Critical Exponents for the Decay Rate of Solutions in a Semilinear Parabolic Equation

Abstract.

This paper is concerned with the Cauchy problem \[ \begin{array}{ll} u _t = u _{xx} - |u| ^{ p-1 } u & \quad \mbox{ in } \R \times (0, \infty), \vspace{5pt} \\ \quad u (x,0) = u_0(x) & \quad \mbox{ in } \R. \end{array} \] A solution u is said to decay fast if $t ^{1/(p-1)} u \rightarrow 0$ as $t \rightarrow \infty$ uniformly in R, and is said to decay slowly otherwise. For each nonnegative integer k, let $\Lambda _k$ be the set of uniformly bounded functions on R which change sign k times, and let $p_k>1$ be defined by $p_k=1+2/(k+1)$. It is shown that any nontrivial bounded solution with $u_0\in\Lambda _k$ decays slowly if $1 < p < p_k$, whereas there exists a nontrivial fast decaying solution with $u_0 \in\Lambda_k$ if $p\gep_k$.