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 $$ \|u(t)\|_{L^r(\Omega)} \leq c \frac{\|u(0)\|^{\gamma_0}_{L^{r_0}(\Omega)}}{t^{\gamma_1}}, \qquad t > 0, $$ where $r$ and $r_0$ are exponents that appear in these integral estimates and $\gamma_1$ and $\gamma_0$ are positive constants that can be expressed in terms of these constants. We will see that in some cases $r > r_0$ (supercontractive estimates) and hence a regularizing effect on $u$ appears while in other cases this improvement of regularity does not appear since $r < r_0$; in any case, we prove that the $L^r(\Omega)$-norm of u decays in time (for t large) since it verifies the previous estimate. We show how to apply this result to obtain new estimates of this kind for the solutions of many nonlinear parabolic equations. This new method allows also to explain the reason of the similar behavior of the solutions of very different parabolic problems. Finally, we study sufficient conditions for extinction in finite time.