Large time behaviour of solutions to a class of non-autonomous, degenerate parabolic equations F. Ragnedda · S. Vernier Piro · V. Vespri Abstract We consider a class of non-autonomous, degenerate parabolic equations and we study the asymptotic behaviour of the solutions. Even if the equation depends explicitly upon the time, we prove that several asymptotic properties, valid for the autonomous case, are preserved in this more general situation. To our knowledge, it is the first time that the asymptotic behaviour of solutions to non-autonomous equations is studied.

We deal with the Cauchy problem associated to a class of quasilinear singular parabolic equations with L∞ coefficients whose prototypes are the p-Laplacian (2N/(N + 1) < p < 2) and the porous medium equation (((N - 2)/N)+ < m < 1). We prove existence of and sharp pointwise estimates from above and from below for the fundamental solutions. Our results can be extended to general non-negative L1 initi

We consider a non-autonomous, degenerate parabolic problem with Dirichlet boundary\ud condition. We study the asymptotic behaviour of solutions, extending an earlier result of\ud the authors, where the forcing term was taken zero

The problem of optimization of contact interaction of a moving rigid punch and an elastic half-space is investigated taking into account friction and wear. The punch shape is accepted as a desirable design variable and the volumetric wear rate under constraints on the friction dissipation power and the total load, applied to the punch, is taken as an optimized quality criterion. The ratio of the volumetric wear rate and the friction dissipation power under constraint on the total load is also considered as a suitable objective functional. A necessary condition for the optimality in quasi-steady state wear process is derived and discussed. The optimization problem is investigated analytically and exact solutions are obtained for the axysimmetric (stamp) punch which has a circular contact region and moves translationally with a constant velocity or rotates with constant angular velocity.