A New Critical Exponent for the Heat and Damped Wave Equations with Nonlinear Memory and Not Integrable Data

In this paper, we discuss the influence of assuming Lm regularity of initial data, instead of L1, on a heat or damped wave equation with nonlinear memory. We find that the interplay between the loss of decay rate due to the presence of the nonlinear memory and to the assumption of initial data in Lm instead of L1, leads to a new critical exponent for the problem, whose shape is quite different from the one of the critical exponent for Lm theory for the corresponding problem with power nonlinearity |u|p. We prove the optimality of the critical exponent using the test function method.