JournalsrlmVol. 18 , No. 1DOI 10.4171/rlm/481

The balance between diffusion and absorption in semilinear parabolic equations

  • Laurent Véron

    Université François Rabelais, Tours, France
  • Andrey Shishkov

    Academy of Sciences of Ukraine, Donetsk, Ukraine
The balance between diffusion and absorption in semilinear parabolic equations cover

Abstract

Let h:[0,)[0,)h:[0,\infty)\mapsto [0,\infty) be continuous and nondecreasing, h(t)>0h(t)>0 if t>0t>0, and m,qm,q be positive real numbers. We investigate the behavior when kk\to\infty of the fundamental solutions u=uku=u_{k} of \prttu\Gdum+h(t)uq=0\prt_{t} u-\Gd u^m+h(t)u^q=0 in \Gw\ti(0,T)\Gw\ti (0,T) satisfying uk(x,0)=k\gd0u_{k}(x,0)=k\gd_0. The main question is wether the limit is still a solution of the above equation with an isolated singularity at (0,0)(0,0), or a solution of the associated ordinary differential equation u+h(t)uq=0u'+h(t)u^q=0 which blows-up at t=0t=0.