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 Θ_t(x) = exp(-x²/(4t))/(4πt)^1/2 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 * Θ_t(x) is a classical solution of the heat equation u₁₁ = u₂. The estimate ∥f * Θ_t∥_∞ ≤ ∥f∥/(πt)^1/2 holds. The Alexiewicz norm is ∥f∥ := sup_I |∫_If|, the supremum taken over all intervals. The initial data is taken on in the Alexiewicz norm, ∥u_t - f∥ → 0 as t → 0⁺. The solution of the heat equation is unique under the assumptions that ∥u_t∥ is bounded and u_t → f in the Alexiewicz norm for some integrable f. The heat equation is also considered with initial data that is the nth derivative of a continuous function and in weighted spaces such that ∫_-∞^∞f(x)exp(-ax²)dx exists for some a > 0. Similar results are obtained.