A probabilistic approach to quasilinear parabolic PDEs with obstacle and Neumann problems

2020 ◽  
Vol 24 ◽  
pp. 207-226
Author(s):  
Lishun Xiao ◽  
Shengjun Fan ◽  
Dejian Tian
Keyword(s):  
Probabilistic Approach ◽  
Neumann Boundary ◽  
Obstacle Problems ◽  
Parabolic Pdes ◽  
Neumann Problems ◽  
Order Coefficient ◽  
And Algebra ◽  
Convexity Constraints ◽  
Fully Coupled ◽  
Quasilinear Parabolic

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.

scholarly journals Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Blow Up Analysis ◽  
Inner Absorption ◽  
Quasilinear Parabolic

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.

scholarly journals Infinitely many solutions for equations of p(x)-Laplace type with the nonlinear Neumann boundary condition

2017 ◽  
Vol 148 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Eun Bee Choi ◽  
Jae-Myoung Kim ◽  
Yun-Ho Kim
Keyword(s):  
Weak Solutions ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Infinitely Many Solutions ◽  
Fountain Theorem ◽  
Neumann Problems ◽  
Neumann Boundary Value Problem ◽  
Cerami Condition ◽  
Image Position ◽  
Laplace Type

We investigate the following nonlinear Neumann boundary-value problem with associated p(x)-Laplace-type operatorwhere the function φ(x, v) is of type |v|p(x)−2v with continuous function p: → (1,∞) and both f : Ω × ℝ → ℝ and g : ∂Ω × ℝ → ℝ satisfy a Carathéodory condition. We first show the existence of infinitely many weak solutions for the Neumann problems using the Fountain theorem with the Cerami condition but without the Ambrosetti and Rabinowitz condition. Next, we give a result on the existence of a sequence of weak solutions for problem (P) converging to 0 in L∞-norm by employing De Giorgi's iteration and the localization method under suitable conditions.

scholarly journals Application of LQR techniques to the adaptive control of quasilinear parabolic PDEs

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 2060013-2060014 ◽  
Author(s):  
Jens Saak ◽  
Peter Benner
Keyword(s):  
Adaptive Control ◽  
Parabolic Pdes ◽  
Quasilinear Parabolic

scholarly journals Dividends Sharing Convertible Bonds Pricing and Numerical Evaluation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xu Guo ◽  
Haiyang Wang
Keyword(s):  
New York ◽  
Convertible Bonds ◽  
Obstacle Problems ◽  
Convertible Bond ◽  
Bond Price ◽  
Parabolic Pdes ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic ◽  
Underlying Stocks ◽  
Discrete Dividends

The convertible bond is becoming one of the most important financial instruments for the company to raise capital fund since it was first issued by American New York Erie Company in 1843. In this paper, it is the first time to study the pricing problem for convertible bond whose underlying stocks pay dividends via the reflected backward stochastic differential equations. Associating the solutions of reflected BSDEs with the obstacle problems for nonlinear parabolic PDEs, we establish the pricing formulas for convertible bonds with continuous and discrete dividends by means of the viscosity solutions for some PDEs. Besides, we also derive the price of convertible bonds with higher borrowing rate which is realistic in the financial market. Then the numerical evaluations are provided by the radial basis functions method. Moreover, we discuss the influence of dividends paying as well as higher borrowing rate on the convertible bond price at last.

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