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Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations. (English) Zbl 0926.35064

Summary: We investigate the close relations existing between certain geometric properties of domains \(\Omega\) of \(\mathbb{R}^N\), the validity of Poincaré inequalities in \(\Omega\), 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 geometric, necessary and sufficient condition on \(\Omega\), for the zero solution to be asymptotically (and exponentially) stable in \(L^r(\Omega)\), \(1< r<\infty\), when \(r\) is supercritical \((r> N(p-1)/2)\). The condition is that the inradius of \(\Omega\) be finite. The result is different for \(r\) critical.
For the equation \(u_t-\Delta u= u^p-\mu|\nabla u|^q\), \(q\geq p>1\), \(\mu>0\), we prove that the finiteness of the inradius is a necessary and sufficient condition for global existence and boundedness of all nonnegative solutions.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B35 Stability in context of PDEs
35B60 Continuation and prolongation of solutions to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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