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

We investigate thc close relations existing between certain geometric properties of domains Ω of RN, the validity of Poincark inequalities in Ω, and the behavior of solutions of semilinear parabolic equations. For the equation ut-△u=|u|p-1 we obtain a purely geometric, necessary and sufficient condition on Ω, for the 0 solution to be asymptotically (and exponentially) stable in Lr(ω)1 N(p-1)/2 . The condition is that the inradius of Ω be finite. The result is different for r critical. For the equation ut-△u=up-μ|u|q,q≥p>1,μ>0 we prove that the finiteness of the inradius is a necessary and sufficient condition for global existence and boundedness of all nonnegative solutions.

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