Dynamical systems of inequalities and nonlinear parabolic equations
Victor A. Galaktionov · Communications in Partial Differential Equations · 1999
A nonlinear second-order uniformly parabolic equation in RR+ u t = F (u; u x ; u xx ); F r (p; q; r) > 0; with suciently smooth right-hand side, is shown to generate a four or two-dimensional dynamical system of inequalities (DSIs) describing evolution properties of a class of solutions u = u(x; t). In general, the two-dimensional DSIs takes into account the joint evolution in time of the one-sided bounds on the solution u(; t) (say, its L 1 x -norm) and on the second spatial derivative u xx (; t). In two dimensions the evolutions orbits of the associated dynamical systems (DSs) form invariant regions for the DSIs. The DSIs and the invariant regions are derived for a general quasilinear heat equation and a fully nonlinear equations. The well-known semiconvexity approach by Aronson-Benilan for the porous medium equation corresponds to a particular case of a discoupled DSIs where the inequalities can be studied separately. Radial solutions of the quasilinear heat equation v t = (...