P. Souplet, Geometry of unbounded domains, Poincare inequalities and stability in semilinear parabolic equations, COMM PART D, 24(5-6), 1999, pp. 951-973
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.