Maximum Principles for Parabolic Equations

Abstract

Let u(x, /) be a smooth function in the domain Q = C2 x (0, L, Q in 0&”, let Du be the spatial gradient of u(x, T) and let vm = (Du, 14). If u(x T) satisfies the parabolic equation F(u, Du, D2u) = ut, we define w(x, T) by g(uO = |vm|_ 1g(vm) (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 am 4- 2h(q2)uluJulJ= W/( 0) we prove That qf(u) Takes its maximum value on the parabolic boundary of Q provided / satisfies a suitable condition. If u(x, T) satisfies the parabolic equation atJ(Du)U(j – b(x, T, u, Du) = ut(b is concave with respect To (x, T, u)) we define C(x, y, T, r) = m(z, 9) – au(x, T) – fiu(y, r) (0 < a, 0 < p, a + p = 1, z = ax 4- py, 9 = at 4- fix) and we prove That if C(x, y, T, r) < 0 when x,y,z e Q and one of /, r = 0, and when T, x e (0, L), and one of x, y, z, e ao, Then it is C(x, y, T, r) < 0 everywhere. © 1994, Australian Mathematical Society. All rights reserved.