An initial-boundary value problem for the one-dimensional non-classical heat equation in a slab

Abstract

Abstract Nonlinear problems for the one-dimensional heat equation in a bounded and homogeneous medium with temperature data on the boundaries x = 0 and x = 1, and a uniform spatial heat source depending on the heat flux (or the temperature) on the boundary x = 0 are studied. Existence and uniqueness for the solution to non-classical heat conduction problems, under suitable assumptions on the data, are obtained. Comparisons results and asymptotic behavior for the solution for particular choices of the heat source, initial, and boundary data are also obtained. A generalization for non-classical moving boundary problems for the heat equation is also given. 2000 AMS Subject Classification: 35C15, 35K55, 45D05, 80A20, 35R35.

Keywords

• Non-classical heat equation
• Nonlinear heat conduction problems
• Volterra integral equations
• Moving boundary problems
• Uniform heat source