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We revisit rescaling methods for nonlinear elliptic and parabolic problemsand show that, by suitable modifications, they may be used for nonlinearitiesthat are not scale invariant even asymptotically and whose behavior can bequite far from power like.  In this enlarged framework, by adapting the doubling-rescaling method from[37, 38], we show that the equivalence found there between universal estimatesand Liouville theorems remains valid. In the parabolic case we also prove aLiouville type theorem for a rather large class of non scale invariantnonlinearities. This leads to a number of new results for non scale invariantelliptic and parabolic problems, concerning space or space-time singularityestimates, initial and final blow-up rates, universal and a priori bounds forglobal solutions, and decay rates in space and/or time.  We illustrate our approach by a number of examples, which in turn giveindication about the optimality of the estimates and of the assumptions.

Universal estimates and Liouville theorems for superlinear problems without scale invariance