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.
Highlights
In this article, we will consider initial and boundary value problems (IBVP), for the one-dimensional non-classical heat equation motivated by some phenomena regarding the design of thermal regulation devices that provides a heater or cooler effect [1,2,3,4,5,6]
Where the unknown function u = u(x,t) denotes the temperature profile for an homogeneous medium occupying the spatial region 0 < x
Non-classical problems like (1.1) to (1.4) are motivated by the modelling of a system of temperature regulation in isotropic media and the source term in (1.1) describes a cooling or heating effect depending on the properties of F which are related to the evolution of the heat ux(0,t)
Summary
We will consider initial and boundary value problems (IBVP), for the one-dimensional non-classical heat equation motivated by some phenomena regarding the design of thermal regulation devices that provides a heater or cooler effect [1,2,3,4,5,6]. Theorem 2 Under assumptions (HA) to (HD), the solution u to problem (P1) in [0,1] × [0,T], given by Theorem 1, is bounded in terms of the initial and boundary data h, f and g.
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