A Tracing of the Fractional Temperature Field

TL;DR

This paper investigates the boundary behavior of solutions to the fractional heat equation, focusing on $L^q$-traces of the fractional temperature field in relation to initial data and source terms.

Contribution

It provides new insights into the $L^q$-trace properties of fractional heat solutions, extending classical results to fractional operators.

Findings

Established $L^q$-trace estimates for fractional heat solutions.

Characterized the conditions for trace regularity in fractional heat equations.

Connected trace properties to initial data and source term regularity.

Abstract

This note is devoted to a study of $L^q$-tracing of the fractional temperature field $u(t,x)$ -- the weak solution of the fractional heat equation $(\partial_t+(-\Delta_x)^\alpha)u(t,x)=g(t,x)$ in $L^p(\mathbb R^{1+n}_+)$ subject to the initial temperature $u(0,x)=f(x)$ in $L^p(\mathbb R^n)$.