Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space

Md. Rabiul Haque; Takayoshi Ogawa; Ryuichi Sato

doi:10.3934/cpaa.2020031

Article Contents

2020, Volume 19, Issue 2: 677-697. Doi: 10.3934/cpaa.2020031

This issue Previous Article Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates Next Article (1+2)-dimensional Black-Scholes equations with mixed boundary conditions

Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space

1.
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
2.
Mathematical Institute/Research Alliance Center of Mathematical Science, Tohoku University, Sendai 980-8578, Japan

^* Corresponding author
^* Corresponding author

* On leaving from Department of Mathematics, University of Rajshahi-6205, Bangladesh

Received: February 28, 2018

Revised: April 30, 2019

Published: October 2019

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Abstract

We consider the local existence and the uniqueness of a weak solution of the initial boundary value problem to a convection–diffusion equation in a uniformly local function space $ L^r_{{\rm uloc}, \rho}( \Omega) $, where the solution is not decaying at $ |x|\to \infty $. We show that the local existence and the uniqueness of a solution for the initial data in uniformly local $ L^r $ spaces and identify the Fujita-Weissler critical exponent for the local well-posedness found by Escobedo-Zuazua [10] is also valid for the uniformly local function class.

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Mathematics Subject Classification: Primary: 35A01, 35K55, 35D30.

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References

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