I'm currently reading Sipser's Introduction to the Theory of Computation, and I'm reading up about linear bounded automata, now we know from Rice's Theorem that whether a TM can accept all strings in a given alphabet is undecidable, I was wondering if the same could apply to linear bounded automata, if not, how would one go about proving the undecidability of such a problem (or its decidabilty?). Thanks in advance.
For Turing machines another problem is also undecidable: whether it accepts any string at all. This is already undecidable for LBA. Basically the proof is by oberving that we can replace a TM accepting string $w$ by a LBA accepting the `trace of the computation' by the TM on $w$. The verification of the trace of the computation does not need more space than the computation itself, and thus can be verified by a LBA.
The question whether a device accepts all strings is already undecidable on a lower level of the Chomsky hierarchy, the context-free languages (CF grammars,or pushdown automata).