Where can I find a proof of the following estimate $$\S(t)v\_{L^p(\Omega)}\leq C_{N,p,q} t^{\frac{N}{2}\left(\frac{1}{q}\frac{1}{p}\right)}\v\_{L^q(\Omega)}, $$ where $1\leq p<q<+\infty$, $\Omega\subset \mathbb{R}^N$ is an open bounded set and $\{S(t)\}_{t\geq 0}$ is the semigroup generated by the heat equation with Dirichlet boundary condition ?
This is standard, but the argument is short enough to fit in an answer. It is not restrictive to assume that $0\in\Omega$. Denote by $Q(t,x,y)$ the heat kernel associated with the Dirichlet Laplacian on $\Omega$ (this is simply the solution to the heat equation on $\Omega$ with Dirichlet b.c. and the delta function as initial data) and by $P(t,x,y)=ct^{n/2}\exp(xy^2/4t)$ the heat kernel on $\mathbb{R}^n$, which is known by Fourier transform. By the maximum principle one obtains easily that $0\le Q\le P$ for all $t>0$ and $x,y\in\Omega$. The estimate follows immediately from this gaussian bound for $Q$.

1$\begingroup$ Carro Piero, it is a bit provocative to call the kernels $p$ and $q$, when these letters denote the indices of the Lebesgue norms. $\endgroup$ Mar 28 '16 at 20:13

1$\begingroup$ You are right, I noticed after I posted :). Let me change the names $\endgroup$ Mar 28 '16 at 20:20