# Heat Equation with Measure as Initial Data

In a paper that I am reading, the author uses measures as initial data for pdes. The setting is as following: Let $$\Omega\subset \mathbb{R}^n$$ be a bounded open set with smooth boundary. Let $$u_0$$ be a nonnegative measure on $$\Omega$$. Let $$u$$ be the solution of the problem \begin{align} u_t - \Delta u &= f(x,t) & &\text{on } \Omega\times(0,\infty),\\ u&=0 & &\text{on }\partial\Omega\times(0,\infty), \\ u &= u_0 & &\text{for a.e. x in \Omega}, \end{align} where $$f$$ is a globally Lipschitz continuous function.

The author uses the properties of $$u$$, as I would if $$u_0\in L^\infty(\Omega)$$. For instance he even uses the convolution of $$u_0$$ with the Green function for the heat equation, i.e., $$\int_\Omega G(x,y,t) u_0(y) dy.$$

But I don't even know how to think of a solution $$u$$ where $$u_0$$ is a measure. How do I show that such a solution $$u$$ exists, is regular, etc. Any help would be appreciated.

• Just use the integral definition of “solution”. It makes sense even if $u_0$ is a measure. Commented Oct 19, 2020 at 10:23
• It makes sense to allow the initial distribution of heat to be a finite measure, for example. Commented Nov 12, 2020 at 23:31
• @DisintegratingByParts but how do I interpret $u(x,0)=u_0(x)$ for almost every $x\in\Omega$? $u_0$ is a measure so it is not defined for single points $x\in \Omega$. Commented Nov 13, 2020 at 13:17
• @SC2020 : You cannot measure heat at a point either. Heat is measured over a small region, and that is well-represented as a measure. So integrating against a measure makes sense: $\int_{\Omega}G(x,y,t)d\mu_0(y)dy$. Then $\int_{\Omega'}u(x,y,t)dxdy$ will have a limit of $\mu(\Omega')$ as $t\downarrow 0$. It is a weak limit of measures. In this context, a heat distribution is represented as a measure, and those measures tend weakly to the original measure at $t=0$ as $t\downarrow 0$. Commented Nov 13, 2020 at 15:28
• @DisintegratingByParts Can I also interpret $u=u_0$ as $\lim_{t\rightarrow 0} \int_\Omega u(x,t) \phi(x) dx = \int_{\Omega} \phi \, u_0(dx)$ for every $\phi \in C_c^\infty(\Omega)$? Commented Nov 13, 2020 at 16:14