On global solvability of non‐linear viscoelastic equations in the analytic category

We prove the global existence of analytic solutions to the non‐linear viscoelastic equations of the following type:\documentclass{article}\pagestyle{empty}\begin{document}$$ u_n \, = \,\phi \,(\,||\,\nabla _x \,u\,||^2)\Delta u\, + \,\Psi \,(||\,D_x^\beta \,1\,u\,||^2,\,.\,.\,.\,\,||\,D_{x^N }^\beta \,||)\,\Delta u_t $$\end{document} where ϕ and ψ are continuous and non‐negative functions satisfying some additional conditions. The method of the analytic energy estimates is used.