On partial regularity of the borderline solution of semilinear parabolic equation with critical growth

Abstract

In this paper, we consider the borderline solution to the semilinear equations with critical growth. A concentration phenomenon of the solution when the time goes to infinity is proved. First, we show that a $\varepsilon$-regularity property holds for an $H^1$ solution to the related elliptic equation, and then give a precise description of the formation of the bubbles. A similar bubbling description is also derived for the harmonic maps on surface. (Cf.~Struwe [22], Qing [19], Qing-Tian [20], Chen-Tian [6], Lin-Wang [13], and Parker [16]).