Pneumatic rubber tire having at least one component of a polybutadiene-rich rubber composition comprised of a blend of a continuous phase of a specialized cis 1,4-polybutadiene rubber and a dispersed phase comprised of a composite of styrene/butadiene elastomer which contains a dispersion of at least partially exfoliated, intercalated, clay platelets, wherein said cis 1,4-polybutadiene rubber and styrene/butadiene elastomer have spatially defined glass transition temperatures, wherein said blend contains amorphous silica and carbon black reinforcement. Such tire component may be, for example, a circumferential tread having a running surface intended to be ground-contacting or a sidewall insert.