Ohmic Heating Phenomena

Abstract

The current chapter considers two main applications associated with Ohmic heating phenomena. Initially we deal with an application from food industry, building up two one-dimensional non-local problems illustrating the evolution of the temperature of the sterilized food. The former model consists of a diffusion-convection equation while the latter of a convection equation with non-local convection velocity. Both of these non-local models are investigated in terms of their stability and the occurrence of finite-time blow-up, where the latter in the current context indicates food burning. Different approaches should be followed though depending on the monotonicity of the nonlinearity appearing in the non-local term, since no maximum principle is available for the non-local parabolic problem when this nonlinearity is increasing. The second part of the chapter is devoted to the study of a non-local parabolic model illustrating the operation of the thermistor device. Notably, conditions under which finite-time blow-up, which here indicates the destruction of the thermistor device, occurs are investigated by using both energy and comparison methods.