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We consider reaction-diffusion equations driven by the $p$-Laplacian onnoncompact, infinite volume manifolds assumed to support the Sobolev inequalityand, in some cases, to have $L^2$ spectrum bounded away from zero, the mainexample we have in mind being the hyperbolic space of any dimension. It isshown that, under appropriate conditions on the parameters involved andsmallness conditions on the initial data, global in time solutions exist andsuitable smoothing effects, namely explicit bounds on the $L^\infty$ norm ofsolutions at all positive times, in terms of $L^q$ norms of the data. Thegeometric setting discussed here requires significant modifications w.r.t. theEuclidean strategies.

Global existence for reaction-diffusion evolution equations driven by the $p$-Laplacian on manifolds