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Robinson, James C. and Rodríguez Bernal, Aníbal
(2022)
*Linear Non-Autonomous Heat Flow in $$L_0^1({{\mathbb {R}}}^{d})$$ and Applications to Elliptic Equations in $${{\mathbb {R}}}^{d}$$.*
Journal of Dynamics and Differential Equations
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ISSN 1040-7294

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Official URL: https://doi.org/10.1007/s10884-022-10195-6

## 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: | Article |
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Additional Information: | CRUE-CSIC (Acuerdos Transformativos 2022) |

Uncontrolled Keywords: | Heat equation, Large solutions, Blow-up, Global solutions, Regularity of elliptic problem |

Subjects: | Sciences > Mathematics |

ID Code: | 75652 |

Deposited On: | 18 Nov 2022 12:52 |

Last Modified: | 18 Nov 2022 13:17 |

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