Degenerate Quasilinear Parabolic Equations Research Articles

Accurate estimates for the amplitude and support of unbounded solutions of the non-linear heat-conduction equation with a source

Accurate (best possible) upper estimates are obtained for the amplitude and length of support of unbounded solutions of the quasilinear degenerate parabolic equation of non-linear heat conduction with a source. The estimates are established by means of a special “comparison by intersections” technique with an exact non-invariant solution of the equation with the same existence time. This comparison, moreover, illustrates the fact that the Cauchy problem for the equation with a delta-function as the initial datum is solvable.

THE FIRST BOUNDARY VALUE PROBLEM FOR STRONGLY DEGENERATE QUASILINEAR PARABOLIC EQUATIONS

This paper concerns with existence of generalized solutions of first boundary value problem for a kind of strongly degenerate quasilinear parabolic equations. The author assumes that the coefficients in the equation satisfy strongly degenerate monotone condition and other weakened conditions comparing with those of Ladyzenskaja's. An existence theorem is given and proved.

Quasilinear Evolutionary Stochastic Systems with Averaging

Previous article Next article Quasilinear Evolutionary Stochastic Systems with AveragingIl. I. GikhmanIl. I. Gikhmanhttps://doi.org/10.1137/1134050PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. I. Gikhman, On an approach to the problem of optimizing stochastic differential equations, Proceedings of the International Conference on Stochastic Optimization, Kiev, 1984, 63–64, (In Russian.) Google Scholar[2] Yv. N. Blagoveshchenskii, Cauchy's problem for degenerate quasilinear parabolic equations, Theory Probab. Appl., 9 (1964), 342–346 10.1137/1109050 LinkGoogle Scholar[3] Hiroshi Tanaka, Local solutions of stochastic differential equations associated with certain quasilinear parabolic equations, J. Fac. Sci. Univ. Tokyo Sect. I, 14 (1967), 313–326 37:3664 0155.24101 Google Scholar[4] M. Freidlin, Existence in the large of smooth solutions of degenerate quasilinear equations, Math. USSR-Sb., 7 (1969), 323–339 0191.11602 CrossrefGoogle Scholar[5] B. L. Rozovskii, Evolutionary Parabolic Systems, Nauka, Moscow, 1983, (In Russian.) Google Scholar[6] V. V. Baklan, A class of stochastic partial differential equationsBehavior of systems in random media (Russian), Akad. Nauk Ukrain. SSR Inst. Kibernet., Kiev, 1976, 3–7 57:17827 0426.60058 Google Scholar[7] P. A. Meyer, Probability and potentials, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966xiii+266 34:5119 0138.10401 Google Scholar[8] I. I. Gikhman and , A. V. Skorokhod, Theory of Stochastic Processes, Vol. 1, Springer-Verlag, New York, Berlin, Heidelberg, 1974 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 34, Issue 3| 1990Theory of Probability & Its Applications History Submitted:10 February 1986Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1134050Article page range:pp. 416-429ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Splitting in a degenerate quasilinear parabolic equation

Splitting in a degenerate quasilinear parabolic equation

Investigation of difference schemes for quasilinear degenerate parabolic equations

Investigation of difference schemes for quasilinear degenerate parabolic equations

Structure of a weak discontinuity of solutions of quasilinear degenerate parabolic equations

Structure of a weak discontinuity of solutions of quasilinear degenerate parabolic equations

Estimates of the H�lder constants of generalized solutions of degenerate parabolic equations

One establishes interior and boundary estimates for arbitrary (not necessarily sign-constant) bounded generalized solutions of quasilinear degenerate parabolic equations with unbounded coefficients.

Concavity of Solutions of the Porous Medium Equation

Abstract : The flow of a gas through a porous medium is governed by a degenerate quasilinear parabolic equation. It is known that the nonnegative solutions to this equation possess a lower bound for the second derivative of the pressure in the spatial variables. This bound plays an important role in the mathematical treatment and is related to the entropy of the flow. Since the solutions exhibit interfaces across which v sub x jumps positively, no upper bound is possible globally for v subxx. Nevertheless it is proven that the concavity of v(.,t) in the region where v is positive is preserved in time. This is in itself an interesting geometric property of the solution. It also allows one to obtain precise information about the asymptotic behaviour of the flow.

Removable singularities of solutions of degenerate quasilinear equations

In this paper we characterize the sets of weighted capacity zero and the weighted null sets, and state analogous properties for the correspondent unweighted sets. Then we provide removable singularity results for weak solutions of degenerate quasilinear parabolic equations and their elliptic equivalents supposing that the singular set is one of the above types of sets. In obtaining these results the basic idea is to connect the capacity (or the nullity indices) of the singular set to the Lebesgue classes of the weight μ, of 1/μ, and of the xi-derivatives. of the weak solution.

A free boundary problem for degenerate quasilinear parabolic equations

Abstract : This paper is concerned with a free boundary problem which arises in the study of a class of degenerate parabolic equations. A thorough treatment is given for a special case which can be reduced to a problem in ordinary differential equations upon introduction of the appropriate similarity variable. Beyond its inherent interest, the solvability of the resulting problem establishes that an analysis by Vol'pert and Hudjaev of jump conditions satisfied by solutions of degenerate parabolic equations is not correct in general. (Author)

Existence and uniqueness of regular solutions of the first boundary-value problem for second-order quasilinear degenerate parabolic equations

One establishes existence and uniqueness theorems for regular generalized solutions of the first boundary-value problem for quasilinear (A,\(\vec b\))-parabolic equations of the divergence form in the cylinder Q=Ω X (T1, T2), where Ω is a bounded domain in ℝn n ⩾ 1, with a sufficiently smooth boundary.

A priori estimates and Harnack's inequality for generalized solutions of degenerate quasilinear parabolic equations

A priori estimates and Harnack's inequality for generalized solutions of degenerate quasilinear parabolic equations

A note on the similarity solutions for degenerate quasilinear parabolic equations

A note on the similarity solutions for degenerate quasilinear parabolic equations

Similarity solutions for degenerate quasilinear parabolic equations

Similarity solutions for degenerate quasilinear parabolic equations

On the convergence of difference methods for certain degenerate quasi-linear parabolic equations

On the convergence of difference methods for certain degenerate quasi-linear parabolic equations

Asymptotic Behavior of Solutions of the Porous Media Equation

A degenerate quasi-linear parabolic equation which arises in the study of fluid flow through porous media is investigated. It is shown that the solutions of a class of mixed initial boundary value problems in the domain $x > 0$, $t > 0$ converge upon a suitably chosen similarity solution as $t \to \infty $. The rate of convergence is discussed.