Qualitative Behavior for a Class of Reaction-Diffusion-Convection Equations

Abstract

Consider the initial and boundary value problem

$$ {v_{t}} = {v_{{xx}}} + {(g(v))_{x}} + f(v)\quad 0 < x < L\quad t > 0 $$

((1))

$$ v(0,t) = v(L,t) = 0\quad t > 0 $$

((2))

$$ v(x,0) = {v_{0}}(x) \geqslant 0\quad 0 < x < L $$

((3))

where f and g are in general nonlinear functions on ℝ⁺, satisfying certain structure conditions. We are interested in describing the structure of the (semi)-dynamical system governed by (1)–(2), namely, what are the equilibrium states, what is the behavior of solutions close to a given equilibrium state, and what else can be said concerning the global description of the set of all possible solutions of (1)–(2).