×

zbMATH — the first resource for mathematics

A nonlinear diffusion equation with nonlinear boundary conditions: Method of lines. (English) Zbl 0664.35049
Existence, uniqueness and some properties of weak solutions of the nonlinear boundary value problem \[ (\beta (u))_ t-\Delta u=f(x,t,u),\quad x\in D,\quad t>0, \]
\[ \partial u/\partial \nu +g(x,t,u)=0,\quad x\in \Gamma,\quad t>0;\quad u(x,0)=u_ 0(x),\quad x\in D, \] are investigated, using the method of lines.
The functions f and g satisfy some smoothness and growth conditions, \(\beta (u)=| u|^ m sign u,\) \(m>0\) \((0<m<1\) slow diffusion, \(m=1\) heat conduction, \(m>1\) fast diffusion).
Reviewer: K.Hawlitschek

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35K20 Initial-boundary value problems for second-order parabolic equations
35A35 Theoretical approximation in context of PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] ARONSON D. G., CRANDALL M. G., PELETIER L. A.: Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Analysis, 6, 1982, 1001-1022. · Zbl 0518.35050
[2] CHZHOU YUI-LIN: The boundary-value problems for the nonlinear parabolic equations. Mat. sb., T47(89), 4, 1959)
[3] FILO J.: On solutions of a perturbed fast diffusion equation. Aplikace Matematiky 32, 1987, 364-380. · Zbl 0652.35064
[4] FRIEDMAN A.: Partial differential equations. Holt, Rinehart and Winston, New York, 1969. · Zbl 0224.35002
[5] FUČÍK S., KUFNER A.: Nonlinear differential equations. SNTL, Praha 1978 · Zbl 0474.35001
[6] JEROME J. W.: Horizontal line analysis of the multidimensional porous medium equation: existence, rate of convergence and maximum principle. Lecture Notes in Math. 506, Springer 1976.
[7] KAČUR J.: Method of Rothe in Evolution Equations. Teubner-Texte zur Mathematik, 80, Leipzig 1985. · Zbl 0582.65084
[8] KAČUR J.: Nonlinear parabolic equations with the mixed nonlinear and nonstationary boundary conditions. Math. Slovaca, 30, 1980, 213-237. · Zbl 0452.35060
[9] KAČUR J.: On boundedness of weak solution for some class of quasilinear partial difìerential equations. Čas. pěst. matem. 98, 1973, 43-55.
[10] KUFNER A., JOHN O., FUČÍK S.: Function Spaces. Academia, Praha 1977.
[11] LADYŽENSKAJA O. A., SOLONIKOV V. A., URAĽCEVA N. N.: Linear and quasilinear equations of parabolic type. Nauka, Moscow 1967
[12] MADDALENA L.: Existence of global solutions for reaction-diffusion systems with density dependent diffusion. Nonlinear Analysis 8, 1984, 1383-1394. · Zbl 0566.35055
[13] OLEJNIK O. A., KALASHNIKOV A. S., CHZHOU YUI-LIN: The Cauchy problem and boundary problems for equations of the type of nonstationary filtration. Izv. Akad. Nauk SSSR, Ser. Mat., 22, 1958, 667-704 · Zbl 0093.10302
[14] SABININA E. S.: On a class of quasilinear parabolic equations not solvable for the time derivative. Sibirsk. Mat. Z., 6, 1965, 1074-1100
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.