Some qualitative properties of solution to a fractional thermo-viscoelastic system with nonlinear sources

Abstract

In the paper, we consider a fractional thermo-viscoelastic system with nonlinear sources and study some of its qualitative properties based on the interaction of the fractional viscoelastic and thermal damping with the external forces. By using the theory of linear Volterra differential-integral equations of convolution type and the Banach fixed point theorem, we first prove the local well-posedness and maximal regularity of the weak solution. Then by using the variational and potential well methods, we give a sufficient condition for the continuity in time of the local weak solution when it starts in the potential wells. Besides that the asymptotic behavior of global solution is also concerned, unlike the classical thermoelasticity where the total energy does not decays uniformly, since the effect of the fractional viscoelastic damping, we show that the total energy shall decay uniformly. In addition, its decay rate is given explicitly and optimally in the sense of Lasiecka et. al.[1]. Finally, since the presence of the nonlinear sources, we show that the blow-up phenomenon may occur in finite time provided that the solution starts outside the potential wells and the relaxation function is small in some sense. Also notice that the effect of the thermal damping is not enough to make the total energy decays to zero, but it could retards the blow-up phenomenon.