Solutions Of Quasilinear Parabolic Equations Research Articles

Global solutions for quasilinear parabolic problems

Results on the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Dirichlet or Neumann boundary conditions are presented. Besides quasilinear parabolic equations, the method is also applicable to some weakly-coupled reaction-diffusion systems and to elliptic equations with nonlinear dynamic boundary conditions.

The motion of a particle on a light viscoelastic bar: asymptotic analysis of its quasilinear parabolic–hyperbolic equation

We study the large longitudinal motion of a nonlinearly viscoelastic bar with one end fixed and the other end attached to a heavy particle. This problem is a precise continuum-mechanical analog of the basic discrete mechanical problem of the motion of a particle on a (massless) spring. This motion is governed by an initial-boundary-value problem for a class of third-order quasilinear parabolic–hyperbolic partial differential equations subject to a nonstandard boundary condition, which is the equation of motion of the particle. The ratio of the mass of the bar to that of the tip mass is taken to be a small parameter ε. We prove that this problem has a unique globally defined solution that admits a valid asymptotic expansion, including an initial-layer expansion, in powers of ε for ε near 0. The validity of the expansion gives a precise meaning to the solution of the reduced problem, obtained by setting ε=0, which curiously is seldom governed by the expected ordinary differential equation. The fundamental constitutive hypothesis that the tension be a uniformly monotone function of the strain rate plays a critical role in a delicate proof that each term of the initial-layer expansion decays exponentially in time. These results depend on new decay estimates for the solution of quasilinear parabolic equations.

Global and blowup solutions of quasilinear parabolic equation with critical Sobolev exponent and lower energy initial value

In this paper, by means ofthe energy method, we first study the existence and asymptotic estimates of global solution of quasilinear parabolic equations involving p-Laplacian (p> 2)and critical Sobolevexponentand lowerenergy initialvalue inaboundeddomain in RV(N> 3),andalsostudythesufficientconditionsoffinitetimeblowupoflocalsolutionby the classical concave method. Finally, we study the asymptotic behavior of any global solutions u(x, t; Uo)whichmaypossesshighenergy initialvalue function Uo(X).Wecanprove that there exists a time subsequence {t,} such that the asymptotic behavior of u(x, t,,; Uo) as t, o is similar to the Palais-Smale sequence of stationary equation of the above parabolic problem.

Space-Time Periodic Solutions and Long-Time Behavior of Solutions to Quasi-linear Parabolic Equations

This paper considers (i) the existence of space-time periodic solutions of quasi-linear parabolic equations and (ii) the convergence, as $t\to\infty$, of space periodic solutions of the initial value problem of quasi-linear parabolic equations to the space-time periodic solutions.

Regularity of a Boundary Point for Degenerate Parabolic Equations with Measurable Coefficients

We investigate the continuity of solutions of quasilinear parabolic equations in the neighborhood of the nonsmooth boundary of a cylindrical domain. As a special case, one can consider the equation \(\frac{{\partial u}}{{\partial t}} - \Delta _p u = 0\) with the p-Laplace operator Δp. We prove a sufficient condition for the regularity of a boundary point in terms of Cp-capacity.

Dead cores and instantaneous compactification of the supports of energy solutions of quasilinear parabolic equations of arbitrary order

A general sufficient condition for the formation of “dead cores” of generalized solutions of a wide class of quasilinear parabolic equations of non-linear diffusion-absorption type is obtained. On that basis a sufficient and close to necessary condition for the instantaneous compactification of the support of an arbitrary local energy solution of the corresponding Cauchy problem is derived, which is expressed in terms of the behaviour at infinity of some integral norm (with respect to balls of fixed radius) of the initial function. A precise upper bound for the compactification radius is obtained.

The hölder continuity of generalized solutions of a class quasilinear parabolic equations

Let A and B satisfy the structural conditions (2), the local Holder continuity interior toQ=G×(0,T) is proved for the generalized solutions of quasilinear parabolic equations as follows:u t -divA(x, t, u, ∇u)+B(x, t, u, ∇u)=0

Global Sobolev Solutions of Quasilinear Parabolic Equations

Global Sobolev Solutions of Quasilinear Parabolic Equations

Existence of extremal periodic solutions for quasilinear parabolic equations

In this paper we consider a quasilinear parabolic equation in a bounded domain under periodic Dirichlet boundary conditions. Our main goal is to prove the existence of extremal solutions among all solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, comparison and truncation techniques and suitably constructed special testfunction.

The support shrinking properties for solutions of quasilinear parabolic equations with strong absorption terms

On etudie par methode d'energie certaines proprietes de support des solutions faibles d'equations paraboliques du second ordre avec absorption. Ces solutions faibles ne sont pas necessairement de signe constant et des proprietes de non-propagation de perturbations de la donnee initiale, retrecissement conique de support et formation de dead-core sont etablies sous hypotheses sur la non linearite et conditions locales sur les donnees.

Transformation of approximate solutions of linear parabolic equations into asymptotic solutions of quasilinear parabolic equations

Transformation of approximate solutions of linear parabolic equations into asymptotic solutions of quasilinear parabolic equations

Pointwise differentiability properties of solutions of quasilinear parabolic equations

Pointwise differentiability properties of solutions of quasilinear parabolic equations

$C_0$-Semigroups in $L^p ({\bf R}^d )$ and $\widehat{C}({\bf R}^d )$ Spaces Generated by the Differential Expression $\Delta + b \cdot \nabla$

Previous article Next article $C_0$-Semigroups in $L^p ({\bf R}^d )$ and $\widehat{C}({\bf R}^d )$ Spaces Generated by the Differential Expression $\Delta + b \cdot \nabla$V. F. Kovalenko and Yu. A. SemenovV. F. Kovalenko and Yu. A. Semenovhttps://doi.org/10.1137/1135064PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] D. G. Aronson and , James Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal., 25 (1967), 81–122 39:5952 0154.12001 CrossrefGoogle Scholar[2] V. F. Kovalenko, , M. A. Perelmuter and , Yu. A. Semenov, Schrödinger operators with $L\sp{1/2}\sb{W}({\bf R}\sp{l})$-potentials, J. Math. Phys., 22 (1981), 1033–1044 10.1063/1.525009 83b:35121 0463.47027 CrossrefGoogle Scholar[3] Kôsaku Yosida, Functional analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965xi+458, Berlin 31:5054 0126.11504 CrossrefGoogle Scholar[4] M. Reed and , B. Simon, Methods of Modern Mathematical Physics, Vol. 2, Academic Press, New York, 1972 Google Scholar[5] R. S. Phillips, Semi-groups of positive contraction operators, Czechoslovak Math. J., 12 (87) (1962), 294–313 26:4195 0113.09901 Google Scholar[6] P. E. Sobolevskii, Equations of parabolic type in a Banach space, Trudy Moskov. Mat. Obšč., 10 (1961), 297–350, (In Russian.) 25:5297 Google Scholar[7] O. A. Ladyzenskaja, , V. A. Solonnikov and , N. N. Uraltseva, Linear and Quasi Nonlinear Equations, Amer. Math. Soc. Transl., New York, 1968 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Regularity of solutions to Kolmogorov equation with Gilbarg–Serrin matrixJournal of Evolution Equations, Vol. 22, No. 1 | 8 March 2022 Cross Ref Markov semi-groups generated by elliptic operators with divergence-free driftJournal of Mathematical Analysis and Applications, Vol. 499, No. 1 | 1 Jul 2021 Cross Ref On the Lp-theory of C0-semigroups associated with second-order elliptic operators with complex singular coefficientsProceedings of the London Mathematical Society, Vol. 115, No. 4 | 24 July 2017 Cross Ref Progress in the Problem of the L p -Contractivity of Semigroups for Partial Differential OperatorsAround the Research of Vladimir Maz'ya III | 7 November 2009 Cross Ref Criteria for the L p -dissipativity of Partial Differential OperatorsAnalysis, Partial Differential Equations and Applications | 1 Jan 2009 Cross Ref Local estimates on two linear parabolic equations with singular coefficientsPacific Journal of Mathematics, Vol. 223, No. 2 | 1 Feb 2006 Cross Ref Criterion for the Lp -dissipativity of second order differential operators with complex coefficientsJournal de Mathématiques Pures et Appliquées, Vol. 84, No. 8 | 1 Aug 2005 Cross Ref On the L-Theory of C0-Semigroups Associated with Second-Order Elliptic Operators, IJournal of Functional Analysis, Vol. 193, No. 1 | 1 Aug 2002 Cross Ref On the L-Theory of C0-Semigroups Associated with Second-Order Elliptic Operators, IIJournal of Functional Analysis, Vol. 193, No. 1 | 1 Aug 2002 Cross Ref Volume 35, Issue 3| 1991Theory of Probability & Its Applications411-623 History Submitted:10 February 1988Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1135064Article page range:pp. 443-453ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Local Existence And Uniqueness of Solutions of Degenerate Parabolic Equations

We consider a class of degenerate parabolic equaitons on a bounded domain with mixed boundary conditions. These problems arise, for example, in the study of flow through porous media. Under appropriate hypotheses, we establish the existence of a nonegative solution which is obtainable as a monotone limit of solutions of quasilinear parabolic equations. This construction is used establish uniqueness, cinparison, and L1 continuous dependence theorems, as well some results on blow up of solutions in finite time

On large deviations of solutions of nonlinear stochastic equations

This paper proves the existence and uniqueness of a strong solution of quasilinear parabolic partial differential equations with white noise. It is proved that the solutions continuously depend on the trajectories of the Wiener process. The main result is exponential estimates for the probabilities of large deviations of the solutions of quasilinear parabolic equations with white noise. These probabilities are estimated via the action functional. Estimates of two types are established, a lower bound on the probability that the solution is in a neighborhood of a fixed trajectory and an upper bound on the probability of large deviation of the solution from the set of trajectories with bounded action. These results generalize the estimates established by Venttsel' and Freidlin for ordinary differential equations to the case of parabolic partial differential equations.

ON THE METHOD OF STATIONARY STATES FOR QUASILINEAR PARABOLIC EQUATIONS

A method is presented for investigating the space-time structure of unbounded nonnegative solutions of quasilinear parabolic equations of the form , where is a nonlinear elliptic operator. Three examples are considered in detail: the Cauchy problem for the equation where 0$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img4.gif/> and 1$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img5.gif/> are constants; the boundary value problem in 0\,\}$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img6.gif/> 0,\quad x\in\Omega;$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img7.gif/> 0,\quad x_3=0;\qquad \alpha=\textrm{const}>0;$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img8.gif/> and the Cauchy problem for the system It is assumed that at the point the solution grows without bound as . The derivation of an estimate of the solution near , is based on an analysis of an appropriate family of stationary solutions : , , 0$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img17.gif/> a parameter. It is shown that the behavior of a solution as depends to large extent on the structure of the envelope 0}U_\lambda(x)$ SRC=http://ej.iop.org/images/0025-5734/67/2/A08/tex_sm_2091_img19.gif/>. In particular, if , then grows without bound as at points arbitrarily far from . If for , then determines a lower bound for in a neighborhood of , .Bibliography: 28 titles.

Convergence to a stationary state and stability for solutions of quasilinear parabolic equations

In this paper some aspects of the asymptotic behavior of solutions of quasilinear (generally nonautonomous) parabolic equations are considered. Specifically a result of convergence to a stationary state is given and, under more restrictive conditions, some sharper descriptions of converging solutions are obtained. Finally a saddle point situation is examined. The employed techniques are abstract and inspired by the papers of Sobolevskii and Da Prato-Grisvard.

A method of investigating unbounded solutions of quasilinear parabolic equations

The quasilinear parabolic equation of heat conduction with a source is considered, in regard to unbounded solutions existing in a finite time interval. A method is proposed for investigating the effect of localization of solutions. This method is applicable for arbitrary values of the energy release parameter.

Global solutions of abstract quasilinear parabolic equations

Local and global existence and uniqueness for strict solutions of abstract quasilinear parabolic equations are studied. Applications to quasilinear parabolic partial differential equations are also given.

Relations between Travelling Wave Solutions of Quasilinear Parabolic Equations

Relations between Travelling Wave Solutions of Quasilinear Parabolic Equations