Existence and uniqueness for parabolic equations with Robin BCs

Question

I'm following Evans's book for PDEs, and the existence and uniqueness for parabolic problems is analised for Dirichlet BCs. I'm trying to analise this for Robin BCs, but in the weak formulation a term involving a boundary integral appears. Consequently, it is not posible to apply the Theorem 3 (section 7.1.2 c).

I suppose that I'm not the first one facing this problem. Is there any approach that I should follow?

Thanks in advance.

Answer 1

I believe you will essentially have to reprove 7.1.2 a-c with this new boundary condition. If you let $\frac{\partial u}{\partial n}=\alpha(x,t)u$ on $\partial U\times[0,T]$, then you still get the weak formulation $(u',v) + B[u,v,t] = (f,v)$, but there are 2 big differences.

1. Your space is now $H^1(U)$, not $H_0^1(U)$, since functions can take on any value on $\partial U$ (in the trace sense). This means that you have to be careful using Poincare's Inequality depending on what version you are used to seeing, as some forms of it only apply to function that are 0 on the boundary.

2. Your bilinear form is now $$B[u,v,t] = \int_U\sum_{i=1,j=1}^Na_{ij}(x,t)u_{x_i}v_{x_i}+\sum_{i=1}^Nb_i(x,t)u_{x_i}v+c(x,t)uv\ dx+\int_{\partial U}\alpha(x,t)uv\ dS.$$

If you look at the proofs of 7.1.2 b and c, they require bounds on $B$ that were derived in the section on elliptic equations for Dirichlet BCs (6.2.2). If you can prove similar bounds using this new bilinear form, you should be able to plug them into the proofs for 7.1.2 a-c and things will work out.

I haven't proved this for myself, but my intuition is that if you let $\alpha\in:L^{\infty}(\partial U_T)$, then you can probably bound that surface integral by the $L^2$ norms of the functions and its contribution to the bound will just result in a larger constant $\gamma$, but there is sure to be some subtleties in relating the boundary values of these functions and their norms and will definitely involve the constant from the trace operator somewhere.

I didn't provide a full answer but I hope this helped you understand what steps you can take and what tools to use.

edit: There is also a discussion of a similar question here: Poisson's equation with Robin boundary conditions