A concentration theorem for the heat equation (Q5945905)

The paper deals with the Cauchy problem for the heat equation \[ (1) \quad\partial_t u-\Delta u=f(x,t), \qquad (2)\quad u(x,0)=0 \] with \(x\in \mathbb{R}^d\) \((d=1,2,3)\), \(t\geq 0\) and \(f(\cdot,t)\in L^1(\mathbb{R}^d)\). Let \(U(x,t)\) be the solution of (1), (2) with \(f\) replaced by a concentrated source \(\delta_{(\overline x)}\), where \(\overline x\) is a given point in an open domain \(\Omega \subset\mathbb{R}^d\). The main theorem claims that \[ \int_\Omega \bigl|u(x,t) \bigr|^k dx\leq C\int_\Omega \bigl|U(x,t)\bigr |^k dx\tag{3} \] with \(C=C(\Omega, \overline x,t)\) and \(k\) integer. As a result some a priori estimates on the solution \(u\) are obtained. Sufficient conditions are given for (3) to be true with an optimal constant \(C=1\).