Linear Non-Autonomous Heat Flow in L 10 ( R d ) and Applications to Elliptic Equations in R d

We study solutions of the equation u t − (cid:2) u + λ u = f , for initial data that is ‘large at infinity’ as treated in our previous papers on the unforced heat equation. When f = 0 we characterise those ( u 0 , λ) forwhichsolutionsconvergeto0 as t → ∞ ,asnotevery λ > 0 isabletoachieve that for all initial data. When f (cid:4)= 0 we give conditions to guarantee that the solution is given by the usual ‘variation of constants formula’ u ( t ) = e − λ t S ( t ) u 0 + (cid:2) t 0 e − λ( t − s ) S ( t − s ) f ( s ) d s , where S ( · ) istheheatsemigroup.Weusetheseresultstotreattheellipticproblem − (cid:2) u + λ u = f when f is allowed to be ‘large at infinity’, giving conditions under which a solution exists that is given by convolution with the usual Green’s function for the problem. Many of our results are sharp when u 0 , f ≥ 0.