An approximation scheme for a nonlinear degenerate parabolic equation with a second-order differential Volterra operator
Motivated by the porous media equation, as well as by problems of thermoelasticity and viscoelasticity, the author presents an analysis of Rothe's method applied to the (variational form of the) degenerate parabolic problem with memory term, \[ \partial_t\beta(u(t))-\Delta u(t)- \int^t_0 a(t,s)\Delta u(s)\,ds= f(t)\quad\text{in }(0,T)\times \Omega, \] with bounded \(\Omega\subset \mathbb{R}^d\) and subject to initial and (homogeneous) boundary conditions. The function \(\beta\) is assumed to be Lipschitz continuous (with Lipschitz constant \(C\)) and such that \[ \beta(0)= 0,\quad \beta'\geq 0\quad\text{a.e. in }\mathbb{R},\quad |\beta(s)|\geq C_1| s|- C. \] The convergence analysis is carried out for problems with convolution kernels \(a(t, s)= a(t- s)\) that are either smooth or weakly singular and completely monotonic.
degenerate parabolic integrodifferential equation
scientific article; zbMATH DE number 2091481
scientific article