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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
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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
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