Linear Non-Autonomous Heat Flow in $$L_0^1({{\mathbb {R}}}^{d})$$ and Applications to Elliptic Equations in $${{\mathbb {R}}}^{d}$$

Robinson, James C.Rodríguez Bernal, Aníbal2023-06-222023-06-222022-10-111040-729410.1007/s10884-022-10195-6https://hdl.handle.net/20.500.14352/72681CRUE-CSIC (Acuerdos Transformativos 2022)We study solutions of the equation ut−Δu+λu=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 (u0,λ) for which solutions converge to 0 as t→∞, as not every λ>0 is able to achieve that for all initial data. When f≠0 we give conditions to guarantee that the solution is given by the usual ‘variation of constants formula’ u(t)=e−λtS(t)u0+∫t0e−λ(t−s)S(t−s)f(s)ds, where S(⋅) is the heat semigroup. We use these results to treat the elliptic problem −Δu+λu=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 u0,f≥0.engAtribución 3.0 Españahttps://creativecommons.org/licenses/by/3.0/es/Linear Non-Autonomous Heat Flow in $$L_0^1({{\mathbb {R}}}^{d})$$ and Applications to Elliptic Equations in $${{\mathbb {R}}}^{d}$$journal articlehttps://doi.org/10.1007/s10884-022-10195-6-0open accessHeat equationLarge solutionsBlow-upGlobal solutionsRegularity of elliptic problemMatemáticas (Matemáticas)12 Matemáticas