We study a class of parabolic equations having first-order terms with superlinear (and subquadratic) growth. The model problem is the so-called viscous Hamilton–Jacobi equation with superlinear Hamiltonian. We address the problem of having unbounded initial data and we develop a local theory yielding well-posedness for initial data in the optimal Lebesgue space, depending on the superlinear growth. Then we prove regularizing effects, short and long time decay estimates of the solutions. Compared to previous works, the main novelty is that our results apply to nonlinear operators with just measurable and bounded coefficients, since we totally avoid the use of gradient estimates of higher order. By contrast we only rely on elementary arguments using equi-integrability, contraction principles and truncation methods for weak solutions.

Magliocca, M., Porretta, A. (2019). Local and global time decay for parabolic equations with super linear first-order terms. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 118(3), 473-512 [10.1112/plms.12187].

Local and global time decay for parabolic equations with super linear first-order terms

MAGLIOCCA, MARTINA;Porretta A.
2019-01-01

Abstract

We study a class of parabolic equations having first-order terms with superlinear (and subquadratic) growth. The model problem is the so-called viscous Hamilton–Jacobi equation with superlinear Hamiltonian. We address the problem of having unbounded initial data and we develop a local theory yielding well-posedness for initial data in the optimal Lebesgue space, depending on the superlinear growth. Then we prove regularizing effects, short and long time decay estimates of the solutions. Compared to previous works, the main novelty is that our results apply to nonlinear operators with just measurable and bounded coefficients, since we totally avoid the use of gradient estimates of higher order. By contrast we only rely on elementary arguments using equi-integrability, contraction principles and truncation methods for weak solutions.
2019
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Magliocca, M., Porretta, A. (2019). Local and global time decay for parabolic equations with super linear first-order terms. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 118(3), 473-512 [10.1112/plms.12187].
Magliocca, M; Porretta, A
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/215296
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