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This paper is concerned with the transient dynamics described by thesolutions of the reaction-diffusion equations in which the reaction termconsists of a combination of a superlinear power-law absorption and atime-independent point source. In one space dimension, solutions of theseproblems with zero initial data are known to approach the stationary solutionin an asymptotically self-similar manner. Here we show that this conclusionremains true in two space dimensions, while in three and higher dimensions thesame conclusion holds true for all powers of the nonlinearity not exceeding theSerrin critical exponent. The analysis requires dealing with solutions thatcontain a persistent singularity and involves a variational proof of existenceof ultra-singular solutions, a special class of self-similar solutions in theconsidered problem.

Eventual self-similarity of solutions for the diffusion equation with nonlinear absorption and a point source