2021/04/30 by Yasuhito Miyamoto, Miyamoto, Yasuhito, Masamitsu Suzuki +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #advanced mathematical theories #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2104.14773
Let N≥ 1 and let f∈ C[0,∞) be a nonnegative nondecreasing function and u0 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 equation \begincases ∂tu=Δu+f(u) amp; \textrmin ℝN×(0,T),
u(x,0)=u0(x) amp; \textrmin ℝN \endcases under mild assumptions on f. A relationship between a growth of f and an integrability of u0 is studied in detail. Our existence theorem gives a sharp integrability condition on u0 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)=u1+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 Lr\rm ul(ℝN). Main technical tools are a monotone iterative method, Lp-Lq estimates, Jensen's inequality and differential inequalities.