ON CLASSES OF WELL-POSEDNESS FOR QUASILINEAR DIFFUSION EQUATIONS IN THE WHOLE SPACE

Well-posedness classes for degenerate elliptic problems in R-N under the form u = Delta phi(x,u) + f(x), with locally (in u) uniformly continuous nonlinearities, are explored. While we are particularly interested in the L-infinity setting, we also investigate about solutions in L-loc(1) and in weighted L-1 spaces. We give some sufficient conditions in order that the uniqueness and comparison properties hold for the associated solutions; these conditions are expressed in terms of the moduli of continuity of u bar right arrow phi(x,u). Under additional restrictions on the dependency of phi on x, we deduce the existence results for the corresponding classes of solutions and data. Moreover, continuous dependence results follow readily from the existence claim and the comparison property. In particular, we show that for a general continuous non-decreasing nonlinearity phi : R bar right arrow R, the space L-infinity (endowed with the L-loc(1) topology) is a well-posedness class for the problem u = Delta phi(u) + f(x).