Consider a second order linear parabolic PDE: \begin{eqnarray} u_t &=& -Lu\\ &=& -\bigg[ p(x)u_{xx} + q(x)u_{x}+r(x)u \bigg] \end{eqnarray} with boundary conditions \begin{eqnarray} u(0,x)=f(x), \\ u(t,0)=C, \\ u(t,\infty)=0 \end{eqnarray} for some constant $C$.

Then, it is well known that there exists an solution $u(t,x)\in C^{1,2}\bigg([0,T]\times[0,\infty]\bigg)$. (for example, black scholes pde)

How to prove the smoothness property of $u$?

Please let me know relevant books or paper. Thank you!

  • $\begingroup$ What conditions do have on $p, q$, and $r$? $\endgroup$
    – mcd
    Commented Nov 6, 2016 at 2:12

1 Answer 1


The general method is using semigroup theory to pass from elliptic operators to the associated parabolic equation. A good reference for the general theory with some applications for bounded domains is the following:

A. Pazy - Semigroups of Linear Operators and Applications to Partial Differential Equations

The book by Renardy & Rogers has a chapter on semigroups and applications to parabolic pde on (spatially) bounded domains.


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