2019 Elsevier Masson SAS We investigate pointwise upper bounds for nonnegative solutions u(x,t) of the nonlinear initial value problem 0( t ) uu in R n R,n1, u=0in R n (,0) where and are positive constants. To do this we first give a definitiontailored for our study of (0.1), (0.2)of fractional powers of the heat operator ( t ) :YX where X and Y are linear spaces whose elements are real valued functions on R n R and 0<< 0 for some 0 which depends on n, X and Y. We then obtain, when they exist, optimal pointwise upper bounds on R n (0,) for nonnegative solutions uY of the initial value problem (0.1), (0.2) with particular emphasis on those bounds as t0 + and as t.
Journal de Mathmatiques Pures et Appliques
- Pointwise Bounds
- Fractional Heat Operator