A variational approach to self-similar solutions for semilinear heat equations

Abstract

Such self-similar solutions are global in time and often used to describe the large time behavior of global solutions to (1), see, e.g., [14, 15, 5, 21]. If w(x, t) is a self-similar solution of (1.1) and has an initial value A(x), then we easily see that A has the form A(x) = A(x/|x|)|x|−2/(p−1). Then the problem of existence of self-similar solutions is essentially depend on the solvablity of the Cauchy problem (1)-(2)λ. In this talk we consider the existence of self-similar solutions of the problem (1)-(2)λ. The idea of constructing self-similar solutions by solving the initial value problem for homogeneous initial data goes back to the study by Giga and Miyakawa [12] for the Navier-Stokes equation in vorticity form. It is well known by Fujita [9] that if 1 (N + 2)/N is necessary for the existence of positive self-similar solutions of (1).