I am looking for a treatment of fourth order parabolic equations in Holder spaces. More precisely fourth order analogues of Theorems 5.1, 5.2, and 10.1 in Chapter IV of Linear and quasilinear equations of parabolic type. These are simply the parabolic Schauder estimates. To be more concrete I am only really concerned with IBVPs, roughly, of the following form:

$$\begin{align} \partial_t u + \partial_x^4 u &= f \mbox{ for } (t,x) \in (0,T) \times (0,1), \\ u(0,x) &= u_0(x) \mbox{ for } x \in (0,1) \\ u(t,0) &= \partial_x u(t,0) = 0, \mbox{ and } \\ u(t,1) &= \partial_x u(t,1) = 0 \mbox{ for } t \in (0,T). \end{align}$$

I have done some searching already but couldn't find anything close to what I needed. Any help would be much appreciated.

  • $\begingroup$ On a formal level, you will need a bit more boundary conditions... $\endgroup$ Oct 7 '12 at 14:46

I do not have access to the book you cite, but Section 3.2 in Lunardi's book contains a broad overview on higher order parabolic problems with lots of references, including the most important Hölder estimates.


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