Event № 62
In this talk I will discuss a transient dynamics describedby the solutions of the reaction-diffusion equations in which thereactionterm consists of a combination of a superlinear power-law absorptionand a time-independent point source. In one space dimension,solutions of these problems with zero initial data are known toapproach the stationary solution in an asymptotically self-similarmanner. Here I will show that this conclusion remains true in twospace dimensions, while in three and higher dimensions the sameconclusion holds true for all powers of the nonlinearity not exceedingthe Serrin critical exponent. The analysis requires dealing withsolutions that contain a persistent singularity and involves avariational proof of existence of ultra-singular solutions, aspecial class of self-similar solutions in the considered problem.