3
$\begingroup$

Suppose we have the differential inequality $$ |\partial_t{u}+\Delta{u}|\leq C(|u|+|\nabla u|) $$ in $\mathbb{R}^n\backslash B_R\times[0,1]$, where $B_R=\{x\in\mathbb{R}^n,|x|\leq R\}$. Then do we have that $$ \||u|+|\nabla u|\|_{L^2(B_{3R}\backslash B_{2R}\times[0,\frac{1}{2}])}\leq C\|u\|_{L^\infty(B_{4R}\backslash B_{R}\times[0,1])} $$ Maybe this is something standard, but I can't find a reference right now. So my question is how to prove such kind of inequalites.

Thanks for any comment and reference.

$\endgroup$
1
  • 3
    $\begingroup$ For the elliptic case such inequalities can be found using integration by parts; for example if $|\Delta u| < C(|u| + |\nabla u|)$ then take the test function $\varphi = \eta^2|u|$ in the inequality $\int_{B_1} \nabla u \cdot \nabla \varphi \leq C\int_{B_1}(|u| + |\nabla u|)\varphi$ and use Cauchy-Schwarz. (This is known as a Caccioppoli inequality). I would imagine the parabolic case is much the same, using parabolic cylinders instead of balls. $\endgroup$ Nov 13, 2013 at 3:38

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.