In this article, it is shown that there exists a unique viscosity solution of the Cauchy problem for a degenerate parabolic equation with non-divergence form.
The uniqueness of solutions for Cauchy problem of the formis studied.It is proved that if u ∈BVx and A(u) is strictly increasing,the solution is unique.
In this paper,we study the surface instability of a cylindrical pore in the absence of stress.This instability is called the Rayleigh-Plateau instabilty.We consider the model developed by Spencer et al.[18],Kirill et ...In this paper,we study the surface instability of a cylindrical pore in the absence of stress.This instability is called the Rayleigh-Plateau instabilty.We consider the model developed by Spencer et al.[18],Kirill et al.[10]and Boutat et al.[2]in the case without stress.We obtain a nonlinear parabolic PDE of order four.We show the local existence and uniqueness of the solution of this problem by using Faedo-Galerkin method.The main results are the global existence of the solution and the convergence to the mean value of the initial data for long time.Numerical tests are also presented in this study.展开更多
This paper deals with the problems on the equation in the following three aspects: i) The existence of a continuous weak solution for the Cauchy problem, Cauchy-Dirichlet problem and the first boundary and initial val...This paper deals with the problems on the equation in the following three aspects: i) The existence of a continuous weak solution for the Cauchy problem, Cauchy-Dirichlet problem and the first boundary and initial values problem, ii) the uniqneness of the the solutions of the above mentioned three problems, iii) the properties of the solutions. The study on this equation, which has many backgrounds such as the filtrations in porous media as water in soil, is itself important in mathematics as well as in practice.展开更多
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 whic...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.展开更多
This article studies the Cauchy problem for a class of doubly nonlinear degenerate parabolic equations au/at = div（A（｜△↓B（u）｜）△↓B（u））. Under certain conditions, the author considers its regularized probl...This article studies the Cauchy problem for a class of doubly nonlinear degenerate parabolic equations au/at = div（A（｜△↓B（u）｜）△↓B（u））. Under certain conditions, the author considers its regularized problem and establishes some estimates. On the basis of the estimates, the existence and uniqueness of the generalized solutions in BV space are proved.展开更多
1 Introduction We want to investigate the following boundary value problem:where m】1, n】1, p≥1 and m】p. As for a, the following properties will be assumed: (A.1) a(r)∈C^1 ([0, ∞)) and a(r)】0 for any r∈(0, ∞);...1 Introduction We want to investigate the following boundary value problem:where m】1, n】1, p≥1 and m】p. As for a, the following properties will be assumed: (A.1) a(r)∈C^1 ([0, ∞)) and a(r)】0 for any r∈(0, ∞); (A.2) there exists α】0 such that (r-α)a(r)≥0 for any r∈[0, ∞). The initial value problem corresponding to (1.1) is the following Cauchy problem:展开更多
Let Γ be a portion of a C^(1,α) boundary of an n-dimensional domain D. Letu be a solution to a second order parabolic equation in D x (-T, T) and assume that u = 0 on Γ x(-T, T), 0 ∈ Γ. We prove that u satisfies ...Let Γ be a portion of a C^(1,α) boundary of an n-dimensional domain D. Letu be a solution to a second order parabolic equation in D x (-T, T) and assume that u = 0 on Γ x(-T, T), 0 ∈ Γ. We prove that u satisfies a three cylinder inequality near Γ x (—T, T). As aconsequence of the previous result we prove that if u(x,t) = O (|x|~k ) for every t ∈ (-T,T) andevery k ∈ N, then u is identically equal to zero.展开更多
基金Supported in part by Dalian Nationalities University (20076209)Departmentof Education of Liaoning Province (2009A152)National Natural Science Foundation of China (10471156,10901030)
文摘In this article, it is shown that there exists a unique viscosity solution of the Cauchy problem for a degenerate parabolic equation with non-divergence form.
基金Project supported by the National Natural Science Foundation of China.
文摘The uniqueness of solutions for Cauchy problem of the formis studied.It is proved that if u ∈BVx and A(u) is strictly increasing,the solution is unique.
基金Supported by LMCM created by Professor Mohamed Boulanouar and PLB-K Program
文摘In this paper,we study the surface instability of a cylindrical pore in the absence of stress.This instability is called the Rayleigh-Plateau instabilty.We consider the model developed by Spencer et al.[18],Kirill et al.[10]and Boutat et al.[2]in the case without stress.We obtain a nonlinear parabolic PDE of order four.We show the local existence and uniqueness of the solution of this problem by using Faedo-Galerkin method.The main results are the global existence of the solution and the convergence to the mean value of the initial data for long time.Numerical tests are also presented in this study.
文摘This paper deals with the problems on the equation in the following three aspects: i) The existence of a continuous weak solution for the Cauchy problem, Cauchy-Dirichlet problem and the first boundary and initial values problem, ii) the uniqneness of the the solutions of the above mentioned three problems, iii) the properties of the solutions. The study on this equation, which has many backgrounds such as the filtrations in porous media as water in soil, is itself important in mathematics as well as in practice.
基金supported by the National Natural Science Foundation of China(Nos.11061018,11261029)the Youth Foundation of Lanzhou Jiaotong University(No.2011028)+1 种基金the Long Yuan Young Creative Talents Support Program(No.252003)the Joint Funds of the Gansu Provincial Natural Science Foundation of China(No.1212RJZA043)
文摘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.
文摘This article studies the Cauchy problem for a class of doubly nonlinear degenerate parabolic equations au/at = div（A（｜△↓B（u）｜）△↓B（u））. Under certain conditions, the author considers its regularized problem and establishes some estimates. On the basis of the estimates, the existence and uniqueness of the generalized solutions in BV space are proved.
基金Project supported by the National Natural Science Foundation of China.
文摘1 Introduction We want to investigate the following boundary value problem:where m】1, n】1, p≥1 and m】p. As for a, the following properties will be assumed: (A.1) a(r)∈C^1 ([0, ∞)) and a(r)】0 for any r∈(0, ∞); (A.2) there exists α】0 such that (r-α)a(r)≥0 for any r∈[0, ∞). The initial value problem corresponding to (1.1) is the following Cauchy problem:
基金This work is partially supported by MURST,Grant No.MM01111258
文摘Let Γ be a portion of a C^(1,α) boundary of an n-dimensional domain D. Letu be a solution to a second order parabolic equation in D x (-T, T) and assume that u = 0 on Γ x(-T, T), 0 ∈ Γ. We prove that u satisfies a three cylinder inequality near Γ x (—T, T). As aconsequence of the previous result we prove that if u(x,t) = O (|x|~k ) for every t ∈ (-T,T) andevery k ∈ N, then u is identically equal to zero.
