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Zbl 0697.73033
Engler, Hans
Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity. (English)
[J] Math. Z. 202, No.2, 251-259 (1989). ISSN 0025-5874; ISSN 1432-1823/e

The following boundary and initial data problem $$ \partial\sp 2\sb tu(x,t)-\partial\sb t\Delta u(x,t)-div\sb x(g(\vert \nabla u(x,t)\vert\sp 2)\nabla u(x,t)=f(x,t),\quad x\in \Omega,\quad 0<t<T, $$ $$ u(x,t)=0,\quad x\in \partial \Omega,\quad 0<t<T, $$ $$ u(x,0)=u\sb 0(x),\quad \partial\sb tu(x,0)=u\sb 1(x),\quad x\in \Omega, $$ 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.
[G.Ciobanu]

MSC 2000:: *74D10 Nonlinear constitutive equations
35A25 Other special methods (PDE)
35B65 Smoothness of solutions of PDE
35A05 General existence and uniqueness theorems (PDE)
35D10 Regularity of generalized solutions of PDE
74S30 Other numerical methods

Keywords: third order partial differential equation; boundary and initial data problem; existence of unique global smooth solutions; large data; general domains in two dimensions

Cited in: Zbl 1121.26019 Zbl 1014.26015 Zbl 0870.26006 Zbl 0789.26009 Zbl 0751.26005

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