Thresholds on growth of nonlinearities and singularity of initial functions for semilinear heat equations

Abstract

Let N ≥ 1 and let f ∈ C[0, ∞) be a nonnegative nondecreasing function and u 0 be a possibly singular nonnegative initial function. We are concerned with existence and nonexistence of a local in time nonnegative solution in a uniformly local Lebesgue space of a semilinear heat equationunder mild assumptions on f . A relationship between a growth of f and an integrability of u 0 is studied in detail. Our existence theorem gives a sharp integrability condition on u 0 in a critical and subcritical cases, and it can be applied to a regularly or rapidly varying function f . In a doubly critical case existence and nonexistence of a nonnegative solution can be determined by special treatment. When f (u) = u 1+2/N [log(u + e)] β , a complete classification of existence and nonexistence of a nonnegative solution is obtained. We also show that the same characterization as in Laister et. al.[11] is still valid in the closure of the space of bounded uniformly continuous functions in the space L r ul (R N ). Main technical tools are a monotone iterative method, L p -L q estimates, Jensen's inequality and differential inequalities.