scholarly journals Existence and probabilistic representation of the solutions of semilinear parabolic PDEs with fractional Laplacians

2021 ◽  
Author(s):  
Guillaume Penent ◽  
Nicolas Privault
Keyword(s):  
Probabilistic Representation ◽  
Parabolic Pdes ◽  
Semilinear Parabolic

scholarly journals An Approximation Scheme for Semilinear Parabolic PDEs with Convex and Coercive Hamiltonians

2020 ◽  
Vol 58 (1) ◽  
pp. 165-191
Author(s):  
Shuo Huang ◽  
Gechun Liang ◽  
Thaleia Zariphopoulou
Keyword(s):  
Approximation Scheme ◽  
Parabolic Pdes ◽  
Semilinear Parabolic

Computational Solution of Blow-Up Problems for Semilinear Parabolic PDEs on Unbounded Domains

2010 ◽  
Vol 31 (6) ◽  
pp. 4478-4496 ◽  
Author(s):  
Hermann Brunner ◽  
Xiaonan Wu ◽  
Jiwei Zhang
Keyword(s):  
Blow Up ◽  
Unbounded Domains ◽  
Parabolic Pdes ◽  
Semilinear Parabolic ◽  
Computational Solution

scholarly journals Fluctuations Around a Homogenised Semilinear Random PDE

2020 ◽  
Author(s):  
Martin Hairer ◽  
Étienne Pardoux
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Parabolic Partial Differential Equation ◽  
Parabolic Pdes ◽  
Large Numbers ◽  
Regularity Structures ◽  
Parabolic Pde ◽  
The Right ◽  
Semilinear Parabolic

Abstract We consider a semilinear parabolic partial differential equation in $$\mathbf{R}_+\times [0,1]^d$$ R + × [ 0 , 1 ] d , where $$d=1, 2$$ d = 1 , 2 or 3, with a highly oscillating random potential and either homogeneous Dirichlet or Neumann boundary condition. If the amplitude of the oscillations has the right size compared to its typical spatiotemporal scale, then the solution of our equation converges to the solution of a deterministic homogenised parabolic PDE, which is a form of law of large numbers. Our main interest is in the associated central limit theorem. Namely, we study the limit of a properly rescaled difference between the initial random solution and its LLN limit. In dimension $$d=1$$ d = 1 , that rescaled difference converges as one might expect to a centred Ornstein–Uhlenbeck process. However, in dimension $$d=2$$ d = 2 , the limit is a non-centred Gaussian process, while in dimension $$d=3$$ d = 3 , before taking the CLT limit, we need to subtract at an intermediate scale the solution of a deterministic parabolic PDE, subject (in the case of Neumann boundary condition) to a non-homogeneous Neumann boundary condition. Our proofs make use of the theory of regularity structures, in particular of the very recently developed methodology allowing to treat parabolic PDEs with boundary conditions within that theory.

scholarly journals Flatness of Semilinear Parabolic PDEs—A Generalized Cauchy–Kowalevski Approach

2013 ◽  
Vol 58 (9) ◽  
pp. 2277-2291 ◽  
Author(s):  
Birgit Schorkhuber ◽  
Thomas Meurer ◽  
Ansgar Jungel
Keyword(s):  
Parabolic Pdes ◽  
Semilinear Parabolic

Numerical Blow-up of Semilinear Parabolic PDEs on Unbounded Domains in ℝ2

2011 ◽  
Vol 49 (3) ◽  
pp. 367-382 ◽  
Author(s):  
Jiwei Zhang ◽  
Houde Han ◽  
Hermann Brunner
Keyword(s):  
Blow Up ◽  
Unbounded Domains ◽  
Parabolic Pdes ◽  
Semilinear Parabolic
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