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

Question

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!

Answer 1

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 $x=L$.

Answer 2

Specifying the terminal value for the heat equation is difficult: One has backward uniqueness, that is there is at most one solution of the (backward) heat equation. In order to have existence of solutions, the terminal value has to be very smooth, as the heat equation is smoothing. This makes the problem ill-posed.