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Journal of Convex Analysis 28 (2021), No. 2, 689--710
Copyright Heldermann Verlag 2021

Uniqueness and Estimates for a Parabolic Equation with L1 Data

Maria Michaela Porzio
Dip. di Matematica, Sapienza UniversitÓ, 00185 Roma, Italy

We investigate on the uniqueness property of the solutions to a parabolic problem with the data and the coefficient of zero order term only summable functions. Despite all this lack of regularity and without using any notion of entropy or renormalized solutions or the property to be a solution obtained by approximations we prove an uniqueness result. Then, we study the behavior in time of the unique solution. Finally, we estimate the distance between the unique solution and the solutions of other parabolic or stationary problems showing cases when the same asymptotic behavior in time appears.

Keywords: Uniqueness, asymptotic behavior, regularity of solutions, linear parabolic equations.

MSC: 35A02, 35B45, 35B40.

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