Abstract
Consider the nonuniformly parabolic operator n n Lu = , a&(x, t)u~Xfj+ : bi(x, t)u2. c(x, t, u)ut + d(x, t)u, i.j=1 i=1 where u, aii, bi, c, d are bounded, real-valued functions defined on a domain D=Qx [0, T]cRn+l. Assume that c(x, t, u) is Lipschitz continuous in I * I' of Ca(D), and that c(x, t, u)_O on D. Sufficient conditions on c are found which guarantee existence of a unique solution u e C2+, to the first initial-boundary value problem Lu=f(x, t), u=?p, on the normal boundary of D, where ' E C2+X. Existence is proved by direct application of a fixed point theorem due to Schauder using existence of a solution to the linear problem as well as a priori estimates.
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