Electron. J. Differential Equations, Vol. 2019 (2019), No. 118, pp. 1-12.

An application of global gradient estimates in Lorentz-Morrey spaces for the existence of stationary solutions to degenerate diffusive Hamilton-Jacobi equations

Minh-Phuong Tran, Thanh-Nhan Nguyen

In mathematics and physics, the Kardar-Parisi-Zhang equation or quasilinear stationary version of a time-dependent viscous Hamilton-Jacobi equation in growing interface and universality classes is also known as the quasilinear Riccati type equation. The existence of solutions to this type of equations still remains an interesting open problem. In previous studies [36,38], we obtained global bounds and gradient estimates for quasilinear elliptic equations with measure data. The main goal of this article is to obtain the existence of a renormalized solution to the quasilinear stationary solution for the degenerate diffusive Hamilton-Jacobi equation with finite measure data in Lorentz-Morrey spaces.

Submitted May 22, 2019. Published November 11, 2019.
Math Subject Classifications: 35K55, 35K67, 35K65.
Key Words: Degenerate diffusive Hamilton-Jacobi equation; stationary solution; quasilinear Riccati type equation; Lorentz-Morrey space; uniformly thickness.

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Minh-Phuong Tran
Applied Analysis Research Group
Faculty of Mathematics and Statistics
Ton Duc Thang University
Ho Chi Minh city, Vietnam
email: tranminhphuong@tdtu.edu.vn
Thanh-Nhan Nguyen
Department of Mathematics
Ho Chi Minh City University of Education
Ho Chi Minh city, Vietnam
email: nguyenthnhan@hcmup.edu.vn

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