MAXIMUM-PRINCIPLES FOR PARABOLIC EQUATIONS

Let u(x, t) be a smooth function in the domain Q = OMEGA x (0, L], OMEGA in R(n), let Du be the spatial gradient of u(x, t) and let delu = (Du, u(t)). If u(x, t) satisfies the parabolic equation F(u, Du, D2u) = u(t), we define w(x, t) by g(w) = \delu\-1G(delu) (g is positive and decreasing, G is concave and homogeneous of degree one) and we prove that w(x, t) attains its maximum value on the parabolic boundary of Q. If u(x, t) satisfies the equation DELTAu + 2h(q2) u(i)u(j)u(ij) = u(t) (q2 = \Du\2, 1+ 2q2h(q2) > 0) we prove that qf(u) takes its maximum value on the parabolic boundary of Q provided f satisfies a suitable condition. If u(x, t) satisfies the parabolic equation a(ij)(Du)u(ij) - b(x, t, u, Du) = u(t) (b is concave with respect to (x, t, u)) we define C (x, y, t, tau) = u(z, theta) - alphau (x, t) - betau(y, tau) (0 < alpha, 0 < beta, alpha + beta = 1, z = alphaX + betay, theta = alphat + betatau) and we prove that if C(x, y, t, tau) less-than-or equal-to 0 when x, y, z is-an-element-of OMEGA and one of t, tau = 0, and when t, tau is-an-element-of (0, L], and one of x, y, z, is-an-element-of partial-derivative OMEGA, then it is C(x, y, t, tau) less-than-or-equal-to 0 everywhere.