Locally Uniform Convergence to an Equilibrium for Nonlinear Parabolic Equations on ℝN

<p>We consider bounded solutions of the Cauchy problem</p> <table><tr><td>{</td><td><i>U_t</i> - Δ<i>u</i> = ƒ_<i>u,</td><td> x</i> ∈ℝ<i>^N, t</i> > 0,</td></tr><tr><td> </td><td><i>u</i>(O, <i>x</i>) = <i>u</i>_o (<i>x</i>), </td><td> x</i> ∈ℝ<i>^N,</td></tr></table> </p>where <i>u</i>₀ is a non-negative function with compact support and ƒ is a <i>C</i>¹ function on ℝ with ƒ (O) = 0. Assuming that ƒ' is locally Hölder continuous, and that ƒ satisfies minor nondegeneracy condition we prove that as <i>t</i> → ∞ the solution <i>u(∙, t</i>) converges to an equilibriun <i>φ</i> locally uniformly in ℝ<i>^N</i>. Moreover, either the limit function <i>φ</i> is a constant equilibrium, or there is a point <i>x</i>_o ∈ ℝ<i>^N</i> such that <i>φ</i> is radically symmetric and radially decreasing about <i>x</i>_o, and it approaches a constant equilibrium as |<i>x - x</i>_o| → ∞. The nondegeneracy condition only concerns a specific set of zeros of ƒ 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 <i>N</i> ⋝ 2. Moreover, because of the symmetries of the equation there are always continua of such equilibria. Our result shows that the assumption "<i>u</i>_o has compact support" is powerful enough to guarantee that, first, the equilibria that can possibly be observed in the <i>w</i>-limit set of <i>u</i> have a rather simple structure; and, second, exactly one of them is selected. Our convergence result remains valid if Δ<i>u</i> is replaced by a general elliptic operator of the form ∑_<i>i₇ja_iju_{x_ix_j} with constant coefficients <i>a_ij</i>.</p>