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We consider reaction-diffusion equations of porous medium type, withdifferent kind of reaction terms, and nonnegative bounded initial data. For allthe reaction terms under consideration there are initial data for which thesolution converges to 1 uniformly in compact sets for large times. We willcharacterize for which reaction terms this happens for all nontrivialnonnegative initial data, and for which ones there are also solutionsconverging uniformly to 0. Problems in this family have a unique (up totranslations) travelling wave with a finite front and we will see how its speedgives the asymptotic velocity of all the solutions with compactly supportedinitial data. We will also prove in the one-dimensional case that solutionswith bounded compactly supported initial data converging to 1 do so approachinga translation of this unique traveling wave. We will prove a similar result fornon-compactly supported initial data in a certain class.

Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour