In this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum u is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of u, immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.
Regularity and time behavior of the solutions to weak monotone parabolic equations / Porzio, M. M.. - In: JOURNAL OF EVOLUTION EQUATIONS. - ISSN 1424-3199. - (2021). [10.1007/s00028-021-00709-y]
Regularity and time behavior of the solutions to weak monotone parabolic equations
Porzio M. M.
2021
Abstract
In this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum u is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of u, immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.File | Dimensione | Formato | |
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