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John Meyer, David Needham^{*}
Research output: Contribution to journal › Article › peer-review
In this paper, we consider a L^{∞} functional derivative estimate for the first spatial derivatives of bounded classical solutions u:R^{N}×[0,T]→R to the Cauchy problem for scalar second order semi-linear parabolic partial differential equations with a continuous nonlinearity f:R→R and initial data u_{0}:R^{N}→R, of the form, maxi=1,…,N(supx∈R^{N}|u_{xi }(x,t)|)≤F_{t}(f,u_{0},u)∀t∈[0,T]. Here F_{t}:A_{t}→R is a functional as defined in §1 and x=(x_{1},x_{2},…,x_{n})∈R^{N}. We establish that the functional derivative estimate is non-trivially sharp, by constructing a sequence (f_{n},0,u^{(n)}), where for each n∈N, u^{(n)}:R^{N}×[0,T]→R is a solution to the Cauchy problem with zero initial data and nonlinearity f_{n}:R→R, and for which there exists α>0 such that maxi=1,…,N(supx∈R^{N}|u_{xi } ^{(n)}(x,T)|)≥α, whilst limn→∞(inft∈[0,T](maxi=1,…,N(supx∈R^{N}|u_{xi } ^{(n)}(x,t)|)−F_{t}(f_{n},0,u^{(n)})))=0.
Original language | English |
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Pages (from-to) | 3345-3362 |
Journal | Journal of Differential Equations |
Volume | 265 |
Issue number | 8 |
Early online date | 14 Jun 2018 |
DOIs | |
Publication status | Published - 15 Oct 2018 |
Research output: Contribution to journal › Article › peer-review
Research output: Book/Report › Book