Uniformly analytic solutions to a class of singular partial differential equations

John Paolo O. Soto; Jose Ernie C. Lope; Mark Philip F. Ona; John Paolo O. Soto; Jose Ernie C. Lope; Mark Philip F. Ona

doi:10.3934/math.2022580

AIMS Mathematics

2022, Volume 7, Issue 6: 10400-10421. doi: 10.3934/math.2022580

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Uniformly analytic solutions to a class of singular partial differential equations

Institute of Mathematics, University of the Philippines Diliman, C.P. Garcia Avenue, U.P. Campus, Diliman, Quezon City 1001, Philippines

Received: 10 October 2021 Revised: 21 March 2022 Accepted: 22 March 2022 Published: 28 March 2022
MSC : 35A01, 35A02, 35F20

We study the singular nonlinear partial differential equation $ t\partial_tu = F(t, x, u, \partial_xu) $, where $ (t, x)\in\mathbb{R}\times\mathbb{R}^n $. Under some growth conditions on the coefficients of the partial Taylor expansion of $ F $, we construct the unique solution that is continuous in $ t $ and $ C^\infty $ in $ x $.
- uniformly analytic functions,
- singular partial differential equation
Citation: John Paolo O. Soto, Jose Ernie C. Lope, Mark Philip F. Ona. Uniformly analytic solutions to a class of singular partial differential equations[J]. AIMS Mathematics, 2022, 7(6): 10400-10421. doi: 10.3934/math.2022580

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Abstract

We study the singular nonlinear partial differential equation $ t\partial_tu = F(t, x, u, \partial_xu) $, where $ (t, x)\in\mathbb{R}\times\mathbb{R}^n $. Under some growth conditions on the coefficients of the partial Taylor expansion of $ F $, we construct the unique solution that is continuous in $ t $ and $ C^\infty $ in $ x $.

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