Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 118, pp. 1-12.

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

Authors: Minh-Phuong (Tran Ton Duc Thang Univ., Ho Chi Minh city, Vietnam)
         Thanh-Nhan Nguyen (Ho Chi Minh City Univ. of Education, Vietnam(

Abstract:
 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.