Weak solutions of a class of quasilinear hyperbolic integro-differential equations describing viscoelastic materials
Some equations arising in nonlinear viscoelasticity are studied. These equations can be transformed in the form \(u_{tt}(x,t)=div_ x\sigma (x,t)+f(x,t)\text{ on } \Omega \times (0,T],\sigma (x,t)=g(\nabla u(x,t))+\int^{t}_{-\infty}a(t-s)u(\nabla u(x,t),\nabla u(x,s))ds,\) with homogeneous boundary conditions and suitable initial conditions. The following types of results are proved: (i) there is global existence of solutions satisfying an entropy inequality for arbitrary data; (ii) local perturbations from the rest state propagate with finite speed; (iii) the solutions gain regularity if a is sufficiently singular at zero; (iv) solutions converge to a rest state as time goes to infinity in some cases. No uniqueness result is included. Space does not permit to go into details, but the paper is well written and interesting to read.
quasilinear hyperbolic integro-differential equations
scientific article; zbMATH DE number 4192387
scientific article