SEMILINEAR PARABOLIC EQUATIONS, DIFFUSIONS, AND SUPERDIFFUSIONS

The semilinear equation (u) over dot + L(0)u = u(alpha), where L-0 is a second-order elliptic differential operator without zero-order term and 1 < alpha less than or equal to 2, has been studied by the author in [4] and [5] by using superdiffusions. In the present paper, we appl y superdiffusions to a more general equation (u) over dot + Lu = psi(u ), where Lu = L(0)u + cu (with a bounded coefficient c) and psi belong s to a convex class which contains ku(alpha) with 1 < alpha less than or equal to 2 and positive locally bounded coefficient ic. We also cov er a substantially wider class of functions psi which do not correspon d to any superdiffusion (for instance, ku(alpha) with alpha > 1). Rela ted problems are treated with the help of diffusion processes. This ap proach is useful even in the linear theory. For instance, the first bo undary value problem for equation (u) over dot + Lu = -f can be invest igated for a class of domains described in probabilistic terms which i s substantially larger than the class considered in the literature on PDEs. (C) 1998 Academic Press.