Robinson, J. C. and Rodriguez-Bernal, A. (2022) Linear non-autonomous heat flow in L^1_0(R^d) and applications to elliptic equations in R^d. Journal of Dynamics and Differential Equations . doi:10.1007/s10884-022-10195-6 ISSN 1040-7294. (In Press)

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Abstract

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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Heat equation, Differential equations, Elliptic -- Numerical solutions
Journal or Publication Title:	Journal of Dynamics and Differential Equations
Publisher:	Springer
ISSN:	1040-7294
Official Date:	11 October 2022
Dates:	Date Event 11 October 2022 Available 18 July 2022 Accepted
DOI:	10.1007/s10884-022-10195-6
Status:	Peer Reviewed
Publication Status:	In Press
Access rights to Published version:	Open Access (Creative Commons)
Date of first compliant deposit:	20 July 2022
Date of first compliant Open Access:	12 October 2022
RIOXX Funder/Project Grant:	Project/Grant ID RIOXX Funder Name Funder ID CEX2019-000904-S Instituto de Ciencias Matemáticas UNSPECIFIED UNSPECIFIED Ministerio de Ciencia e Innovación UNSPECIFIED UNSPECIFIED Ministerio de Ciencia e Innovación UNSPECIFIED
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