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We study the interior H\"older regularity problem for weak solutions of theporous medium equation with external forces. Since the porous medium equationis the typical example of degenerate parabolic equations, H\"older regularityis a delicate matter and does not follow by classical methods.Caffrelli-Friedman, and Caffarelli-Vazquez-Wolansky showed H\"older regularityfor the model equation without external forces. DiBenedetto and Friedman showedthe H\"older continuity of weak solutions with some integrability conditions ofthe external forces but they did not obtain the quantitative estimates. Thequantitative estimates are important for studying the perturbation problem ofthe porous medium equation. We obtain the scale invariant H\"older estimatesfor weak solutions of the porous medium equations with the external forces. Asa particular case, we recover the well known H\"older estimates for the linearheat equation.

H\"older estimates for solutions of the Cauchy problem for the porous medium equation with external forces