Almost automorphically-forced flows on S^(1) or R in one-dimensional almost periodic semilinear heat equations

In this paper,we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost periodically-forced scalar reaction-diffusion equation:u_(t)=u_(xx)+f(t,u,u_(x)),t>0,0<x<L(0.1)with the periodic boundary condition u(t,0)=u(t,L),u_(x)(t,0)=u_(x)(t,L),(0.2)where f is uniformly almost periodic in t.In particular,we study the topological structure of the limit sets of the skew-product semiflow.It is proved that any compact minimal invariant set(throughout this paper,we refer to it as a minimal set)can be residually embedded into an invariant set of some almost automorphically-forced flow on a circle S^(1)=R/LZ(see Definition 2.4 for“residually embedded”).Particularly,if f(t,u,p)=f(t,u,-p),then the flow on a minimal set can be embedded into an almost periodically-forced minimal flow on R(see Definition 2.4 for“embedded”).Moreover,it is proved that the ω-limit set of any bounded orbit contains at most two minimal sets that cannot be obtained from each other by phase translation.In addition,we further consider the asymptotic dynamics of the skew-product semiflow generated by(0.1)with the Neumann boundary condition u_(x)(t,0)=u_(x)(t,L)=0 or the Dirichlet boundary condition u(t,0)=u(t,L)=0.For such a system,it has been known that theω-limit set of any bounded orbit contains at most two minimal sets.By applying the new results for(0.1)+(0.2),under certain direct assumptions on f,we prove in this paper that the flow on any minimal set of(0.1)with the Neumann boundary condition or the Dirichlet boundary condition can be embedded into an almost periodically-forced minimal flow on R.Finally,a counterexample is given to show that even for quasi-periodically-forced equations,the results we obtain here cannot be further improved in general.

Electrocaloric devices part Ⅰ:Analytical solution of one-dimensional transient heat conduction in a multilayer electrocaloric system

The analytical solution is reported for one-dimensional(1D)dynamic conduction heat transfer within a mulilayer system that is the typical structure of electrocaloric devices.Here,the multilayer structure of typical electrocaloric devices is simplified as four layers in which two layers of electrocaloric materials(ECMs)are sandwiched between two semi infinite bodies representing the thermal sink and source.The temperature of electrocaloric layers can be instantaneously changed by extemal electic field to establish the initial temperature profile.The analytical solution includes the temperatures in four bodies as a function of both time and location and heat flux through each of the three interfaces as a function of time.Each of these analytical solutions includes five infinite series.It is proved that each of these series is convergent so that the sum of each series can be calculated using the first N terms of the series.The formula for calculating the value of N is presented so that the simulation of an electrocaloric device,such as the temperature distribution and heat transferred from one body to another can be performed.The value of N is dependent on the thickness of electrocaloric material layers,the time of heat conduction,and thermal properties of the materials used.Based on a case study,it is concluded that the N is mostly less than 20 and barely reaches more than 70.The application of the analytical solutions for the simulation of real electrocaloric devices is discussed.