Exponential Stability Of Abstract Coleman–gurtin Equations In Hilbert Spaces

The classical theory of heat conduction originated in 1822 when French mathematician and physicist J.Fourier studied the temperature distribution function of special continuum.Compared with the classical heat conduction model dominated by Fourier’s law,B.D.Coleman and M.E.Gurtin proposed a heat conduction model for thermoviscoelastic materials in 1967.They believed that in real life,the thermal conductivity and unit heat capacity of materials are not constants but functions with respect to time.It is for this reason that the thermoviscoelasticity of the material is considered,which makes this model have some memory behavior.In this paper,we study well-posedness and exponential stability of a class of integrodifferential equations in Hilbert spaces,which fits the heat conduction model of Coleman and Gurtin and is called the Coleman-Gurtin equations.The thermoviscoelasticity of materials leads to the appearance of the convolution term in equations.Accordingly,this abstracted integro-differential equation is more general than the Cauchy problem.In addition,ColemanGurtin equations can be used to simulate non-Fickian diffusion of materials with complex molecular structures,and can also be applied to population dynamics,ecology and atomic reaction dynamics.Because there is the unbounded operator in the convolution term,it brings some difficulty to the problem discussed.We consider its inhomogeneous form,embed it into a product Hilbert space,and then reformulate it into a Cauchy problem.In this case,the solutions of the Coleman-Gurtin equation are given by the first coordinate of solutions of this Cauchy problem.Therefore,well-posedness and exponential stability of the Coleman-Gurtin equation can be deduced by those of this Cauchy problem respectively.According to the semigroup theory,if the system operator of this Cauchy problem generates an operator semigroup,this Cauchy problem is well-posed.On this basis,according to the results given by L.Gearhart in1978,the some boundedness of resolvent of its system operator can ensure its exponential stability.Next,we discuss the special case of exponentially decaying kernels and establish a sufficient condition to judge the exponential stability.Finally,a finite dimensional example and three infinite dimensional examples are given,and the feasibility and effectiveness of the proposed method are verified by Matlab.