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Single-point blowup for nonlocal parabolic problems. (English) Zbl 0952.74028

Summary: The formation of shear bands in materials has important implications to a variety of physical processes. These bands are observed in very thin zones and are generally regarded as a precursor to material failure. Shear band formation is caused by the heat generated in regions with highest strain rate. With insufficient time for diffusion of this heat, a localized thermal softening of the material occurs which enhances plastic flow in a thin zone. This adiabatic strain localization can be modelled as nonlinear thermally-activated reaction-diffusion equations. This leads to a class of nonlocal parabolic problems and their associated time-independent steady-state counterparts. A discussion of the models and an analysis of the problems will be presented.

MSC:

74H35 Singularities, blow-up, stress concentrations for dynamical problems in solid mechanics
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
74R20 Anelastic fracture and damage
35K99 Parabolic equations and parabolic systems
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