Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:

$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \partial\Omega\right\}$$,

where $b\in L^{\infty}(\partial\Omega)$ (can be taken positive if needed). Denote by $T(t)_{t\geq 0}$ the semigroup generated by $\Delta_R$.

I read in some article that it can be shown that for any $1\leq q\leq p\leq +\infty$ there is a constant $C=C(\Omega,p,q)>0$ (depending only on $\Omega,p,q$) such that following estimate hold:

$$\Vert T(t)\phi\Vert_{L^p(\Omega)}\leq C t^{-\frac{N}{2}\left (\frac{1}{q}-\frac{1}{p}\right )}\Vert\phi\Vert_{L^q(\Omega)},\ \forall\ \phi\in L^q(\Omega).$$

How can we prove that inequality?

I read the proof for Dirichlet boundary conditions in T. Cazenave & A. Haraux - An introduction to Semilinear Evolution Equations,1998, page 44. But how can it be done for Neumann or Robin boundary conditions?

Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:

$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \partial\Omega\right\}$$,

where $b\in L^{\infty}(\partial\Omega)$ (can be taken positive if needed). Denote by $T(t)_{t\geq 0}$ the semigroup generated by $\Delta_R$. Here $\Omega\subseteq\mathbb{R}^N$ is an open and bounded set with uniform Lipschitz boundary.

I read in some article that it can be shown that for any $1\leq q\leq p\leq +\infty$ there is a constant $C=C(\Omega,p,q)>0$ (depending only on $\Omega,p,q$) such that following estimate hold:

$$\Vert T(t)\phi\Vert_{L^p(\Omega)}\leq C t^{-\frac{N}{2}\left (\frac{1}{q}-\frac{1}{p}\right )}\Vert\phi\Vert_{L^q(\Omega)},\ \forall\ \phi\in L^q(\Omega).$$

How can we prove that inequality?

I read the proof for Dirichlet boundary conditions in T. Cazenave & A. Haraux - An introduction to Semilinear Evolution Equations,1998, page 44. But how can it be done for Neumann or Robin boundary conditions?

Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:

$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \partial\Omega\right\}$$,

where $b\in L^{\infty}(\partial\Omega)$ (can be taken positive if needed). Denote by $T(t)_{t\geq 0}$ the semigroup generated by $\Delta_R$.

I read in some article that it can be shown that for any $1\leq q\leq p\leq +\infty$ there is a constant $C=C(\Omega,p,q)>0$ (depending only on $\Omega,p,q$) such that following estimate hold:

$$\Vert T(t)\phi\Vert_{L^p(\Omega)}\leq C t^{-\frac{N}{2}\left (\frac{1}{p}-\frac{1}{q}\right )}\Vert\phi\Vert_{L^q(\Omega)},\ \forall\ \phi\in L^q(\Omega).$$

How can we prove that inequality?

I read the proof for Dirichlet boundary conditions in T. Cazenave & A. Haraux - An introduction to Semilinear Evolution Equations,1998, page 44. But how can it be done for Neumann or Robin boundary conditions?

Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:

$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \partial\Omega\right\}$$,

where $b\in L^{\infty}(\partial\Omega)$ (can be taken positive if needed). Denote by $T(t)_{t\geq 0}$ the semigroup generated by $\Delta_R$.

I read in some article that it can be shown that for any $1\leq q\leq p\leq +\infty$ there is a constant $C=C(\Omega,p,q)>0$ (depending only on $\Omega,p,q$) such that following estimate hold:

$$\Vert T(t)\phi\Vert_{L^p(\Omega)}\leq C t^{-\frac{N}{2}\left (\frac{1}{q}-\frac{1}{p}\right )}\Vert\phi\Vert_{L^q(\Omega)},\ \forall\ \phi\in L^q(\Omega).$$

How can we prove that inequality?

I read the proof for Dirichlet boundary conditions in T. Cazenave & A. Haraux - An introduction to Semilinear Evolution Equations,1998, page 44. But how can it be done for Neumann or Robin boundary conditions?

Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacean defined on:

$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \partial\Omega\right\}$$,

where $b\in L^{\infty}(\partial\Omega)$ (can be taken positive if needed). Denote by $T(t)_{t\geq 0}$ the semigroup generated by $\Delta_R$.

I read in some article that it can be shown that for any $1\leq q\leq p\leq +\infty$ there is a constant $C=C(\Omega,p,q)>0$ (depending only on $\Omega,p,q$) such that following estimate hold:

$$\Vert T(t)\phi\Vert_{L^p(\Omega)}\leq C t^{-\frac{N}{2}\left (\frac{1}{p}-\frac{1}{q}\right )}\Vert\phi\Vert_{L^q(\Omega)},\ \forall\ \phi\in L^q(\Omega).$$

How can we prove that inequality?

I read the proof for Dirichlet boundary conditions in T. Cazenave& A. Haraux - An introduction to Semilinear Evolution Equations,1998, page 44. But how can it be done for Neumann or Robin boundary conditions?

Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:

$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \partial\Omega\right\}$$,

where $b\in L^{\infty}(\partial\Omega)$ (can be taken positive if needed). Denote by $T(t)_{t\geq 0}$ the semigroup generated by $\Delta_R$.

I read in some article that it can be shown that for any $1\leq q\leq p\leq +\infty$ there is a constant $C=C(\Omega,p,q)>0$ (depending only on $\Omega,p,q$) such that following estimate hold:

$$\Vert T(t)\phi\Vert_{L^p(\Omega)}\leq C t^{-\frac{N}{2}\left (\frac{1}{p}-\frac{1}{q}\right )}\Vert\phi\Vert_{L^q(\Omega)},\ \forall\ \phi\in L^q(\Omega).$$

How can we prove that inequality?

I read the proof for Dirichlet boundary conditions in T. Cazenave & A. Haraux - An introduction to Semilinear Evolution Equations,1998, page 44. But how can it be done for Neumann or Robin boundary conditions?