We obtain two-sided sub-Gaussian estimates of heat kernels for strongly
local Dirichlet forms on intervals, equipped with self-similar measures
generated by iterated function systems (IFS's) that do not satisfy the open
set condition (OSC) and have overlaps. We first give a framework for heat
kernel estimates on intervals, and then consider examples of self-similar
measures to illustrate this phenomenon. These examples include the infinite
Bernoulli convolution associated with the golden ratio, and a family of
convolutions of Cantor-type measures. We make use of Strichartz second-order
identities defined by auxiliary IFS's to compute measures of cells on
different levels. These auxiliary IFS's do satisfy the OSC and are used to
define new metrics. The walk dimensions obtained under these new metrics are
strictly greater than 2 and are closely related to the spectral dimension
of fractal Laplacians.