The one-dimensional heat equation in the Alexiewicz norm

Abstract

A distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock--Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let $\Theta_t(x)=\exp(-x^2/(4t))/\sqrt{4\pi t}$ be the heat kernel. With initial data $f$ that is the distributional derivative of a continuous function, it is shown that $u_t(x):=u(x,t):=f\ast\Theta_t(x)$ is a classical solution of the heat equation $u_{11}=u_2$. The estimate $\|f\ast\Theta_t\|_\infty\leq\|f\|/\sqrt{\pi t}$ holds. The Alexiewicz norm is $\|f\|=\sup_I|\int_If|$, the supremum taken over all intervals. The initial data is taken on in the Alexiewicz norm, $\|u_t-f\|\to 0$ as $t\to 0^+$. The solution of the heat equation is unique under the assumptions that $\|u_t\|$ is bounded and $u_t\to f$ in the Alexiewicz norm for some integrable $f$. The heat equation is also considered with initial data that is the $n$th derivative of a continuous function and in weighted spaces such that $\int_{-\infty}^\infty f(x)\exp(-ax^2)\,dx$ exists for some $a>0$. Similar results are obtained.