A maximum principle for semilinear parabolic systems and applications

A generalization of the maximum principle to nonlinear parabolic systems

“…Their technique is based upon analytic semigroup approach and the Tikhonov Fixed Point Theorem. As a most recent publication dealing with such systems we mention [4].…”

Local existence and uniqueness of solutions of a degenerate parabolic system

“…When m 1 = m 2 = 1, the system (1.1) models the Newtonian fluids, which is couples with Laplace equations. For various initial boundary problems to this kind system, many articles have been devoted to the existence of the solutions and blowup properties of the solutions [7][8][9].…”

Global existence for parabolic systems by Lyapunov functions

“…In certain cases one can use maximum principles to argue that the existence of suitable super/sub-solutions to (1.1) prevents finite-time blow-up. Note that, in contrast with the scalar case, the usual forms of the maximum principle do not hold, in general, for systems -for a simple counterexample we refer to [9]. However, maximum principles for parabolic systems as (1.1) are available when f (t, x, u, v, ξ) and g(t, x, u, v, ξ) are nondecreasing functions in each of the (u, v)-variables (see [9]).…”

“…Note that, in contrast with the scalar case, the usual forms of the maximum principle do not hold, in general, for systems -for a simple counterexample we refer to [9]. However, maximum principles for parabolic systems as (1.1) are available when f (t, x, u, v, ξ) and g(t, x, u, v, ξ) are nondecreasing functions in each of the (u, v)-variables (see [9]). When applicable, approaches based on maximum principles can yield global existence results for nonlinearities with superlinear growth (see [9] and references therein).…”

“…However, maximum principles for parabolic systems as (1.1) are available when f (t, x, u, v, ξ) and g(t, x, u, v, ξ) are nondecreasing functions in each of the (u, v)-variables (see [9]). When applicable, approaches based on maximum principles can yield global existence results for nonlinearities with superlinear growth (see [9] and references therein). However, the construction of super/sub-solutions is intricate.…”