Degenerate Initial Boundary Value Problem Research Articles

Mathematical Study of Medicine Propagation in Biological Tissue and Some of Its Applications

The paper deals with the problem of the distribution of the medicine (enzyme) in the damaged biological tissue where the reaction enzyme—substrat takes place. The biological problem is reduced to a singular degenerate initial-boundary value problem for two coupled ordinary differential equations. Analytical solution of the singular degenerated IBV-problem was obtained by power series. The solution demonstrates the real situation and found suitable to depict the degeneration of singular system, caused by low concentration of the enzyme.

On the existence of solutions to a one-dimensional degenerate nonlinear wave equation

This paper is concerned with the degenerate initial–boundary value problem to the one-dimensional nonlinear wave equation utt=((1+u)aux)x which arises in a number of various physical contexts. The global existence of smooth solutions to the degenerate problem was established under relaxed conditions on the initial–boundary data by the characteristic decomposition method. Moreover, we show that the solution is uniformly C1,α continuous up to the degenerate boundary and the degenerate curve is C1,α continuous for α∈(0,min⁡{a1+a,11+a}).

Existence of smooth solutions to a one-dimensional nonlinear degenerate variational wave equation

This paper is focused on a one-dimensional nonlinear variational wave equation which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. We establish the global existence of smooth solutions to its degenerate initial–boundary value problem under relaxed conditions on the initial–boundary data. Moreover, we show that the solution is uniformly C1,α continuous up to the degenerate boundary and the degenerate curve is C1,α continuous for α∈(0,12).

Global solvability of one degenerate semilinear differential operator equation

We consider the Cauchy problem for the abstract semilinear differential equation \(\frac{d}{{dt}}\left( {Au\left( t \right)} \right) + Bu\left( t \right) = f\left( {t,u\left( t \right)} \right),t_0 - T < t < t_0 + T\) where A and B are linear closed, generally speaking, degenerate operators acting from a Banach space X into a Banach space Y and f(t, u) is a continuously differentiable function. We assume that the resolvent (A + μB)−1 has a pole of at most second order at the point μ = 0. Global conditions for the existence and uniqueness of a solution of the Cauchy problem are obtained. The results are applied to a nonlinear degenerate initial boundary-value problem with partial derivatives and to a system of differential-algebraic equations of a nonlinear electric circuit.

Distributed Capacitance Mictostructure in Conductors

A new model for distributed capacicance in a conduction medium is introduced as a system of local RC diffusion equations coupled by a global elliptic equation. This model contains the local geometry of the distributed capacitors on which charge is stored and the exchange of current flux on their interface with the medium. The resulting degenerate initial—boundary—value problem in shown to be well posed and certain singular limits are characterized.