Asymptotic expansions for degenerate parabolic equations

We prove asymptotic convergence results for some analytical expansions of solutions to degenerate PDEs with applications to financial mathematics. In particular, we combine short-time and global-in-space error estimates, previously obtained in the uniformly parabolic case, with some a priori bounds on “short cylinders”, and we achieve short-time asymptotic convergence of the approximate solution in the degenerate parabolic case.     

Asymptotic Behavior of Solutions of Inverse Problems for Degenerate Parabolic Equations

We obtain theorems on the proximity as t → +∞ between the solution of the inverse problem for a second-order degenerate parabolic equation with one spatial variable and the solution of the inverse problem for a second-order degenerate ordinary differential equation under an additional integral observation condition. The conditions imposed on the input data admit oscillations of the functions on the right-hand side in the parabolic equation under study.     

Singular and degenerate anisotropic parabolic equations with a nonlinear source

The Dirichlet problem in arbitrary domain for degenerate and singular anisotropic parabolic equations with a nonlinear source term is considered. We state conditions that guarantee the existence and uniqueness of a global weak solution to the problem. A similar result is proved for the parabolic p-Laplace equation.     

Convergence to equilibrium for a parabolic problem with mixed boundary conditions in one space dimension

We prove that any bounded non-negative solution of a degenerate parabolic problem with Neumann or mixed boundary conditions converges to a stationary solution.     

Positive solutions of the porous medium equation with hysteresis

The problem under consideration is a degenerate parabolic equation with hysteretic terms. We establish existence and uniqueness of solutions with given initial data and investigate their main properties. An example of explicit self-similar solution is presented.     

On a degenerate parabolic equation arising in pricing of Asian options

We study a certain one-dimensional, degenerate parabolic partial differential equation with a boundary condition which arises in pricing of Asian options. Due to degeneracy of the partial differential operator and the non-smooth boundary condition, regularity of the generalized solution of such a problem remained unclear. We prove that the generalized solution of the problem is indeed a classical solution.     

The unique solvability of a certain nonlocal nonlinear problem with a spatial operator strongly monotone with respect to the gradient

We consider a nonlinear degenerate parabolic equation whose spatial operator depends on a nonlocal characteristic of the solution. We prove the uniqueness of the solution in the class of vector-valued functions that take on values in Sobolev spaces.     

Regularization of outflow problems in unsaturated porous media with dry regions

We study a porous medium with saturated, unsaturated, and dry regions, described by Richards' equation for the saturation s and the pressure p. Due to a degenerate permeability coefficient k(x,s) and a degenerate capillary pressure function pc(x,s), the equations may be of elliptic, parabolic, or of ODE-type. We construct a parabolic regularization of the equations and find conditions that guarantee the convergence of the parabolic solutions to a solution of the degenerate system. An example shows that the convergence fails in general. Our approach provides an existence result for the outflow problem in the case of x-dependent coefficients and a method for a numerical approximation.     

Existence for a degenerate Cauchy problem

We prove the existence of a solution to the degenerate parabolic Cauchy problem with a possibly unbounded Radon measure as an initial data. To accomplish this, we establish a priori estimates and derive a compactness result. We also show that the result is optimal in the Euclidian setting.     

Conditions for the Time Global Existence and Nonexistence of a Compact Support of Solutions to the Cauchy Problem for Quasilinear Degenerate Parabolic Equations

We study a quasilinear degenerate parabolic equation with a nonhomogeneous density. We establish that depending on the behavior of the density at infinity either a solution to the Cauchy problem possesses the property that the speed of propagation of perturbations is finite or the support blows up in finite time.