ON SOLVABILITY OF A TIME-FRACTIONAL SEMILINEAR HEAT EQUATION, AND ITS QUANTITATIVE APPROACH TO THE CLASSICAL COUNTERPART

We are concerned with the following time-fractional semilinear parabolic problem in the N-dimensional whole space R-N with N >= 1, (P)(alpha )(c)partial derivative(alpha)(t)u - Delta u = u(p), t > 0, x is an element of R-N, u(0) = mu in R-N, where (c)partial derivative(alpha)(t) denotes the Caputo derivative of order alpha is an element of (0,1), p > 1, and mu is a nonnegative Radon measure on R-N. The case of alpha = 1 formally gives the Fujita-type equation (P)(1) partial derivative(t)u - Delta u = u(p). In particular, we mainly focus on the Fujita critical case p = pF : =1 + 2/N. It is well known that the Fujita exponent pF separates the ranges of p for the global-in-time solvability of (P)(1). In particular, (P)(1) with p = pF possesses no global-in-time solutions, and is not locally-in-time solvable in its scale critical space L-1(R-N). It is also known that the exponent p(F) plays the same role for the global-in-time solvability for (P)(alpha). However, the problem (P)(alpha) with p = p(F) is globally-in-time solvable, and exhibits local-in-time solvability in its scale critical space L-1(R-N). The purpose of this paper is to clarify the collapse of the global-in-time solvability and the local-in-time solvability of (P)(alpha) as alpha approaches 1-0.