A sufficient condition for slow decay of a solution to a semilinear parabolic equation

Imen Ben Arbi (1) , Alain Haraux (1)

1 LJLL - Laboratoire Jacques-Louis Lions

We consider the equation $ \psi_t -\Delta \psi + c | \psi |^{p-1} \psi=0$ with Neumann boundary conditions in a bounded smooth open connected domain of ${\R}^{n} $ with $p>1, c>0$ . We show that if the initial condition is small enough and if the absolte value of its average overpasses a certain multiple of the $p$th power of its $ L^{\infty}$ norm, then $ \psi(t,\cdot)$ decreases like $t^{-\frac{1}{(p-1)}} .$

Domaines: Analyse numérique [cs.NA]

Imen Ben Arbi, Alain Haraux. A sufficient condition for slow decay of a solution to a semilinear parabolic equation. 2010. ⟨hal-00518113⟩