A doubly critical semilinear heat equation in the L1 space

We study the existence and nonexistence for a Cauchy problem of the semilinear heat equation: partial differential {partial derivative(t)u = Delta u + vertical bar u vertical bar(p-1) u in R-N x (0, T), u(x, 0) = phi (x) in R-N in L-1 (R-N). Here, N >= 1, p = 1+ 2/N and phi is an element of L-1 (R-N) is a possibly sign-changing initial function. Since N(p - 1)/2 = 1, the L1 space is scale critical and this problem is known as a doubly critical case. It is known that a solution does not necessarily exist for every phi is an element of L-1 (R-N). Let X-q := {phi is an element of L-loc(1), (R-N) vertical bar /integral N-R vertical bar phi vertical bar [log(e + vertical bar phi vertical bar)(q) dx < infinity)}(subset of L-1 (R-N)). In this paper, we construct a local-in-time mild solution in L1 (RN) for phi is an element of X-q if q N/2. We show that, for each 0 <= q < N/2, there is a nonnegative initial function phi is an element of X-q such that the problem has no nonnegative solution, using a necessary condition given by Baras Pierre (Ann Inst Henri Poincare Anal Non Lineaire 2:185-212, 1985). Since X-q subset of X-N/2 for q >= N/2, XN/) becomes a sharp integrability condition. We also prove a uniqueness in a certain set of functions which guarantees the uniqueness of the solution constructed by our method.