Partial Differential Equations
PDEs arising in a majority of real-world applications are nonlinear. Despite its complexity, in the theory of nonlinear PDEs, one can observe key equations or systems which are essential for the development of the theory for a particular class of equations. One of those key equations is the nonlinear degenerate/singular parabolic equation, the so-called nonlinear diffusion equation. In a series of publications a general theory of the nonlinear degenerate and/or singular parabolic equations in general non-smooth domains under the minimal regularity assumptions on the boundary was developed. This development was motivated with numerous applications, including flow in porous media, heat conduction in a plasma, free boundary problems with singularities, etc. Related publications are listed below.
Well-posedness of the Dirichlet Problem for the Nonlinear Diffusion Equation in Non-smooth Domains, Transactions of Amer. Math. Soc., 357, 1, 2005, 247-265.
Reaction-diffusion in nonsmooth and closed domains, Boundary Value Problems, Special issue: Harnack estimates, Positivity and Local Behavior of Degenerate and Singular Parabolic Equations, Vol. 2007 (2007).
On the Dirichlet problem for the nonlinear diffusion equation in non-smooth domains, J. Math. Anal. Appl., 246, 2, 2001, 384-403.
Reaction-diffusion in Irregular Domains, J. Differential Equations, 164, 2000, 321-354.
Reaction-diffusion in a closed domain formed by irregular curves, J. Math. Anal. Appl., 246, 2000, 480-492.
Interface Development and Local Solutions to Reaction-Diffusion Equations, SIAM J. Math. Anal., 32, 2, 2000, 235-260.
Evolution of interfaces and explicit asymptotics at infinity for the fast diffusion equation with absorption, Nonlinear Analysis, 50, 4, 2002, 541-560.
Local structure of solutions of the Dirichlet problem of N-dimensional reaction-diffusion equations in bounded domains, Advance in Differential Equations, Volume 4, Number 2, 1999, 197-224.
Evolution of interfaces for the nonlinear parabolic p-Laplacian type reaction-diffusion equations, European Journal of Applied Mathematics, Volume 28, Issue 5, 2017, 827-853.
Evolution of Interfaces for the Nonlinear Double Degenerate Parabolic Equation of Turbulent Filtration with Absorption, Mathematics and Computers in Simulation, 153 (2018), 59-82.
Evolution of interfaces for the non-linear parabolic p -Laplacian type reaction-diffusion equations. II. Fast diffusion vs. strong absorption. European Journal of Applied Mathematics, March 2019
Interface development for the nonlinear degenerate multidimensional reaction-diffusion equations, Nonlinear Differential Equations and Applications NoDEA, February 2020, 27:3.
Instantaneous shrinking of the support of a solution of a nonlinear degenerate parabolic equation. (Russian) Mat. Zametki 63(1998), no.3,323-331; translation in Math. Notes 63(1998), no.3-4,285-292.
On sharp local estimates for the support of solutions in problems for nonlinear parabolic equations. (Russian) Mat. Sbornik 186(1995), no.8,3-24; translation in Sb. Math. 186(1995), no.8,1085-1106.
One of the oldest problems in the theory of PDEs is the problem of finding geometric conditions on the boundary manifold for the regularity of the solution of the elliptic and parabolic PDEs. There is a deep connection between this problem and the delicate problem of asymptotics of the corresponding Wiener processes. In the paper
First Boundary Value Problem for the Diffusion Equation. I. Iterated Logarithm Test for the Boundary Regularity and Solvability, SIAM J. Math. Anal., 34, 6, 2003, 1422-1434.
a geometric iterated logarithm test for the boundary regularity of the solution to the heat equation was proved. In addition, an exterior hyperbolic paraboloid condition for the boundary regularity, which is the parabolic analogy of the exterior cone condition for the Laplace equation is established. In fact, for the characteristic top boundary point of the symmetric rotational boundary surfaces, the necessary and sufficient condition for the regularity coincides with the well-known Kolmogorov-Petrovsky test for the local asymptotics of the multi-dimensional Brownian motion trajectories:
Multidimensional Kolmogorov-Petrovsky test for the boundary regularity and irregularity of solutions to the heat equation, Boundary Value Problems, 2, 2005, 181-199.
In the case when symmetric rotational boundary surfaces extend to t=-∞, the regularity of the point at ∞ precisely characterize the uniqueness of the bounded solutions. Geometric necessary and sufficient condition for the regularity of ∞ was proved. In the probabilistic context the result coincides with the Kolmogorov-Petrovsky test for the asymptotics of the multi-dimensional Brownian motion trajectories at infinity. Related publications are listed below.
Necessary and sufficient condition for the existence of a unique solution to the first boundary value problem for the diffusion equation in unbounded domains, Nonlinear Analysis, 64, 5(2006), 1012-1017.
Kolmogorov problem for the heat equation and its probabilistic counterpart, Nonlinear Analysis, 63, 5-7, 2005, 712-724.