Green's function and infinite-time bubbling in the critical nonlinear heat equation

Abstract

Let Omega be a smooth bounded domain in R-n, n >= 5. We consider the classical semilinear heat equation at the critical Sobolev exponent. ut =Delta u + un+2/n-2 in Omega x (0, infinity), u = 0 on partial derivative Omega x (0, infinity). Let G (x, y) be the Dirichlet Green function of 1 in similar to Delta in Omega and H(x, y) its regular part. Let q(j) is an element of Omega, j = 1,,,,, k, be points such that the matrix

[h(q1, q2) -G(q1, q2) ... -G(q1, q2) -G(q1, q2) H(q1, q2) -G(q1, q2) -G(q1, q2)

-G(q1, qk) ... -G(q(k-1), qk) J(qk, qk)

is positive definite. For any k >= 1 such points indeed exist. We prove the existence of a positive smooth solution u.x; t/ which blows up by bubbling in infinite time near those points. More precisely, for large time t, u takes the approximate form.

u(x, t) approximate to Sigma(alpha n)(j=1)(mu(j)(t)/mu(j)(t)(2) + vertical bar x -xi(j) (t)vertical bar(2))((n-2)/2.).

Here xi(j).(t) -> q(j) and 0 < mu j(t) -> 0 as t -> infinity. We find that mu(j) (t)/ similar to t(-1/(n-4)) as t -> infinity when, n >= 5.