Regularity of solutions to a linear degenerate parabolic pde - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T15:38:13Z http://mathoverflow.net/feeds/question/82710 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82710/regularity-of-solutions-to-a-linear-degenerate-parabolic-pde Regularity of solutions to a linear degenerate parabolic pde Thomas Richard 2011-12-05T15:56:18Z 2011-12-09T16:57:40Z <p>I've encountered the following problem which is causing me some trouble :</p> <p>Let $(M^2,g)$ be a smooth compact Riemannian surface (with constant curvature for instance) and consider a smooth function $u:M\times [0,1) \to \mathbb{R}$ which is a solution of the following PDE : $\partial_t u(x,t)=A(x,t)\Delta u(x,t)$ where $A$ is a function which is smooth on $M\times [0,1)$ and satisfies : $C_1|1-t|^\alpha \leq A(x,t)\leq C_2$.</p> <p>It is not hard to see that $u$ is uniformly bounded with the maximum principle, a simple computation also shows that $\int_M |\nabla u(x,t)|^2dv_g$ decreases with $t$, which gives $L^p$ convergence of $u(.,t)$ as $t$ goes to $1$.</p> <p>My question is : does $u$ have a continuous extension to $M\times[0,1]$ ?</p> <p>Any insight or reference on this kind of problem is welcome.</p> <p>Thanks.</p> <p>EDIT :</p> <p>Some further observations :</p> <ul> <li><p>For the application I have in mind, I would be happy with just a sequence $t_i$ going to $1$ such that $u(.,t_i)$ uniformly converges.</p></li> <li><p>The statement in the previous item is true if we replace $M$ by an interval, thanks the Rellich-Kondrachov compactness theorem, in fact, with $M$ of dimension 2,we are exactly at the critical exponent given by Rellich-Kondrachov Theorem.</p></li> <li><p>The statement is true if $A$ depends only on $t$ (not on $x$), just put $t'=\int_0^tA(\tau)d\tau$ and you get that $u(x,t')$ satisfies the usual heat equation.</p></li> </ul> <p>Thanks again.</p>