Line method approximations to the Cauchy problem for nonlinear parabolic differential equations

Summary The Cauchy problemu_t=f(x, t, u, u_x, u_xx),u(x, o)=(x),xR, is treated with the longitudinal method of lines. Existence, uniqueness, monotonicity and convergence properties of the line method approximations are investigated under the classical assumption that satisfies an inequality |(x)|<=conste^Bx2. We obtain generalizations of the works of Kamynin [4], who got similar results in the case of the one dimensional heat equation when is allowed to grow likee^Bx2–, >0, and of Walter [11], who proved convergence in the case of nonlinear parabolic differential equations under the growth condition |(x)|<=conste^B|x|