Abstract:We show that nonnegative solutions of $$ \hbox {} \vcenter \bgroup \def \errhelp \defaulthelp@ \errmessage {AmS-TeX error: Invalid use of \vspace}##1{\crcr \noalign {\vskip ##1\relax }}\relax \let =\cr \afterassignment \advance \lineskip \dimen@ \advance \baselineskip \dimen@ \advance \lineskiplimit \dimen@ \dimen@ =\jot \everycr {}\tabskip \z@skip \halign \bgroup \hfil \copy \strutbox@ $\mathsurround \z@ \displaystyle {##}$& $\mathsurround \z@ \displaystyle {{}##}$\hfil \crcr & u_{t}-u_{xx}+f(u)=0,\hskip 1em\relax x\in \Bbb R,\hskip 1em\relax t>0, & u=\alpha \mathaccent "7016 u,\hskip 1em\relax x\in \Bbb R,\hskip 1em\relax t=0, \hskip 1em\relax supp\mathaccent "7016 u \hbox { compact } \crcr \egroup \egroup $$ either converge to zero, blow up in $L^{2}$-norm, or converge to the ground state when $t\to \infty $, where the latter case is a threshold phenomenon when $\alpha >0$ varies. The proof is based on the fact that any bounded trajectory converges to a stationary solution. The function $f$ is typically nonlinear but has a sublinear growth at infinity. We also show that for superlinear $f$ it can happen that solutions converge to zero for any $\alpha >0$, provided $supp\mathaccent "7016 u$ is sufficiently small.
Keywords: parabolic equation, stationary solution, convergence
AMS Subject Classification: 35B40, 35K55, 35B05