On nonnegative solutions of elliptic and parabolic equations

A theorem on the nonexistence of a nonnegative nontrivial generalized solution inR ⁿ is proved for general quasilinear second-order degenerate elliptic equations. Analogous results are obtained for a large class of systems of partial differential equations, second-order parabolic and inverse parabolic equations, which are nonlinear and may be degenerate.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 114–136, 1992.     

On the solvability of the inverse problem for determining the right-hand side of a degenerate parabolic equation with integral observation

Existence and uniqueness theorems and theorems on stability under perturbations of the input data for solutions of the inverse problem for a degenerate parabolic equation in the plane with integral observation are obtained. The cases of bounded and unbounded coefficients are studied. Estimates of the solution with constants explicitly written out in terms of the input data of the problem are obtained.     

On the stability and the attraction domain of the stationary solution of a singularly perturbed parabolic equation with a multiple root of the degenerate equation

We construct and justify the asymptotics of a boundary layer solution of a boundary value problem for a singularly perturbed second-order ordinary differential equation for the case in which the degenerate (finite) equation has an identically double root. A specific feature of the asymptotics is the presence of a three-zone boundary layer. The solution of the boundary value problem is a stationary solution of the corresponding parabolic equation. We prove the asymptotic stability of this solution and find its attraction domain.     

Viscosity solutions to a Cauchy problem of a phase-field model for solid-solid phase transitions

We investigate a partial differential equation which models solid-solid phase transitions. This model is for martensitic phase transitions driven by configurational force and its counterpart is for interface motion by mean curvature. Mathematically, this equation is a second-order nonlinear degenerate parabolic equation. And in multidimensional case, its principal part cannot be written into divergence form . We prove the existence and uniqueness of viscosity solution to a Cauchy problem for this model.     

On a linear inverse problem for a second-order parabolic equation

The question is studied of existence and uniqueness of a solution to the inverse problem of finding the right-hand side of a special kind for a second-order parabolic equation.     

On a singularly perturbed system of two second-order equations in the case of intersecting roots of the degenerate equation

For a singularly perturbed system of two second-order differential equations (one rapid and one slow), we prove the existence of a solution and obtain its asymptotics for the case in which the degenerate equation has two intersecting roots. In addition, we show that this solution is an asymptotically stable stationary solution of the corresponding parabolic problem and find its local attraction domain.     

On long-time asymptotics of solutions of parabolic equations with increasing leading coefficients

Sharp sufficient conditions on the coefficients of a second-order parabolic equation are examined under which the solution of the corresponding Cauchy problem with a power-law growing initial function stabilizes to zero. An example is presented showing that the found sufficient conditions are sharp. Conditions on the coefficients of a parabolic equation are obtained under which the solution of the Cauchy problem with a bounded initial function stabilizes to zero at a power law rate.     

Neumann problem with the integro-differential operator in the boundary condition

The Neumann problem for a second-order parabolic equation with integro-differential operator in the boundary condition is considered. A well-posedness theorem is proved, in particular, the integral representation of the solution is obtained, estimates for the derivatives of the solution are established, and the kernel of the inverse operator of the problem is explicitly expressed.     

Degenerate first-order inverse problems in Banach spaces

We are concerned with an inverse problem for a degenerate linear evolution equation of first-order. Both hyperbolic and parabolic cases will be considered. We indicate sufficient conditions for the existence and uniqueness of a solution. All the results can be applied to inverse problems for equations from mathematical physics. As a possible application of the abstract theorems, some examples of partial differential equations are given.     

Asymptotic integration of linear parabolic problems with high-frequency coefficients in the critical case

For a linear second-order parabolic equation with high-frequency coefficients, the complete asymptotics of time-periodic solutions is constructed and justified. The limit averaged stationary problem is assumed degenerate.     

An inverse backward problem for degenerate parabolic equations

This work studies the inverse problem of reconstructing an initial value function in the degenerate parabolic equation using the final measurement data. Problems of this type have important applications in the field of financial engineering. Being different from other inverse backward parabolic problems, the mathematical model in our article may be allowed to degenerate at some part of boundaries, which may lead to the corresponding boundary conditions missing. The conditional stability of the solution is obtained using the logarithmic convexity method. A finite difference scheme is constructed to solve the direct problem and the corresponding stability and convergence are proved. The Landweber iteration algorithm is applied to the inverse problem and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown initial value is recovered very well.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1900–1923, 2017     

On the stabilization to zero of the solutions of the inverse problem for a degenerate parabolic equation with two independent variables

Mathematical Notes - Theorems on the stabilization to zero as t → +∞ of solutions of the inverse problem of determining the unknown right-hand side of a degenerate parabolic equation...     

Uniqueness of the bounded solution to a strongly degenerate parabolic problem

This paper concerns the uniqueness of the bounded solution to a strongly degenerate parabolic problem. The equation considered may have two kinds of strong degeneracies and there is no restriction on the relation between the two degeneracies. By using Holmgren’s approach, we prove that the bounded solution of the associated initial–boundary value problem is unique under some essentially necessary condition on the growth of the convection.     

Determination of source terms in a degenerate parabolic equation from a locally distributed observation

The aim of this Note is to prove a Lipschitz stability and uniqueness result for an inverse source problem relative to a one-dimensional degenerate parabolic equation. We use the method introduced by Imanuvilov and Yamamoto in 1998, with the help of some recent Carleman estimate for degenerate equations obtained by Cannarsa, Martinez and Vancostenoble.     

An Inverse Problem of Identifying the Radiative Coefficient in a Degenerate Parabolic Equation

The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.     

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.     

An inverse problem of identifying the coefficient of first-order in a degenerate parabolic equation

This work studies an inverse problem of determining the first-order coefficient of degenerate parabolic equations using the measurement data specified at a fixed internal point. Being different from other ordinary parameter identification problems in parabolic equations, in our mathematical model there exists degeneracy on the lateral boundaries of the domain, which may cause the corresponding boundary conditions to go missing. By the contraction mapping principle, the uniqueness of the solution for the inverse problem is proved. A numerical algorithm on the basis of the predictor-corrector method is designed to obtain the numerical solution and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown function is recovered very well. The results obtained in the paper are interesting and useful, and can be extended to other more general inverse coefficient problems of degenerate PDEs.     

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.     

Linear parabolic problem: High-frequency asymptotics in the critical case

A second-order linear parabolic problem with high-frequency terms is considered. The elliptic operator of the corresponding limiting (averaged) problem is assumed to be degenerate. A complete formal asymptotic expansion of a time-periodic solution of the perturbed problem is constructed.     

Quasilinear parabolic and elliptic equations with nonlinear boundary conditions

This paper is concerned with a class of quasilinear parabolic and elliptic equations in a bounded domain with both Dirichlet and nonlinear Neumann boundary conditions. The equation under consideration may be degenerate or singular depending on the property of the diffusion coefficient. The consideration of the class of equations is motivated by some heat-transfer problems where the heat capacity and thermal conductivity are both temperature dependent. The aim of the paper is to show the existence and uniqueness of a global time-dependent solution of the parabolic problem, existence of maximal and minimal steady-state solutions of the elliptic problem, including conditions for the uniqueness of a solution, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solutions. Applications are given to some heat-transfer problems and an extended logistic reaction–diffusion equation.