FULLY NONLINEAR PARABOLIC BOUNDARY-VALUE-PROBLEMS IN ONE SPACE DIMENSION .1.

Of concern is the parabolic equation ∂u ∂t(x,t) = F (x,u, ∂u ∂x, ∂2u ∂x2) for t > 0, x ε{lunate} [0, 1], and with the initial condition u(x, 0) = u0(x) and the nonlinear Robin boundary condition ( ∂u ∂x)(0, t) ε{lunate} β0(u(0, t)), ( -∂u dx)(1, t) ε{lunate} β1(u(1, t)). Here β0 and β1 are maximal monotone graphs in R × R with 0 ε{lunate} β0(0)∩β1(0). We solve it by using nonlinear semigroup theory in the space C[0, 1]. © 1990.