We consider bounded solutions of the Cauchy problem where u0 is a non-negative function with compact support and f is a C1 function on R with f (0) = 0. Assuming that fis locally H Nolder continuous, and that f satisfies a minor nondegeneracy condition, we prove that, as the solution u( E, t) converges to an equilibrium. locally uniformly in RN. Moreover, either the limit function. is a constant equilibrium, or there is a point x0 RN such that. is radially symmetric and radially decreasing about x0, and it approaches a constant equilibrium as |x. x0|. The nondegeneracy condition only concerns a specific set of zeros of f, and we make no assumption whatsoever on the nonconstant equilibria. The set of such equilibria can be very complicated, and indeed a complete understanding of this set is usually beyond reach in dimension N. 2. Moreover, because of the symmetries of the equation, there are always continua of such equilibria. Our result shows that the assumption gu0 has compact support h is powerful enough to guarantee that, first, the equilibria that can possibly be observed in the β-limit set of u have a rather simple structure; and, second, exactly one of them is selected. Our convergence result remains valid if.u is replaced by a general elliptic operator of the form Pi,j with constant coefficients aij.
- Bounded solutions
- Semilinear parabolic equations