# Second order PDE: what are the restrictions on boundary conditions?

For a second order PDE with, say one derivative in time and two in space, can I define two boundary conditions in time and one is space? Is that a well-defined problem?

For example, consider the heat equation

$$\frac{\partial c(x,t)}{\partial t} = \frac{\partial^2 c(x,t)}{\partial x^2}$$ on e.g. $(x,t) \in [0,L]\times[0,T]$.

A standard set of boundary conditions are $$\{c(x,0)=c_0(x), \quad c(0,t)=a, \quad c(L,t)=b \}$$ or $$\{c(x,0)=c_0(x), \quad \partial_x c(0,t)=a, \quad \partial_x c(L,t)=b \}$$

or combinations thereof.

My questions is this: Is the problem still well-defined by these conditions:

$$\{c(0,t)=c_0(t), \quad c(x,0)=a, \quad c(x,T)=b \}$$ (or with derivatives)?

I'm not able to solve that system, so my guess is no. But I would like a certain answer. Thank you in advance!

No, in general not. Multiply the heat equation by the solution and integrate in space: \begin{align} \int_0^L c c_t \text{d}x &= \int_0^L c c_{xx} \text{d}x \\ \Leftrightarrow \frac{1}{2} \| c \|_t^2 &= cc_x \big|_0^L - \int_0^L c_x^2 \text{d}x \\ \Leftrightarrow \| c \|_t^2 + 2\| c_x \|^2 &= 2cc_x \big|_0^L. \end{align} In order for the problem to be well-posed (have a unique solution bounded by the data of the problem) the left hand side has to be bounded. This cannot be guaranteed unless you impose boundary conditions both at $x=0$ and $x=L$.