-\frac{1}{q}\right )}\Vert\phi\Vert_{L^p(\Omega)},\ \forall\ \phi\in L^p(\Omega),\ \forall\ t\geq 0 \end{equation}

P.S. It's a natural question, since in many other books like Cazenave & Haraux - An introduction to semilinear evolution equations (page 44) or Barbu Viorel - Analysis and Control of Nonlinear Infinite Dimensional Systems (page 31) the above property is proved for any $t>0$ in the case of Dirichlet Laplacean.

I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on semigroups of linear operators, I found on many places properties of the Neumann Laplacian.

In W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, Handbook of Differential Equations: Evolutionary Equations. Vol. 1. North-Holland, 2002, pages 1-85 (it can be seen here),

on page 69 we found an assertion that the Neumann Laplacian generates an ultracontractive semigroup. This means that the below property is satisfied only for any $1\geq t>0$ (see the definition from page 65).

Here is my question: How can we prove that the property is true for any $t\in (0,\infty)$? And why the definition of ultracontractivity is only for $t\in (0,1]$?

For an open, bounded, connected and with an uniform Lipschitz boundary $\Omega\subseteq\mathbb{R}^2$ consider the semigroup of linear operators $S(t)_{t\geq 0}$ generated by the Neumann Laplacian:

\begin{equation} \Delta_N:D(\Delta_N)\to L^2(\Omega),\ D(\Delta_N)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}=0, \ \mathcal{H}^{1}\text{- a.e. on}\ \partial\Omega\right\}. \end{equation}

Then for any $1\leq p\leq q\leq +\infty$ there is a constant $c=c(\Omega,p,q)$ that possess the following property (called ultracontractivity):

\begin{equation} \Vert S(t)\phi\Vert_{L^q(\Omega)}\leq c t^{-\frac{N}{2}\left (\frac{1}{p}-\frac{1}{q}\right )}\Vert\phi\Vert_{L^p(\Omega)},\ \forall\ \phi\in L^p(\Omega),\ \forall\ t\geq 0 \end{equation}

P.S. It's a natural question, since in many other books like Cazenave & Haraux – An introduction to semilinear evolution equations (page 44) or Barbu Viorel - Analysis and control of nonlinear infinite dimensional systems (page 31) the above property is proved for any $t>0$ in the case of Dirichlet Laplacian.