Connecting orbits in scalar reaction diffusion equations
P. Brunovský, Bernold Fiedler · Dynamics reported · 1988
We consider the flow of a one-dimensional reaction diffusion equation 1.1 $$ {u_t} = {u_{xx}} + f(u),x \in (0,1)$$ with Dirichlet boundary conditions 1.2 $$ u(t,0) = u(t,1) = 0{\text{ }}$$ Let v, w denote stationary, i.e. t-independent solutions. We say that v connects to w, if there exists an orbit u(t, x) of (1.1), (1.2) such that 1.3 $$ \mathop {\lim }\limits_{t \to - \infty } \;u\left( {t, \cdot } \right) = \upsilon \;\mathop {\lim }\limits_{t \to - \infty } \;u\left( {t, \cdot } \right) = w$$ i.e. u(t, ·) is a heteroclinic orbit connecting v to w. In this report we address the following question: (*) Given v, which stationary solutions w does it connect to?