J Austral Math Soc Ser A 56 pp41--52, 1994.

Maximum Principles for Parabolic Equations

Giovanni Porru and Salvatore Serra

Abstract

Let u(x, t) be a smooth function in the domain Q = W×(0, L], W in Rⁿ, let Du be the spatial gradient of u(x, t) and let Ñu = (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) = |Ñu|^-1G(Ñu) (g is positive and decreasing, G is concave and homogenous 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 Du + 2h(q²)u_iu_ju_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, t) = u(z, q) - au(x, t) - bu(y, t) (0 < a, 0 < b, a+ b = 1, z = ax + by, q = at + bt) and we prove that if C(x, y, t, t) £ 0 when x, y, z Î W and one of t, t = 0, and when t, t Î (0, L], and one of x, y, z Î ¶W, then it is C(x, y, t, t) £ 0 everywhere.

1991 AMS Subject Classification: 35B50

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Authors

Giovanni Porru

Dipartimento di Matematica, Universita degli studi di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy.

Salvatore Serra

Dipartimento di Matematica, Universita degli studi di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy.