Geometry of unbounded domains, Poincare inequalities and stability in semilinear parabolic equations

We investigate the close relations existing between certain ge ometric prop erties of domains Omega of R-N, the validity of PoincarC inequalities in Om ega, and the behavior of solutions of semilinear parabolic equations. For the equation u(t) - Delta u = \u\(p-1) u, p > 1, we Obtain a purely geo metric, necessary and sufficient condition on Omega, for the 0 solution to be asymptotically (and exponentially) stable in L-r(Omega), 1 < r < infinit y, when r is supercritical (r > N (p - 1)/2). The condition is that tho inr adius of Omega be finite. The result is different for 1 critical. For the equation u(t) - Delta u = u(p) - mu\del u\(q), q greater than or eq ual to p > 1, mu > 0, we prove that the finiteness of the inradius is a nec essary and sufficient condition for global existence and boundedness of all nonnegative solutions. AMS CLASSIFICATION: 35K60, 35B35, 35B60, 46E35.