# Lecture 4

Dear participants,

now it is time for the fourth lecture of this Internet Seminar: the first lecture on the heat equation. The analysis of the heat equation is the starting point for the analysis of more general parabolic equations in the whole space or in domains with suitable boundary conditions. The mean feature of the heat equation in the whole space is that an explicit formula for the solution of the associated Cauchy problem with initial condition $u(0,\cdot)=f\in C_b(\mathbb R^d)$ (the space of bounded and continuous functions over $\mathbb R^d$) is available and it is given by the convolution of $f$ with the so-called Gaussian kernel. This, clearly, considerably simplifies the analysis of the heat equation.

We will associate a semigroup of bounded operators with the heat equation: the so-called Gauss-Weierstrass semigroup, and we will study some of its smoothing effects. In view of the applications of the next lectures, particular attention is paid to the behaviour as $t$ tends to $0^+$ of the uniform norm of the derivatives of the function $T(t)f$ when $f$ belongs to $C_b(\mathbb R^d)$ or to some proper subspaces. We will also show that the Gauss-Weierstrass semigroup is analytic but not strongly continuous in $C_b(\mathbb R^d)$. If $d=1$ its infinitesimal generator is the second-order derivative with $C^2_b(\mathbb R)$ as as domain. In the multidimensional case, the infinitesimal generator is a proper extension of the Laplacian, defined in $C^2_b(\mathbb R^d)$ (the space of twice continuously differentiable functions $f:\mathbb R^d\to\mathbb R$ which are bounded together with their derivatives).