A nonlinear diffusion equation with reaction localized in the half-line

Abstract

<abstract><p>We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t = (u^m)_{xx}+a(x) u^p, $\end{document} </tex-math></disp-formula></p> <p>$ m, p &gt; 0 $ and $ a(x) = 1 $ for $ x &gt; 0 $, $ a(x) = 0 $ for $ x &lt; 0 $. We first characterize the global existence exponent $ p_0 = 1 $ and the Fujita exponent $ p_c = m+2 $. Then we pass to study the grow-up rate in the case $ p\le1 $ and the blow-up rate for $ p &gt; 1 $. In particular we show that the grow-up rate is different as for global reaction if $ p &gt; m $ or $ p = 1\neq m $.</p></abstract>