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1. Degenerate linear parabolic equations in divergence form on the upper half space

   https://doi.org/10.1090/tran/8892  

   Dong, Hongjie; Phan, Tuoc; Tran, Hung (June 2023, Transactions of the American Mathematical Society)

   We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems. 

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2. External optimal control of fractional parabolic PDEs

   https://doi.org/10.1051/cocv/2020005  

   Antil, Harbir; Verma, Deepanshu; Warma, Mahamadi (January 2020, ESAIM: Control, Optimisation and Calculus of Variations)

   In [Antil et al. Inverse Probl. 35 (2019) 084003.] we introduced a new notion of optimal control and source identification (inverse) problems where we allow the control/source to be outside the domain where the fractional elliptic PDE is fulfilled. The current work extends this previous work to the parabolic case. Several new mathematical tools have been developed to handle the parabolic problem. We tackle the Dirichlet, Neumann and Robin cases. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the control/source either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control/source in the exterior. We introduce the notions of weak and very-weak solutions to the fractional parabolic Dirichlet problem. We present an approach on how to approximate the fractional parabolic Dirichlet solutions by the fractional parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones. 

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3. Neural Network Method for Integral Fractional Laplace Equations

   https://doi.org/10.4208/eajam.010122.210722  

   Hao, Zhaopeng; Park, Moongyu; Cai, Zhiqiang (January 2023, East Asian Journal on Applied Mathematics)

   A neural network method for fractional order diffusion equations with integral fractional Laplacian is studied. We employ the Ritz formulation for the corresponding fractional equation and then derive an approximate solution of an optimization problem in the function class of neural network sets. Connecting the neural network sets with weighted Sobolev spaces, we prove the convergence and establish error estimates of the neural network method in the energy norm. To verify the theoretical results, we carry out numerical experiments and report their outcome. 

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4. Global Sobolev persistence for the fractional Boussinesq equation with zero diffusivity

   Kukavica, Igor; Wang, Weinan (January 2020, Pure and applied functional analysis)

   We prove the persistence of regularity for the 2D alpha-franctional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e. for (u_0,rho_0)\in W^s,q(R^2)\times W^s,q(R^2), where s>1 and 2 

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5. A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs

   https://doi.org/10.3934/dcdss.2022012  

   Antil, Harbir; Gal, Ciprian G.; Warma, Mahamadi (January 2022, Discrete and Continuous Dynamical Systems - S)

   We consider optimal control of fractional in time (subdiffusive, i.e., for \begin{document}$$ 0<\gamma <1 $$\end{document}) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we \begin{document}$$\mathsf{first\;show}$$\end{document} the existence and regularity of solutions to the forward and the associated \begin{document}$$\mathsf{backward\;(adjoint)}$$\end{document} problems. In the second part, we prove existence of optimal \begin{document}$$\mathsf{controls }$$\end{document} and characterize the associated \begin{document}$$\mathsf{first\;order}$$\end{document} optimality conditions. Several examples involving fractional in time (and some fractional in space diffusion) equations are described in detail. The most challenging obstacle we overcome is the failure of the semigroup property for the semilinear problem in any scaling of (frequency-domain) Hilbert spaces. 

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