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Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity. (English) Zbl 0697.73033

The following boundary and initial data problem

${\partial }_{t}^{2}u\left(x,t\right)-{\partial }_{t}{\Delta }u\left(x,t\right)-di{v}_{x}{\left(g\left(|\nabla u\left(x,t\right)|}^{2}\right)\nabla u\left(x,t\right)=f\left(x,t\right),\phantom{\rule{1.em}{0ex}}x\in {\Omega },\phantom{\rule{1.em}{0ex}}0
$u\left(x,t\right)=0,\phantom{\rule{1.em}{0ex}}x\in \partial {\Omega },\phantom{\rule{1.em}{0ex}}0
$u\left(x,0\right)={u}_{0}\left(x\right),\phantom{\rule{1.em}{0ex}}{\partial }_{t}u\left(x,0\right)={u}_{1}\left(x\right),\phantom{\rule{1.em}{0ex}}x\in {\Omega },$

is considered. The equation of the problem is describing the antiplane shear motion of certain viscoelastic solids. For the above problem the author obtains results regarding the existence of unique global smooth solutions for large data in general domains in two dimensions, by imposing some natural conditions on the function g. The mechanical interpretations of these results are pointed out. Some other boundary conditions are considered.

Reviewer: G.Ciobanu

##### MSC:
 74D10 Nonlinear constitutive equations (materials with memory) 35A25 Other special methods (PDE) 35B65 Smoothness and regularity of solutions of PDE 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000) 74S30 Other numerical methods in solid mechanics
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