The following is from Evans 7.4.2, second-order parabolic PDE.
We assume $L $ has the divergence form, which means $ Lu= -\sum _{i,j=1} ^n (a^{i,j}(x,t)u_{x_i})_{x_j} + \sum_{i=1} ^n b^i(x,t)u_{x_i} +c(x,t)u $ and has smooth coefficients.
Suppose $U$ is a bounded open set with smooth boundary.
Set $ X=L^2 (U), D(A) = H_0^1(U) \cap H^2(U) $ and define $Au= - Lu $, $u \in D(A)$.
The book writes $A$ is an unbouned linear operator on $X$. But I don't know how to check this.
How can I get the unboundness of $A$? Please help me.
This is just a matter of definition. We do not view $A$ as an operator from $D(A)\to X$, but when we write "an unbounded operator on $X$'', we interpret $A$ as an operator from $X\to X$ that is only bounded on a subset of $X$, that set being $D(A)$. So now the only thing you have to check is whether or not $A$ is defined on the whole space $X$, which it clearly is not.