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By using the analytic tools of Dirichlet forms, we initiate a study of somenon-linear parabolic equations on Sierpinski gasket, motivated by modellings offluid flows along a fractal (which can be considered as a simplified roughporous medium). Unlike the regular spaces case, such equation involving aconvection term must take a quite different form and the convection term mustbe singular to the "linear part" which is determined the heat semigroup. Inorder to study these parabolic type equations, a new kind of Sobolevinequalities for the Dirichlet form on the gasket will be established as aneffective tool for our study. These Sobolev inequalities, which are interestingby their own and in contrast to the case of Euclidean spaces, involve two $L^p$norms with respect two mutually singular measures. By examining properties of asingular convolution of the associated heat semigroup, we derive the space-timeregularity of solutions to the parabolic equations under a few technicalconditions. The Burgers equations on the Sierpinski gasket are also studied,for which a maximum principle for solutions is derived using techniques frombackward stochastic differential equations, and the existence and uniqueness aswell as the regularity of solutions are obtained.

Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket