Initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations

We consider an initial mixed-boundary value problem for anisotropic fractional type degenerate parabolic equations posed in bounded domains. Namely, we consider that the boundary of the domain splits into two parts. In one of them, we impose a Dirichlet boundary condition and in the another one a Neumann condition. Under this mixed-boundary condition, we show the existence of solutions for measurable and bounded non-negative initial data. The nonlocal anisotropic diffusion effect relies on an inverse of a s−fractional type elliptic operator, and the solvability is proved for any s ∈ (0, 1).

[1]  C. Imbert.  Finite speed of propagation for a non-local porous medium equation , 2014, 1411.4752.

[2]  Gerd Grubb,et al.  Regularity of spectral fractional Dirichlet and Neumann problems , 2014, 1412.3744.

[3]  E. Mainini,et al.  A Gradient Flow Approach to the Porous Medium Equation with Fractional Pressure , 2016, 1606.06787.

[4]  J. Lions,et al.  Problèmes aux limites non homogènes et applications , 1968 .

[5]  M. Chipot,et al.  On some class of problems with nonlocal source and boundary flux , 2001, Advances in Differential Equations.

[6]  Herbert Amann,et al.  Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems , 1993 .

[7]  J. Vázquez,et al.  Regularity of solutions of the fractional porous medium flow with exponent 1/2 , 2014 .

[8]  J. Vázquez,et al.  A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators , 2014 .

[9]  Evgeniy Panov,et al.  Strong Traces for Conservation Laws with General Nonautonomous Flux , 2018, SIAM J. Math. Anal..

[10]  J. Málek.  Weak and Measure-valued Solutions to Evolutionary PDEs , 1996 .

[11]  J. Vázquez,et al.  Nonlinear Porous Medium Flow with Fractional Potential Pressure , 2010, 1001.0410.

[12]  Mircea Sofonea,et al.  Elements of Nonlinear Analysis , 2013 .

[13]  J. V'azquez,et al.  Transformations of self-similar solutions for porous medium equations of fractional type , 2014, 1402.6877.

[14]  Piotr Biler,et al.  The Nonlocal Porous Medium Equation: Barenblatt Profiles and Other Weak Solutions , 2013, 1302.7219.

[15]  P. R. Stinga,et al.  Fractional elliptic equations, Caccioppoli estimates and regularity , 2014, 1409.7721.

[16]  Yannick Sire,et al.  Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains , 2014, 1404.6195.

[17]  W. Neves,et al.  Initial-boundary value problem for a fractional type degenerate heat equation , 2018 .

[18]  P. Alam.  ‘G’ , 2021, Composites Engineering: An A–Z Guide.

[19]  J. Vázquez,et al.  Regularity of solutions of the fractional porous medium flow , 2012, 1409.8190.

[20]  J. V'azquez,et al.  Porous medium equation with nonlocal pressure , 2018, 1801.04244.

[21]  J. Lions,et al.  Lectures on elliptic partial differential equations , 1957 .

[22]  E. Davies,et al.  Heat kernels and spectral theory , 1989 .