JournalsrlmVol. 18, No. 1pp. 59–96

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
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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.

Cite this article

Laurent Véron, Andrey Shishkov, The balance between diffusion and absorption in semilinear parabolic equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 18 (2007), no. 1, pp. 59–96

DOI 10.4171/RLM/481