Existence and uniform rates of decay of contant problems in viscoelasticity

Abstract

We Consider the unilateral contact problem for the viscoelastic equation in one dimension and we show existence of weak solutions. Moreover, we shown that the solutions decay uniformly to zero with rates depending onthe rate of decay of the relaxation function. That is, denoting by E(t) the first order energy associated with the viscoelastic contant inequality,we show that there exist positive constant C and γ satisfying:

provided the relaxation function decays exponentially.When the rate of decay of the relaxation function is polynomial, then the energy decays also polynomically. That is, if the kernel g Satisfies

then the first order energy decays as .

Keywords:

• Viscoelasticity
• contant problem
• asymptotic behavior
• initial boundary value problems

^*Supported by a grant of CNPq 305406/88-4

^†supported by a grant of CAPES

^*Supported by a grant of CNPq 305406/88-4

^†supported by a grant of CAPES

Notes

^*Supported by a grant of CNPq 305406/88-4

^†supported by a grant of CAPES