A correct boundary-value problem for a third-order degenerate operator equation

Translated from Matematicheskie Zametki, Vol. 46, No. 6, pp. 101–109, December, 1989.     

ON THE CAUCHY PROBLEM OF NONLINEAR DEGENERATE PARABOLIC EQUATION

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Approximate stabilization for a nonlinear parabolic boundary-value problem

For the problem of optimal stabilization of solutions of a nonlinear parabolic boundary-value problem with small parameter in the nonlinear term, we substantiate the form of approximate regulator on the basis of the formula of optimal synthesis of the corresponding linear-quadratic problem.     

Stefan problem for a weakly degenerate parabolic equation

In a domain with free boundary, we consider the inverse problem of determination of the coefficient of the first derivative of the unknown function in a parabolic equation with weak power degeneration. The Stefan condition and the integral condition are used as overdetermination conditions. The conditions for existence and uniqueness of the classical solution of the posed problem are established.     

Nonlocal Neumann problem for a degenerate parabolic equation

In the spaces of classical functions with power weight, we prove the existence and uniqueness of a solution of the nonlocal Neumann problem for nonuniformly parabolic equations without restrictions on the power order of coefficient degeneration. We find an estimate of the solution of this problem in the spaces considered. Chernovtsy State University, Chernovtsy. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1232–1243, September, 1999.     

On the Cauchy problem for a doubly nonlinear degenerate parabolic equation with strongly nonlinear sources

In this article, we consider the existence of local and global solution to the Cauchy problem of a doubly nonlinear equation. By introducing the norms |||_f||| _h and 〈f〉_h, we give the sufficient and necessary conditions on the initial value to the existence of local solution of doubly nonlinear equation. Moreover some results on the global existence and nonexistence of solutions are considered. This work was supported by the National Natural Science Foundation of China (Grant No. 10531020)     

Liouville-type theorems for nonlinear degenerate parabolic equation

We study Liouville-type theorems for degenerate parabolic equation of the form \({u_t-{\rm div}(|\nabla u|^{m-2}\nabla u) = u^p}\) where \({m > 2}\) and \({p > m - 1}\). We prove the optimal Liouville-type results in dimension \({N = 1}\), and for radial solutions in any dimension. We also provide some partial results for non-radial solutions in dimension \({N \geq 2}\). Our proofs are based on a generalized Gidas–Spruck technique, combined with the idea of Serrin and Zou (Acta Math 189(1):79–142, 2002) and of Bidaut-Véron (Équations aux dérivées partielles et applications. Elsevier, Paris, pp 189–198, 1998). Finally, we clarify and correct some of the previous results on this topic.     

The oblique boundary-value problem for a quasilinear parabolic equation

The solvability of the oblique boundary-value problem for quasilinear parabolic nondivergent equations with singularities is studied. Bibliography: 9 titles. To Olga Ladyzhenskaya on the occasion of the jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992. pp. 118–131. Translated by V. I. Ochkur.     

Implicit difference method for solving the first boundary-value problem for a second-order nonlinear parabolic equation

Journal of Mathematical Sciences -     

Symmetrization in a nonlinear degenerate parabolic problem

Summary Bounds for the solution of a nonlinear degenerate parabolic problem are given by means of the solution of a «symmetrized» problem.This work was performed under a national program of Italian M.P.I. (60%, 1985).     

一类含非局部源的非线性退化抛物型方程

The global existence and finite time blow up of the positive solution for a nonlinear degenerate parabolic equation with non-local source are studied.