Local Solutions to Quasilinear Parabolic Equations without Growth Restrictions

Abstract

The paper deals with quasilinear parabolic boundary value problems where the nonlinear coefficients and right-hand side are defined with respect to the unknown function $u=u(x;t)$ only in a neighbourhood of the initial function. The quasilinear parabolic problem is approximated by linear elliptic problems by means of semidiscretization in time. Itis proved that the approximations converge uniformly although the data are not continuous functions, and the limit turns out to be the weak solution of the parabolic problem for sufficiently small time $t$. The crucial points of the paper are $L_{\infty }$-estimates to ensure that the approximations belong to the domain of non-linearities and uniform estimates of the discrete time derivatives in a Lebesgue space in order to obtain uniform convergence.