Volume 24, 2020
|Page(s)||207 - 226|
|Published online||18 March 2020|
A probabilistic approach to quasilinear parabolic PDEs with obstacle and Neumann problems*
Department of Epidemiology and Biostatistics, Xuzhou Medical University,
221004, PR China.
2 School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, PR China.
** Corresponding author: email@example.com
Accepted: 6 November 2019
In this paper, by a probabilistic approach we prove that there exists a unique viscosity solution to obstacle problems of quasilinear parabolic PDEs combined with Neumann boundary conditions and algebra equations. The existence and uniqueness for adapted solutions of fully coupled forward-backward stochastic differential equations with reflections play a crucial role. Compared with existing works, in our result the spatial variable of solutions of PDEs lives in a region without convexity constraints, the second order coefficient of PDEs depends on the gradient of the solution, and the required conditions for the coefficients are weaker.
Mathematics Subject Classification: 60H30 / 60H10 / 35K59
Key words: Quasilinear PDE / viscosity solution / Neumann boundary condition / obstacle problem / forward-backward stochastic differential equation
L. Xiao is supported by the Research Initiation Fundation of Xuzhou Medical University (No. D2018002) and the National Natural Science Foundation of China (Nos. 11601509 and 31801957), S. Fan is supported by the National Fund for Study Abroad (No. 201806425013) and D. Tian is supported by the National Natural Science Foundation of China (No. 11601509).
© EDP Sciences, SMAI 2020
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