Problem For Degenerate Parabolic Equation Research Articles

$L^1$ Contraction for Bounded (Nonintegrable) Solutions of Degenerate Parabolic Equations

We obtain new $L^1$ contraction results for bounded entropy solutions of Cauchy problems for degenerate parabolic equations. The equations we consider have possibly strongly degenerate local or nonlocal diffusion terms. As opposed to previous results, our results apply without any integrability assumption on the solutions. They take the form of partial Duhamel formulas and can be seen as quantitative extensions of finite speed of propagation local $L^1$ contraction results for scalar conservation laws. A key ingredient in the proofs is a new and nontrivial construction of a subsolution of a fully nonlinear (dual) equation. Consequences of our results are maximum and comparison principles, new a priori estimates, and, in the nonlocal case, new existence and uniqueness results.

Identification problems for degenerate parabolic equations

This paper deals with multivalued identification problems for parabolic equations. The problem consists of recovering a source term from the knowledge of an additional observation of the solution by exploiting some accessible measurements. Semigroup approach and perturbation theory for linear operators are used to treat the solvability in the strong sense of the problem. As an important application we derive the corresponding existence, uniqueness, and continuous dependence results for different degenerate identification problems. Applications to identification problems for the Stokes system, Poisson-heat equation, and Maxwell system are given to illustrate the theory.

ЗАДАЧА ОПТИМАЛЬНОГО СТАРТОВОГО КЕРУВАННЯ ВИРОДЖЕНИМ ПАРАБОЛІЧНИМ РІВНЯННЯМ

An optimal control problem for degenerate parabolic equation with mixed boundary conditions are considered. Having applied the Hardy - Poincare inequality, it is shown that this problem has a unique optimal solution in the correspondence weighted Sobolev space. The necessary optimality conditions are derived and substantiated.

On the Regularity of Solutions to FENE Models

In the present paper, we focus on the microscopic description in finite extensible nonlinear elasticity (FENE) systems for polymeric fluids. Based on study of an initial boundary value problem for degenerate parabolic equations we prove the existence of global smooth solutions in some weighted spaces to the microscopic FENE model if the nondimensional parameter $b>2$. The regularity obtained here is intrinsic and might be the best one.

Uniqueness and existence for anisotropic degenerate parabolic equations with boundary conditions on a bounded rectangle

We study the comparison principle for anisotropic degenerate parabolic–hyperbolic equations with initial and nonhomogeneous boundary conditions. We prove a comparison theorem for any entropy sub- and super-solution, which immediately deduces the L 1 contractivity and therefore, uniqueness of entropy solutions. The method used here is based upon the kinetic formulation and the kinetic techniques developed by Lions, Perthame and Tadmor. By adapting and modifying those methods to the case of Dirichlet boundary problems for degenerate parabolic equations we can establish a comparison property. Moreover, in the quasi-isotropic case the existence of entropy solutions is proved.

Local and global estimates of the solutions of the Cauchy problem for quasilinear parabolic equations with a nonlinear operator of Baouendi-Grushin type

In this paper, the qualitative properties of the solutions of the Cauchy problem for degenerate parabolic equations with a nonlinear operator of Baouendi-Grushin type are studied. Sharp local and global (with respect to the spatial and temporal variables) estimates of the solution are obtained. The property of the finiteness of the support of the solution is established.

Inverse problem for the strongly degenerate heat equation in a domain with free boundary

In a domain with free boundary, we establish conditions for the existence and uniqueness of a solution of the inverse problem of finding the time-dependent coefficient of heat conductivity. We study the case of strong degeneration where the unknown coefficient tends to zero as t → +0 as a power function t β , where β ≥ 1. Problems for degenerate parabolic equations, free-boundary problems, and inverse problems arise in the course of investigation of many processes important from the practical point of view. Each of these types of problems is fairly well investigated, but their combination within a single problem, in fact, has not been studied. The inverse problem for the one-dimensional heat equation in a domain with free boundary was investigated in [1]. Conditions for the existence and uniqueness of a solution of the inverse problem for a parabolic equation with strong power degeneration were established in [2]. The inverse problems for hyperbolic and elliptic equations with degeneration were studied in [3] and [4], respectively. The free-boundary problem for a degenerate parabolic system was investigated in [5]. In the present paper, we study the one-dimensional inverse problem for the heat equation with strong power degeneration in the case where a part of the boundary is unknown. An analogous problem for the case of weak degeneration was considered in [6].

- estimates of solutions of the Cauchy problem for an anisotropic degenerate parabolic equation with double non-linearity and growing initial data

The Cauchy problem for a degenerate parabolic equation with anisotropic -Laplacian and double non-linearity is considered. For increasing initial data local estimates for the -norm of a solution are obtained, which yield a precise characterization of the growth of solutions at infinity. An estimate for the order of the length of the time interval on which a solution is defined is found in its dependence on the initial data. Bibliography: 12 titles.

Theorems on the existence and nonexistence of solutions of the Cauchy problem for degenerate parabolic equations with nonlocal source

We consider the Cauchy problem for a doubly nonlinear degenerate parabolic equation with nonlocal source under the assumption that the initial function is integrable. We establish the existence and nonexistence of time-global solutions of the problem.

Analytic regularity of solutions to the Cauchy problem for degenerate parabolic equations

Analytic regularity of solutions to the Cauchy problem for degenerate parabolic equations

Symmetry in a free boundary problem for degenerate parabolic equations on unbounded domains

We use the method of Alexandroff-Serrin to establish the spherical symmetry of the ground domain and of the weak solution to a free boundary problem for a class of quasi-linear parabolic equations in an unbounded cylinder $\Omega \times (0,T)$, where $\Omega = (\mathbb {R}^{n} \backslash \overline {\Omega _{1}})$, with $\Omega _{1}\subset \mathbb R^n$ a simply connected bounded domain. The equations considered are of the type $u_{t} - div (a(u,|Du|)Du) = c(u,|Du|)$, with $a$ modeled on $|Du|^{p-2}$. We consider a solution satisfying the boundary conditions: $u(x,t)=f(t)$ for $(x,t)\in \partial \Omega _{1} \times (O,T)$, and $u(x,0)=0$, $u\rightarrow 0$ as $|x|\rightarrow \infty$. We show that the overdetermined co-normal condition $a(u,|Du|)\frac {\partial u}{\partial \nu }=g(t)$ for $(x,t)\in \partial \Omega _{1} \times (O,T)$, with $g(\overline T) > 0$ for at least one value $\overline T \in (0,T)$, forces the spherical symmetry of the ground domain and of the solution.

Cauchy problem for nonlinear parabolic equations

The Cauchy problem for degenerate parabolic equations with bounded mea- surable coecients is studied. The existence and uniqueness of initial trace for nonneg- ative solutions

Estimate of the maximum of modulus of generalized solutions of the first boundary-value problem for degenerate parabolic equations

We study parabolic equations of the divergent form with degeneration. We have obtained an estimate for the maximum of modulus of generalized solutions of the first boundary-value problem with a zero on the parabolic boundary.

On Time Periodic Solutions of the Dirichlet Problem for Degenerate Parabolic Equations of Nondivergence Type

In this paper, we are concerned with the existence of periodic solutions of a quasilinear parabolic equation[formula]with the Dirichlet boundary condition, where Ω is a smoothly bounded domain inRNandfis a given function periodic in time defined on[formula]. Our results depend on the first eigenvalue λ1of −Δ in Ω with the Dirichlet boundary condition. If λ1>1, then there exists a unique positive periodic solution for a positivef(γ∈R). In the case of λ1<1, we construct a nonnegative periodic solution for a negativef(1≤γ<3).

A free boundary problem for quasilinear degenerate parabolic equations with general degeneracy

In this paper, we study the free boundary problem for degenerate parabolic equations (1.1)–(1.4). The existence of generalized solutions inBV1, 1/2 is obtained by the means of parabolic regularization under certain restrictions. The uniqueness and regularity of generalized solutions are also discussed. In addition, a C1+α smoothness for the free boundary is obtained in the parabolic case.

Numerical simulation for degenerate diffusions

Numerical treatment of certain diffusion problems via indirect (analytical) methods requires solving boundary-value problems for degenerate parabolic equations or systems. Degeneracy may arise from coordinate singularity (polar coordinates) or other more intrinsic reasons. Examples of degenerate diffusions can be found in wave propagation in random media, genetics and mathematical theory of evolution. Methods based on suitable finite difference schemes either suggested by the structure of the problems (symmetry, formally obtained limiting-equations, etc.) or by the conservation of certain quantities, as it happens for their continuous analogue, are introduced to compute the solutions with accuracy and stability.

Smoothness of solutions of the Cauchy problem for degenerate parabolic equations

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The Cauchy Problem for Degenerate Parabolic Equations with Discontinuous Drift

The coefficient of the gradient is allowed to be discontinuous but is assumed to satisfy a “one-sided” Lipschitz condition. This condition insures the pathwise uniqueness of the underlying Markov process which in turn yields the existence of a unique stable generalized solution of the parabolic equation. If the data is Lipschitz continuous, then so is the solution.

Uniqueness for the Cauchy problem for degenerate parabolic equations

Uniqueness for the Cauchy problem for degenerate parabolic equations

Cauchy problem for degenerate parabolic equations

Cauchy problem for degenerate parabolic equations