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oai:docta.ucm.es:20.500.14352/726812025-08-01T17:22:09Zcom_20.500.14352_14col_20.500.14352_15

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, Aníbal Heat equation Large solutions Blow-up Global solutions Regularity of elliptic problem Matemáticas (Matemáticas) 12 Matemáticas CRUE-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. Ministerio de Ciencia e Innovación (España) Universidad Complutense de Madrid Depto. de Análisis Matemático y Matemática Aplicada Fac. de Ciencias Matemáticas TRUE pub 2023-06-22T12:29:47Z 2023-06-22T12:29:47Z 2022-10-11 journal article https://hdl.handle.net/20.500.14352/72681 1040-7294 10.1007/s10884-022-10195-6 eng PID2019-103860GB-I00 CEX2019-000904-S GR58/08, UCM (920894) Atribución 3.0 España https://creativecommons.org/licenses/by/3.0/es/ open access application/pdf Springer