On uniform and decay estimates for unbounded solutions of partial differential equations

We prove that if a function u satisfies certain integral estimates (of energy type) then it verifies estimates of the type‖u(t)‖Lr(Ω)≤c‖u(0)‖Lr0(Ω)γ0tγ1,t>0, where r   and r0r0 are exponents that appear in these integral estimates and γ1γ1 and γ0γ0 are positive constants that can be expressed in terms of these constants. We will see that in some cases r>r0r>r0 (supercontractive estimates) and hence a regularizing effect on u   appears while in other cases this improvement of regularity does not appear since r