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A model of viscoelasticity with time-dependent memory kernels. (English) Zbl 1480.45010

Summary: We consider the model equation arising in the theory of viscoelasticity \[ \partial_{tt}u-h_t(0)\Delta u-\int_{0}^\infty h_t'(s)\Delta u(t-s)\mathrm{d}s+f(u)=g. \] Here, the main feature is that the memory kernel \(h_t(\cdot)\) depends on time, allowing for instance to describe the dynamics of aging materials. From the mathematical viewpoint, this translates into the study of dynamical systems acting on time-dependent spaces, according to the newly established theory of F. Di Plinio et al. [Discrete Contin. Dyn. Syst. 29, No. 1, 141–167 (2011; Zbl 1223.37100); Boll. Unione Mat. Ital. (9) 5, No. 1, 19–53 (2012; Zbl 1256.35155)]. In this first work, we give a proper notion of solution, and we provide a global well-posedness result. The techniques naturally extend to the analysis of the longterm behavior of the associated process, and can be exported to cover the case of general systems with memory in presence of time-dependent kernels.

MSC:

45K05 Integro-partial differential equations
35Q70 PDEs in connection with mechanics of particles and systems of particles
74B20 Nonlinear elasticity
74D10 Nonlinear constitutive equations for materials with memory
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